This contribution describes the dynamic simulation of the contact of coils of a valve spring within a multi-body system application. The spring is described by a multi-mass model. Contacting spring coils influence the dynamical properties of a valve spring significantly. The possible interaction between adjacent coils is modeled by means of non-smooth mechanics. Signorini conditions on displacement level are imposed on contact candidates. The set of inequality constraints is transformed into a set of equations by introducing a nonlinear complementarity function, which contains the semi-smooth maximum function. The set of equations of motion together with the contact constraints are integrated in time by a Backward Differentiation Formula (BDF) scheme. In each time step, the resulting nonlinear algebraic equation system is solved by a semi-smooth Newton method. The approach is evaluated by two examples. The first model represents a cylindrical helical spring. The performance of the algorithm is compared to an approach, where the coil contact is modeled by using spring-damper elements in between possible contact nodes. The proposed approach is not only running much faster, but also avoids the need of artificial parameters to calibrate the spring-damper elements. The second example deals with a full model of a single valvetrain system, demonstrating that the valve train dynamics is widely affected by the vibrational characteristics of the valve springs.

We consider an equilibrium problem for 2D elastic body with a thin inclusion crossing an external boundary at zero angle. It is assumed that the inclusion is delaminated, therefore a crack between the inclusion and the body is considered. To prevent a mutual penetration between crack faces, inequality type boundary conditions are imposed at the crack faces. We analyze elastic inclusions as well as rigid inclusions. Passages to limits are investigated as a rigidity parameter of the inclusion goes to infinity. Theorems of existence and uniqueness are proved.

This paper proves some regularity criteria for the Hall-MHD system in terms of the pressure and the magnetic field.

The well-known issue with the absence of conservation of angular momentum in classical particle systems with periodic boundary conditions is addressed. It is shown that conventional theory based on Noether's theorem fails to explain the simplest possible example, notably jumps of angular momentum in the case of single particle moving in a periodic cell. It is suggested to consider the periodic cell as an *open system*, exchanging mass, momentum, angular momentum, and energy with neighboring cells. Then the behavior of the cell is described by balance laws rather than conservation laws. It is shown using the law of angular momentum balance that the variation of the angular momentum in systems with periodic boundary conditions is a consequence of (i) the non-zero flux of angular momentum through the boundaries and (ii) torque acting on the cell due to interactions between particles in the cell with images in the neighboring cells. Three simple examples demonstrating individual and combined effect of these factors are presented. Thus the paper provides a rational explanation for the absence of angular momentum conservation in particle systems with periodic boundary conditions.

In the classical theory of rigid perfectly/plastic solids, the calculation of plane strain characteristic fields is reduced to solving the telegraph equation and subsequent evaluation of ordinary integrals. The telegraph equation can be integrated by the method of Riemann. In the present brief note this approach is extended to pressure-dependent plasticity based on the Coulomb-Mohr yield function. Body forces are neglected and kinematic theories are not considered.

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro-magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves.

Well-known models for rubberlike elasticity with strain-stiffening effects provide good predictions for unlimitedly, rapidly growing stress in a process of approaching a very large strain limit. According to such models, however, unbounded elastic strain energy would be generated as the strain limit is approached. To resolve this issue, a new, explicit approach is proposed to obtain multi-axial elastic potentials based on uniaxial data and shear data. Then, a strain-stiffening elastic potential is given to always yield bounded strain energy. Good agreement is achieved with a number of test data.

The purpose of this paper is to analyze the interaction of thin elastic inclusions with globular defects in a solid structural element and develop technique to determine fracture parameters when the elastic inclusion of the structure is close to a circular hole and/or to its bonding layers. Procedures for determination of fracture parameters are based on the J-integral relation with generalized stress intensity factors (GSIF) recently obtained by the authors and the boundary element method is adopted for studying thin shapes. The developed techniques, dominating GSIF and mutual integral method, are applied to two specific problems: the interaction of a traction-free hole with a nearby, thin inclusion and the interaction of a constrained hole with the inclusion. The direct numerical solution was obtained for the two principal models which represent two different boundary conditions on the edge of the hole of the structural element. The study shows that if the hole is unstressed, the values of GSIF are approximately the same as the corresponding values of the fracture parameters and the presence of the hole and the rigid inclusion have only a small effect on GSIF. But if the hole is constrained along its boundary, the values of GSIF generally decrease with decreasing the distance between the inclusion and the hole. The stress concentration on the hole substantially depends on the presence of the inclusion and varies significantly with respect to its radius, the distance from the inclusion, and the relative rigidity of the inclusion.

In this paper, a method of local perturbations, previously successfully applied to decompose the problem of elasticity in the system of connected thin rods and beams [24], is used to study the asymptotic behavior of the elasticity problem in connected thin plates. A complete decomposition of the problem, i.e. the separation of the original problem into the two-dimensional problem of the theory of plates and local problems is proposed. The local problems describe the three-dimensional stress-strain state in the connected plates and can be solved by numerical methods.

Elastomers take an important role in many industrial applications. In the automotive industries for example, elastomers are used in various bearings, where they inhibit vibration propagation and thereby significantly enhance driving performance and comfort. Several models have been developed to simulate the material's mechanical response to various stresses and strains a component may undergo during its lifetime. So far, these models are commonly developed under isothermal conditions. In this contribution it is shown that the mechanical properties significantly depend on the temperature and that the material heats up under large dynamic deformations. Therefore, an elastomer's behaviour is not described sufficiently with an isothermal approach, a detailed thermo-viscoelastic modelling is required. In this contribution, the behaviour of elastomers is experimentally investigated in order to gain informations about the time- and temperature-dependent mechanical properties. We perform different tests on a natural rubber to emphasize the temperature dependence of the equilibrium stress-strain relation as well as the time-dependent behaviour in relaxation tests. As it is necessary for parameterising a material model, thermal tests are carried out to determine the specific heat capacity, the thermal expansion coefficient and the thermal conductivity. In a second step, we introduce a material model which is able to represent the temperature-dependent viscoelastic material behaviour including large deformations, as well as the self-heating of the material. The model's mechanical parameters are identified on tension tests. In first FE calculations, the applicability of the introduced model is proven by depicting the experimental results of several tension tests at different temperatures. Besides these validations, the self-heating under dynamic load, depending on the loads amplitude and frequency as well as the surrounding temperature is calculated.

Non-linear fracture of Single Cantilever Beam (SCB) is studied theoretically using J-integral. It is assumed that the beam is made of unidirectional fiber reinforced polymer composite which obeys the stress-strain relation of an elastic-perfectly plastic material. The lower crack arm is loaded by an external moment while the upper crack arm is stress free. Closed form analytical solutions of J-integral are found for different magnitudes of the external load corresponding to the different distribution of stresses and strains in the beam cracked and un-cracked portions. For this purpose, a Mechanics of materials based model is used. The validity of the solution is proved by comparison with formula for strain energy release rate in the linear-elastic stage of the work of the material. A numerical example is presented to illustrate the influence of the material non-linearity on J-integral. It is found that the material non-linearity leads to the increase of J-integral values. This is attributed to the strain energy dissipation due to the non-linear deformation.

By using the generalized Almansi's theorem and the Schmidt method, the solution of two 3D rectangular cracks in an orthotropic elastic media is investigated. The problems are solved through 2D Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The effects of the geometric shape of the rectangular crack and the distance between two rectangular cracks on the stress intensity factors (SIFs) in an orthotropic elastic media are concluded.

We consider three Sturm–Liouville boundary value problems (two of them are coercive and the other one is not) in a bounded Lipschitz domain for a perturbed Lamé operator with boundary conditions of Robin type. We prove that the problems are Fredholm in suitable weighted Sobolev type spaces. Conditions, providing the completeness of root functions related to the boundary value problems, are described.

In this paper we study the regularity of solutions to the Stokes and the Navier-Stokes system in polyhedral domains contained in ℝ^{3}. We consider the scale B^{s}_{τ}(L_{τ}), 1/τ = s/3 + 1/2 of Besov spaces which determines the approximation order of adaptive numerical wavelet schemes and other nonlinear approximation methods. We show that the regularity in this scale is large enough to justify the use of adaptive methods. The proofs of the main results are performed by combining regularity results in weighted Sobolev spaces with characterizations of Besov spaces by wavelet expansions.

In this note we show that the relaxed linear micromorphic model recently proposed by the authors can be suitably used to describe the presence of band-gaps in metamaterials with microstructures in which strong contrasts of the mechanical properties are present (e.g. phononic crystals and lattice structures). This relaxed micromorphic model only has 6 constitutive parameters instead of 18 parameters needed in Mindlin- and Eringen-type classical micromorphic models. We show that the onset of band-gaps is related to a unique constitutive parameter, the *Cosserat couple modulus* μ_{c} which starts to account for band-gaps when reaching a suitable threshold value. The limited number of parameters of our model, as well as the specific effect of some of them on wave propagation can be seen as an important step towards indirect measurement campaigns.

The paper deals with the elastic and cohesive interface behavior of pre-cracked bi-material ceramic-metal structures under dynamic time harmonic load. The shear lag model as well as the Fourier method is applied to find the dynamic response of the considered bi-material structure, assuming the cohesive interface behaviour, accompanied before of the elastic-brittle one. In both cases, the growth of debond length is not considered, e.g. at a given loading condition the only corresponding debond length is found. The inertia forces of the already elastic debond parts of the bi-material structure are neglected. Appropriate contact conditions are proposed in order to fit together both elastic and cohesive solutions. The numerical predictions for the cohesive debond length of the bi-material structures is calculated by the aid of the corresponding value of the elastic debond length at the same loading condition. The influence of loading characteristics i.e. frequencies and amplitude fluctuations on the debond length and the interface shear stress distribution is discussed. The parametric analysis of the results obtained is illustrated by examples of the modern ceramic-metal composites on metal substrates and is depicted in figures.

An overview of one specific design of continuously variable transmission (CVT) is presented. An approach to the modeling of contact interactions is described. Several CVT models of different complexity are developed and used in numerical simulations. The results of simulations can be used to estimate local behavior of CVT parts, as well as its global characteristics.

We derive a scaling-relation, for the infimum of the energy

- \[ J_{\varepsilon,\delta}(u,\gamma)=\int\limits_\Omega\frac{1}{\varepsilon}\operatorname{dist}^q\left(\nabla u\left(\mathds{1}-\gamma\vec{e}_1\otimes \vec{e}_2\right),SO(2)\right)+|\gamma|^p d\lambda^2(x,y)+\delta V_y \left(\chi_{\{\gamma=0\}},\Omega\right), \]

for small ε,δ > 0, where p, q ≥ 1, u: Ω ℝ^{2} is a deformation with suitable affine boundary conditions and γ: Ω ℝ is a suitable slip variable. This model is motivated by a two-dimensional single-slip model in finite crystal plasticity. We show, that the infimum of the energy J_{ε,δ} scales as

. This scaling-relation is attained by an asymptotically self-similar branching construction.

Fractal fault systems are analyzed mechanically by means of the fractional calculus. Small elastic deviations from equilibrium are captured by vectorial wave equations which imply elastic energy and conservation of momentum with spatio-temporal isofractality. Laplace and Fourier transformations lead to an eigenvalue problem which enables a diagonalization for the stable range with convex elastic energy. A degenerate fractional wave equation is proposed for a collapse at the verge of stability. The divergence at collapse is limited to small ranges and times. Substituting such a jerk by a stress jump, its propagation into a stable near-field is analyzed with a commuted isofractional wave equation. Novel solutions are presented which capture some features of earthquakes. These findings can be extended with less symmetry than first assumed for the ease of presentation. The outlook comprehends anelastic effects, coupling with pore water and multi-fractality.

A comparison of two different probabilistic techniques in the context of a homogenization of the polymer filled with rubber particles is demonstrated in this work. Homogenization approach is based on deformation energy of the Representative Volume Element containing a single spherical particle, which must be equal for the heterogeneous and equivalent homogenized bodies. Both probabilistic methods are based on a series of the Finite Element Method experiments that leads to determination of effective characteristics as the polynomial forms of the input random variables and it is done by using the optimized Weighted Least Squares Method. These functions are next integrated in the semi-analytical probabilistic approach and embedded into the Taylor series expansions as the derivatives in the framework of the alternative stochastic perturbation-based approach. These two strategies are employed to study a polymeric matrix with rubber particle as the filler because such a composite has enormously large contrast in-between expectations of the two random Young's moduli of its constituents. Numerical optimization concerns the order of the approximating polynomial that needs to maximize correlation with the set of trial points and minimize at the same time both computational error with the least squares standard deviation.

In this paper, based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, the vibration behavior of an elastic oval cylindrical shell with parabolically varying thickness along of its circumference resting on the Winkler-Pasternak foundations is investigated. The theoretical analysis of the governing equations of the shell is formulated to overcome the mathematical difficulties of mode coupling of variable curvature and thickness of shell. Using the transfer matrix of the shell, the vibration equations based on the Winkler-Pasternak foundations are written in a matrix differential equation of first order in the circumferential coordinate and solved numerically. The proposed model is applied to get the vibration frequencies and the corresponding mode shapes of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration characteristics and bending deformations to the Winkler-Pasternak foundations moduli, thickness variation, ovality and orthotropy of the shell is studied for different type-modes of vibration.

This research work considers a more efficient vibro-impacting piezoelectric energy harvester (VIPEH) structure, which is intended both to prevent the device from excessive displacements as well as to increase its operational bandwidth in actual excitation conditions. Multi-physics finite element model of the VIPEH was developed in Comsol with the objective to analyze influence of stopper location on the mechanical and electrical characteristics of the piezoelectric transducer, followed by the experimental study of two device configurations. Numerical and experimental results revealed that stopper location influences the magnitude of generated voltage since, at certain stopper location points, higher vibration modes are excited during the impact.

We consider a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature independent conductivities of their components. Imperfect contact conditions are assumed on the boundaries between the matrix and inclusions. By introducing complex potentials, the corresponding boundary value problem for the Laplace equation is transformed into a special R-linear boundary value problem for doubly periodic analytic functions. The method of functional equations is used for obtaining a solution. Thus, the R-linear boundary value problem is transformed into a system of functional equations which is analysed afterwards. A new improved algorithm for solving this system is proposed. It allows to compute the average property and reconstruct the temperature and the flux at an arbitrary point of the composite. Computational examples are presented.

A transient torsional response of an elastic solid is discussed. Introducing a semi-infinite needle-like torque source, exact closed form expressions for torsional displacement and stress are obtained. Due to the needle-like shape of the source, two waves are produced. One is a cylindrical wave centered at its symmetry axis and the other is a spherical one centered at the edge of the needle source. The cylindrical wave shows the inverse square root singularity at its front, but the spherical wave does a finite jump at the front. Some additional discussions for the static response to the torque source are also carried out. The proposed needle-like torque source might be used as a primary model for the bore drilling technology.

The present paper contributes to the modeling of contacts with viscoelastic materials. An indenter is pressed into an elastomer and slides on its surface with a constant velocity. Contact regions typically experience periodical loading due to surface roughness with characteristic length scales. Loading within a medium frequency range is studied, in which the viscous properties of the elastomer dominate. After passing a transition zone a stationary state is reached. An one-dimensional model is used to determine estimations for the indentation depth and the coefficient of friction. Results for two simply shaped indenters are presented and compared to boundary element simulations.

This contribution presents a method to improve the energy efficiency of walking bipedal robots over 50% in a range of speed from 0.3 to 2.3 m/s by the use of constant elastic couplings. The method consists of modeling the robot as underactuated system – so that it is making use of its natural dynamics instead of fighting against it – controlling its joint-angle trajectories with input-output feedback linearization and optimizing the joint-angle trajectories as well as the elastic couplings numerically. The mechanism of minimizing energy expenditure consists of reducing impact losses by choosing smaller steps, which gets favorable by a higher natural frequency due to elastic couplings. The method is applied to a planar robot with upper body, two stiff legs, two actuators in the hip joints and one simple rotational spring between the legs as elastic coupling. The mechanism of energy expenditure is investigated for the robot with and without elastic coupling between legs in detail.

An energy harvesting concept has been proposed comprising a piezoelectric patch on a vertical cantilever beam with a tip mass. The cantilever beam is excited in the transverse direction at its base. This device is highly nonlinear with two potential wells for large tip masses, when the beam is buckled. For the pre-buckled case considered here, the stiffness is low and hence the displacement response is large, leading to multiple solutions to harmonic excitation that are exploited in the harvesting device. To maximise the energy harvested in systems with multiple solutions the higher amplitude response should be preferred. This paper investigates the amplitude of random noise excitation where the harvester is unable to sustain the high amplitude solution, and at some point will jump to the low amplitude solution. The investigation is performed on a validated model of the harvester and the effect is demonstrated experimentally.

In this research, the axial buckling and postbuckling configurations of single-walled carbon nanotubes (SWCNTs) under different types of end conditions are investigated based on an efficient numerical approach. The effects of transverse shear deformation and rotary inertia are taken into account using the Timoshenko beam theory. The nonlinear governing equations and associated boundary conditions are derived by the virtual displacements principle and then discretized via the generalized differential quadrature method. The small scale effect is incorporated into the model through Eringen's nonlocal elasticity. To obtain the critical buckling loads, the set of linear discretized equations are solved as an eigenvalue problem. Also, to address the postbuckling problem, the pseudo arc-length continuation method is applied to the set of nonlinear parameterized equations. The effects of nonlocal parameter, boundary conditions, aspect ratio and buckling mode on the critical buckling load and postbuckling behavior are studied. Moreover, a comparison is made between the results of Timoshenko beam model and those of its Euler-Bernoulli counterpart for various magnitudes of nonlocal parameter.

In this work, the interaction between a moving anti-plane crack and a screw dislocation in the magneto-electro-elastic material is studied. The exact solutions are derived by the Muskhelishvili theory and the mechanical-electric-magnetic fields are presented. Comparing with the recent researching status, the present work has three prominent features. Firstly, both the remote loads with eight possible combinations and the coupling of the mechanical-electric-magnetic fields are considered. Secondly, the crack face is assumed to have the general boundary condition which is not the single permeable or impermeable case. Finally, the general solution is derived with the consideration of the crack moving velocity and the screw dislocation. Results presented in this paper would have potential applications on the analysis and design of magneto-electro-elastic structures, especially for the influence of microdefects on the initiation and propagation of cracks.

Based on the Mindlin plate theory the eigenfunctions for a through-thickness crack in a bending and twisting plate have been derived. The results are given as power series in terms of deflection, rotation and stresses. By introducing two auxiliary functions in the Mindlin plate theory we obtained two decoupled partial differential equations of the fourth and second order. This system of partial differential equations allows for each crack face to describe all three types of static boundary conditions. The first eigenfunction of the stress state shows the same singular near-tip field at the crack tip known from two- and three-dimensional crack analyses as well as from Reissner's plate theory. The second eigenfunction similarly characterizes the constant stress parallel to the crack as the T-stress in plane elastic problems.

In the paper at hand, co-simulation approaches are analyzed for coupling two solvers. The solvers are assumed to be coupled by algebraic constraint equations. We discuss 2 different coupling methods. Both methods are semi-implicit, i.e. they are based on a predictor/corrector approach. Method 1 makes use of the well-known Baumgarte-stabilization technique. Method 2 is based on a weighted multiplier approach. For both methods, we investigate formulations on index-3, index-2 and index-1 level and analyze the convergence, the numerical stability and the numerical error. The presented approaches require Jacobian matrices. Since only partial derivatives with respect to the coupling variables are needed, calculation of the Jacobian matrices may very easily be calculated numerically and in parallel with the predictor step. For that reason, the presented methods can in a straightforward manner be applied to couple commercial simulation tools without full solver access. The only requirement on the subsystem solvers is that the macro-time step can be repeated once in order to accomplish the corrector step. Within the paper, we introduce methods for coupling mechanical systems. The presented approaches can, however, also be applied to couple arbitrary non-mechanical dynamical systems.

This paper deals with the description of steady-state sub- and superharmonic motion in rotor-stator contact using truncated complex Fourier series. Two different approaches are presented with different stages of simplification. In particular, a kinematic contact condition describing continuous contact is used. The multi-harmonic balance method is applied to solve the differential algebraic system of equations. A further simplification is implemented which uses the triangular inequality to approximate the nonlinear term in the kinematic contact condition. The Fourier coefficients of the nonlinear term are calculated using an integral expression. Reasonable initial values of the Fourier coefficients for the numerical solution are obtained by a hybrid approach. Results show good agreement with calculations by direct numerical integration using a pseudo-linear viscoelastic contact model.

In this extended note a critical discussion of an extension of the Lorentz transformations for velocities faster than the speed of light given recently by Hill and Cox [1] is provided. The presented approach reveals the connection between faster-than-light speeds and the issue of isotropy of space. It is shown if the relative speed between the two inertial frames v is greater than the speed of light, the condition of isotropy of space cannot be retained. It further specifies the respective transformations applying to -∞<v<-c and c<v<+∞. It is proved that such Lorentz-like transformations are improper transformations since the Jacobian is negative. As a consequence, the wave operator, the light-cone and the volume element are not invariant under such Lorentz-like transformations. Also it is shown that such Lorentz-like transformations are not new and already known in the literature.

The investigations fulfilled in this article are founded on two results. The first is experiments of M. Beteno, Y. Duboshinsky. The description of these experiments is adduced in [1]. In these experiments the low frequency oscillations of iron ball suspended on the thin string [1] were obtained (frequency of these oscillations is order to eigen frequency of pendulum). And the second is theoretical conclusions about possible stabilizations of early (without alternating magnetic field) unstable equilibrium positions. This theoretical result was obtained by the asymptotical solution of Lagrange-Maxwell equations of dynamic of electromechanical systems suspended in alternating magnetic field [2].

The phenomenon of brake squeal, which is a type of friction-induced vibration, is analyzed using a pin-on-disc system. For this purpose, a finite element model is derived and its parameters are updated on the basis of experiments. The FEM analysis includes the complex eigenvalue analysis and the transient analysis. As the brake-squeal phenomenon is very sensitive with respect to parametric uncertainty, the two numerical analyses are combined with an uncertainty analysis, which in this study is based on fuzzy arithmetic. The uncertainty analysis enables the determination of both the overall uncertainty of the considered output quantity and the influence of each individual uncertain model parameter on the overall uncertainty of the output. With this information about propagation and influence of parametric uncertainty in the system, the methods of complex eigenvalue analysis and transient analysis can be compared with respect to their appropriateness for predicting the tendency of the brake to squeal.

The manufacturing process of paper machines consists of several steps to produce high quality papers. This is done by sequentially lined-up machines including the head box, the drying sections, the finishing part and the wrapping systems. In the finishing part, the rollers of the paper calender compress the fibrous material involving viscoelastic and plastic deformations. Modern calenders are composed of several roller pairs, each consisting of a soft and a hard roller. The homogenization of the paper density and the refinement of the paper surface is achieved by the compression in the roller pairs. While very high values for the plastic strain occur in the first roller pair, the plastification decreases for the subsequent pairs. Therefore, the force distribution and the occurring vibrations can deviate significantly between the roller pairs. Two main vibration problems are observed in paper calenders caused by the contact and the orthotropic behavior of the paper: wear-induced corrugation on the surface of the soft rollers and sudden instabilities going along with high vibration amplitudes. In this paper the main focus is placed on a simplified modeling of the paper plastification during the calendering process. The restoring forces are non-smooth due to the plasticity and additional considerations have to be included for the derivation of the stability problem.

The AUFS scheme by Sun and Takayama is a flux splitting scheme without breakdown of discrete shock profiles, usually called carbuncle, but still with a good resolution of entropy waves. Unfortunately, numerical tests with this scheme yield that the viscosity on entropy waves is too small while the viscosity on shear waves is too large. In this paper, we prove that both deficiencies are inherent to the construction of the scheme and provide fixes to overcome them.

The paper is devoted to the life and work of Alexander Mikhailovich Ertel, the founder of elastohydrodynamics. He was the first to solve problems of hydrodynamic lubrication accounting for effects of elastic deformation of the bodies in contact as well as for the dependence of the viscosity of the lubricant on pressure and temperature, thereby opening a new branch of tribology – elastohydrodynamics. However, due to complicated historical entanglements, the real authorship remained unknown over many years. On occasion of the 100th anniversary of Alexander Mohrenstein-Ertel, we investigate the circumstances of the creation of elastohydrodynamics and provide a short sketch of the main ideas of the early works of Mohrenstein-Ertel. The biography of Mohrenstein-Ertel is like a criminal novel: the 20th century history is reflected in the hard fate of this scientist and Russian history up to the time of Pushkin can be traced in his genealogy.

We study the stationary interaction between a 2D viscous fluid, governed by the Stokes equation, and a rigid structure that can move following rigid displacements. The displacements of the structure are determined using an algebraic equation. A slip boundary condition of friction type is used on the fluid–solid interface. An existence result is proved and numerical tests are presented.

The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the **MDGKN** type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (**G**-matrix) and circulatory terms (**N**-matrix, which may lead to self-excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (**D**-matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices. Here we present some new results (using a variety of methods of proof) on the influence of the damping terms, which are quite general. Starting from a number of conjectures, they were jointly developed by the authors during recent months.

Degradation mechanisms in Li-ion batteries such as SEI formation, isolation of active material and reduction in electronic conductivity appear to correlate with the increase in surface area of electrode particles caused by particle fracture during charge/discharge cycles. The focus of this study is on the surface cracking of an electrode particle, as large tensile stresses operate on the surface during the delithiation process of charge/discharge cycling. The pre-existing surface flaws act as crack initiators under this scenario and we discuss the extension of these cracks under two different operating conditions. Approximate analytical expressions for the propensity for surface crack growth is derived in terms of stress intensity factor and fracture toughness of the material. Utilizing dimensional analysis, we arrive at fracture limit diagrams to determine fracture-free conditions as design guidelines for prescribed electrode particle size or diffusion boundary conditions related to the charging/discharging process. Another significant result of the fracture analysis is that smaller particles can withstand a wider range of fluctuations in concentration or flux at the boundary for surface fracture-free conditions. This result supplements the conventional understanding that smaller particles show higher structural integrity because of fewer pre-existing defects.

The application of high-frequency vibration processes for intensification of machining requires a control technique for identification, excitation and stabilisation of the nonlinear resonant mode in machining systems with unpredictable variation of processing loads. Such a technique was developed with the use of a self-exciting mechatronic system. This method of control is known as autoresonance. Autoresonant control of ultrasonically assisted drilling machine intended to improve machining process is thoroughly analysed and the simulation results of analysis for both mechanical feedback and electrical feedback are presented together with the application of different filters.

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two-layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

The classical method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions are used to investigate free transverse vibrations of an elastic membrane of elliptical planform with an arbitrarily located elliptical perforation. Subsequently, the elaborated method of eigenfunction expansion is employed to obtain an exact time-domain series solution, in terms of products of angular and radial Mathieu functions, for the forced transverse oscillations of the eccentric membrane. The analytical solution is illustrated through numerical examples including circular/elliptical membranes with a circular perforation or with an elliptical perforation of selected geometric, orientation, and location parameters. The first five natural frequencies are tabulated, and selected vibration mode shapes are presented in graphical form. Also, the displacement responses of representative membranes in a practical loading configuration (i.e., a uniformly distributed step load) are calculated. The accuracy of solutions is ensured through proper convergence studies, and the validity of results is demonstrated with the aid of a commercial finite element package as well as by comparison with the existing data. The set of data reported herein is believed to be the first rigorous attempt on the free/forced vibrational characteristics of eccentric elliptical membranes for a wide range of geometric parameters.

The anti-plane strain elastodynamic problem for a continuously inhomogeneous half-plane with free-surface relief subjected to time-harmonic SH-wave is studied. The computational tool is a boundary integral equation method (BIEM) based on analytically derived Green's function for a quadratically inhomogeneous in depth half-plane. To show the versatility of the proposed BIE method, it is considered SH-wave propagation in an inhomogeneous half-plane with free surface relief presented by a semi-circle, semi-elliptic and triangle canyon. The inhomogeneous in depth half-plane is modeled in two different ways: (i) the material properties vary continuously in depth and BIEM based on Green's function is used; (ii) the material properties vary in a discrete way and the half-plane is presented by a set of homogeneous layers with horizontal interfaces and a hybrid technique based on wave number integration method (WNIM) and BIEM is applied. The equivalence of these two different models is shown. The simulations reveal a marked dependence of the wave field on the material inhomogeneity and the potential of the BIEM based on the Green's function for half-plane to produce highly accurate results by using strongly reduced discretization mesh in comparison with the conventional boundary element technique using fundamental solution for the full plane.

Based upon a one-dimensional nonlinear Klein-Gordon equation with a perturbed one-gap periodic potential, this paper deals with the question as to whether spatially localized structures in periodic media can exist for all times. As it turns out that, given our model equation, the latter question cannot be answered in the affirmative, we show the asymptotic stability of the vacuum state in appropriate dispersive norms and provide an upper bound for the temporal decay rates of the corresponding solutions. This is done by using the dispersive estimates proved in [23]. More precisely, if the perturbed Hill operator associated to our problem has no eigenvalue, we add a power nonlinearity u^{p} with p ∈ {6, 7, 8, …}. In this setting, the convergence to the trivial solution w. r. t. the L^{∞} norm is shown in a canonical way. We obtain the corresponding linear rate. In contrast, if the spatially localized potential creates an eigenvalue in the band gap of the continuous spectrum, then we multiply u^{p} by a spatial weight function and prove an asymptotic stability result w. r. t. a weighted L^{2} norm for p ∈ {3, 4, 5, …}. Now, in the presence of an eigenvalue, there is a strongly reduced decay compared to the associated linearized problem. It is due to the component that belongs to the discrete spectral subspace of L^{2} w. r. t. the perturbed Hill operator. As in [29], this phenomenon is referred to as metastability of the corresponding solutions.

We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory modeling solids with micro-structure. The complexity of physical phenomena appears as encoded in so called material laws. The usefulness of the structural perspective developed is illustrated by showing that many initial boundary value problems in the theory of such elastic solids share the same type of solution theory. Moreover, interconnections of the respective models are discussed via a previously introduced mother/descendant mechanism.

We consider the convergence to (global) equilibrium of a partially dissipative hyperbolic system using entropy methods. An entropy functional is determined and exponential decay for this functional is proven. Moreover, an explicit estimate for the decay rate is given.

Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H^{1} seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C^{0} interior penalty method with improved stability properties in comparison with the Galerkin FEM.

The damage acting on a structure can lead to disproportionate consequences, i.e., the global collapse. This extreme situation has to be avoided and, thus, structural monitoring is requested in those structures where human losses are possible and large economic consequences are expected. Static measurement devices are the most economic instrumental set-ups able to highlight the presence of progressive damages. However, this monitoring system suffers from the structural behaviour under the external loads. In many situations, alternate load paths shown the non-effectiveness of the measurement system since the instrumentation are installed on elements not relevant for the response under the given loads. The same problems occur when the structure is compartmentalized, i.e. the structural responses of the single parts dependent on the loads acting almost only on the same single component. In order to measure the degree of compartmentalization, different novel metrics based on stiffness matrix properties are proposed and their effectiveness discussed. The new idea of this paper is to connect compartmentalization of structures with a sort of distance of the stiffness matrix from the set of diagonal matrices. Few examples are illustrated.

In this contribution the eigenfrequencies of a special linear vibration system are investigated. Based on the properties of the corresponding mass and stiffness matrices of the chain structured mass-spring vibration system with arbitrary n degrees of freedom an algebraic proof for the determination of the eigenfrequencies is given.

This paper deals with the eigenvalues of the Neumann Laplacian on simply-connected Lipschitz planar domains with some rotational symmetry. Our aim is to continue the investigations from Enache and Philippin [7] and derive new isoperimetric estimates for eigenvalues of higher order.

Maxwell's stress tensor is well known from electromagnetic theory. But correct application of it to practical problems is by no means general knowledge even among experts. In this article we present a survey of the electromagnetic stress tensor and of the electromagnetic forces in strongly polarizable materials. We relate the observed ponderomotoric phenomena to the stress tensor and we present a number of applications in modern devices and processes using ferromagnetic colloids, so-called ferrofluids. We emphazise the correct applications of the stress tensor to these examples in contrast to common popular usage of a so-called “electromagnetic pressure”. We predict some new effects, e.g. density variations and pressure measurements in ferrofluids. Our work is based on two preceding articles by the present authors [4, 33] wherein the electromagnetic stress tensor is deduced from conservation laws together with Maxwell's equations and with thermodynamic relations.

This paper concerns optimal design of so-called Neuro-Mechanical Oscillators (NMOs). An NMO is a new type of bio-inspired mechatronic system which consists of an actuated truss with a recurrent neural network (RNN) superimposed onto it. By choosing the entries of the weight matrix of the RNN, an NMO can be designed, using numerical optimization, to generate pre-specified time-varying motions when subject to certain time-varying input signals. However, to rule out possible dependence of the motion on the initial state of the system as well as convergence into limit cycles, some form of constraints must be imposed on the system's design parameters. To derive such constraints, we investigate under what conditions the influence of the initial state eventually vanishes and the motion becomes completely determined by the input signal. Three sufficient criteria are presented for RNNs, but the possibility of large mechanical deformations most likely rule out global system level results. For sufficiently small deformations, however, local results are obtained, and a numerical example provided in the paper indicates that these can be useful for designing practical systems.

A general method for the modal decomposition of the equations of motion of damped multi-degree-of-freedom-systems is presented. The first variant of the presented method in earlier publications of the author, including the complex right eigenvectors, is briefly reviewed first. The second presented variant is also based on the corresponding eigenvalue problem of the damped structure including the complex left and right eigenvectors. After initial partitioning of the equations of motion a real modal transformation matrix is built by a combination of two complex transformations. For the general case of damped structures with non-modal symmetric damping matrix a modal analysis can be performed in real arithmetic. Two numerical examples with 3 and 10 DOF's demonstrate the accuracy and the advantages of the presented modal solution method.

In shift gearboxes, audible vibrations can arise during the process of clutch engagement. Measurements show an unstable axial motion of the gear unit input shaft. In order to explain this effect, an approach which establishes the interaction of the slipping clutch and the gearing is proposed. The suggested model consists of a pressure plate, a clutch disc, a gear unit input shaft and a tooth contact. The gear unit input shaft is rigidly connected to the gearing and the clutch disc and has translational and rotational degrees of freedom. The tooth contact imposes a kinematic constraint on the system. This is why the sliding friction torque is transformed into forces in axial and radial direction of the shaft. Depending on the gearing parameters and the slip, the contact normal force between clutch disc and pressure plate is amplified (motor accelerating) or reduced (motor slowing down) by the toothing. For both shifting situations, a region of flutter instability in the parameter space can be found. The dynamic behavior of the system with parameters in the latter region is analyzed. It is shown that there exists a stable piecewise-continuous limit cycle: During one period, the contact between the clutch disc and the pressure plate opens shortly. A variation of parameters is performed to show the dependence of the limit cycle on the geometry and on the operation conditions.

In this work, we apply the fractional order theory of thermoelasticity to a 1D problem for a half-space overlaid by a thick layer of a different material. The upper surface of the layer is taken to be traction free and is subjected to a constant thermal shock. There are no body forces or heat sources affecting the medium. Laplace transform techniques are used to eliminate the time variable t. The solution in the transformed domain is obtained by using a direct approach. The inverse Laplace transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the fractional order theory are discussed and compared with those for the generalized theory of thermoelasticity. We also study the effect of the fractional derivative parameters of the two media on the behavior of the solution. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the directional deformation is proposed. This leads to a clear division of the deformation and stress states in the direction of anisotropy and a remaining part. The decomposition is explained in detail and a constitutive model of hyper-elasticity is proposed for the case of transversal isotropy, where the behavior of the model is investigated with the aid of simple analytical examples. The model is also investigated for inhomogeneous deformation states using high-order finite elements based on hierarchical shape functions showing the sensitivity of the accuracy of the results in the case of anisotropic media.

This paper is concerned with some properties of a modified periodic two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in H^{s} (��), s > 3/2.

We consider the quasi-Newtonian flow in a domain with a periodic rough bottom Γ_{ε} of period of order the small parameter ε and amplitude δ_{ε}, such that δ_{ε} << ε. The flow is described by the 3D incompressible non-Newtonian Navier-Stokes system where the viscosity is given by the non linear power law which is widely used for dilatant fluids (shear thickening). Assuming that the fluid satisfies the Navier slip condition on Γ_{ε} and letting ε 0, we obtain three different macroscopic models depending on the magnitude of δ_{ε} with respect to ε^{2p-1/p}, with p > 2. In the case δ_{ε}>> ε^{2p-1/p} the effective boundary condition in the limit ε = 0 is the no-slip condition, while if δ_{ε}<< ε^{2p-1/p} there is no roughness-induced effect. In the critical case when δ_{ε} ~ ε^{2p-1/p} we provide a more accurate effective boundary condition of Navier type. Finally, we also obtain corrector result for the pressure and velocity in every cases.

The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Numerical experiments cover one- and two-dimensional situations.

The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space.

The analysis of problem of joined elastic beams is presented in comparison with the engineering and asymptotic approaches. Our analysis is based on three-dimensional elasticity theory model and recently developed method of local perturbation (Gaudiello and Kolpakov, 2011), which seems to be an effective tool for analysis of fields in the vicinity of joint. We demonstrate that the method of local perturbation developed in (Gaudiello and Kolpakov, 2011) for scalar Laplace equation can be modified for vectorial elasticity theory problem. We demonstrate that the elasticity theory problem in joined domains of small diameter can be decomposed into one-dimensional problem describing global deformation of a system of joined beams and three-dimensional problems describing local deformation of singular joints in uniform fields. The first problem is the classical one, which ignores individual properties of joint absolutely. The second problem initiates reminiscence about the cellular problem of the homogenization theory for periodic structure. In spite of some similarities, the mentioned problems differ significantly. In particular, the joint of normal type (the joint similar in dimensions and material characteristics to the joined beams) does not manifest itself on global level. Due to the strong localization of perturbation of solution, computation of local strains and stresses in the vicinity of joint can be realized with standard FEM software.

The analysis of problem of joined elastic beams is presented in comparison with the engineering and asymptotic approaches. The analysis is based on three-dimensional elasticity theory model and recently developed method of local perturbation [Gaudiello and Kolpakov, Int. J. Eng. Sci. **49**, 295–309 (2011)], which seems to be an effective tool for analysis of fields in the vicinity of joint. Due to the strong localization of perturbation of solution, computation of local strains and stresses in the vicinity of joint can be realized with standard FEM software.

In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoelasticity with diffusion of types II and III. This theory permits propagation of both thermal and diffusion waves at finite speeds. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators we establish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally, we investigate the impossibility of the localization in time of solutions.

The fracture behaviour of multi-cracked materials has become a key issue in fracture mechanics and has received large attention recently. In this paper, an infinite plane containing three collinear cracks under biaxial compression has been investigated. Considering crack surface friction and using complex stress function theory, the exact analytical solution of stress intensity factors (SIFs) for an infinite plane containing three collinear cracks is obtained. The corresponding finite element code of Abaqus is employed to validate the theoretical results, and its results agree very well with the theoretical results. The effects of confining stresses, crack distances and the crack surface frictions on SIFs are analyzed through the theoretical results and the Abaqus code. A photoelastic experiment was conducted to validate the theoretical result about the effect of confining stresses.

Considering crack surface friction and using complex stress function theory, the exact analytical solution of stress intensity factors (SIFs) for an infinite plane containing three collinear cracks is obtained. The effects of confining stresses, crack distances and the crack surface frictions on SIFs are analyzed through the theoretical results and the Abaqus code. A photoelastic experiment was conducted to validate the theoretical result about the effect of confining stresses.

Lattice models are powerful tools to investigate damage processes in quasi-brittle material by a microscale perspective. Starting from prior work on a novel rational damage theory for a 2D heterogenous lattice, this paper explores the connection between the series of critical strains at which the microcracks form (i.e. lattice links fail) and the second gradient of the microscale displacement field. Taking a simple tensile test as a representative case study for this endeavour, the analysis of accurate numerical results provides evidence that the second gradient of the microscale displacement field (notably the quantity | ∇ (∂ u^{x}/∂ x)| for the specific example elaborated here) conveys indeed crucial information about the microcracks formation process and can be conveniently used to introduce simplifications of the rational theory that are of relevance by practical purposes as full field strain measurements become routinely possible with digital imaging correlation techniques. Note worthy, the results support the new view that the damage evolution is a three regimes process (I dilute damage, II homogeneous interaction, III localization.) The featured connection with the second gradient of the microscale displacement field is applicable in regions II–III, where microcracks interactions grow stronger and the lattice transitions to the softening regime. The potential impact of these findings towards the formulation of new and physically based CDM models, which are consistent with the reference discrete microscale theory, cannot be overlooked and is pointed out.

Lattice models are powerful tools to investigate damage processes in quasi-brittle material by a microscale perspective. This paper explores the connection between the series of critical strains at which the microcracks form (i.e. lattice links fail) and the second gradient of the microscale displacement field. Note worthy, the results support the new view that the damage evolution is a three regimes process (I dilute damage, II homogeneous interaction, III localization.) The featured connection with the second gradient of the microscale displacement field is applicable in regions II–III, where microcracks interactions grow stronger and the lattice transitions to the softening regime.

We consider the harmonic and eigenvalue problems for piezoelectric nanodimensional bodies with account for surface stresses and surface electric charges. For harmonic problem new mathematical model is suggested, which generalizes the model of the piezoelectric medium with damping properties, boundary conditions of contact type and surface effects. The classical and generalized (weak) statements for harmonic and eigenvalue problems are formulated in the extended and reduced forms. The spectral properties of the eigenvalue problems with account for surface effects are determined. A variational minimal principle is constructed which is similar to the well-known variational principle for problems for pure elastic and piezoelectric media. The discreteness of the spectrum and completeness of the eigenvectors are proved. As a consequence of variational principle, the properties of the natural frequencies increase or decrease are established for changing the mechanical, electric and “surface” boundary conditions and the moduli of piezoelectric body. The finite element approaches are described for determination of the natural frequencies, the resonance and antiresonance frequencies and harmonic behavior of nanosized piezoelectric bodies with account for surface effects.