In this article we apply spectral collocation method to find a numerical solution of stochastic partial differential equations (SPDEs). Spectral collocation method is known to be impressively efficient for PDEs. We investigate this method for numerical solution of SPDEs and we obtain its rate of convergence. At first, the results are expressed for equations with globally Lipschitz coefficient, then we extend it to cases with locally Lipschitz coefficient. The analysis is supported by numerical results for some important SPDEs such as stochastic Kuramoto-Sivashinksy equation.

]]>Stability of fully developed mixed convection flows, with significant viscous dissipation, in a vertical channel bounded by isothermal plane walls having the same temperature and subject to pressure gradient is investigated. It is shown that one of the dual solutions is always unstable and that both are unstable when the total flow rate is big enough. The completely passive natural convection flow is shown to be unstable.

]]>A representation theorem is obtained for an arbitrarily loaded elastic bimaterial solid consisting of an infinite plane containing a circular inhomogeneity. The elastic image method is used for the analysis. The theorem expresses the Airy stress functions that generate the elastic fields for the composite solid explicitly in terms of the Airy stress function for the corresponding homogeneous infinite solid. It shows that if the solution for the homogeneous infinite solid is available, then the solutions for the corresponding bimaterial solid can be deduced by the process of differentiation and integration. The result could provide the important advantage of economy of effort in the determination of the elastic fields for composite planes with circular interfaces.

]]>We revisit, from the standpoint of the modern theory of phase transitions, the classical problem on stretching of a strain-softening bar, considered earlier by Bažant, Belytschko et al. The known solution is singular and predicts localization of deformations at a single point (an interval with zero length) of the bar. We use the model of a phase transforming bar with trilinear stress-strain relation and analytically consider the particular limiting case where the stiffness of a new phase inclusion in the phase-transforming bar is much less than the stiffness of the initial phase. This allows us to construct a regular solution, which converges to the known singular solution in the limiting case of zero new phase stiffness.

]]>Based on the linear piezoelectric theory, three kinds of displacement boundary conditions are used to study the deformations of piezoelectric cantilever beams. The first two conditions are conventional simplified displacement boundary conditions, and the third one is an improved boundary condition determined by the least-squares method. Two load cases and six slenderness ratios of cantilever beams are investigated. Solutions are given by both the conventional boundary conditions and the improved boundary condition, and are then compared with solutions by finite element method. Results from the improved boundary condition are found to be much better than those from the conventional displacement boundary conditions especially for short beams. Among the three displacement boundary conditions of the fixed end, the boundary condition determined by the least-squares method is proved to be the most effective boundary condition.

]]>This paper presents a new analytical approach for sliding contact analysis of laterally graded materials, which allows taking into account the spatial variation of the friction coefficient. The method is developed by considering a sliding frictional contact problem between a laterally graded elastic medium and a rigid flat punch. Governing partial differential equations entailing the displacement components are derived in accordance with the theory of plane elasticity. General solutions are determined and boundary conditions are implemented by the use of Fourier transformation; and the problem is reduced to a singular integral equation of the second kind. Both the shear modulus and the coefficient of friction are assumed to be a functions of the lateral coordinate in the derivations. The singular integral equation is solved numerically by means of an expansion-collocation technique in which the primary unknown is represented as a series in terms of Jacobi polynomials. Outlined procedures yield the stresses at the half-plane surface and the tangential contact force required for sliding. Proposed techniques are verified by making comparisons to the contact stresses computed for constant-friction type sliding contact problems involving homogeneous and laterally graded materials. Parametric analyses are presented so as to demonstrate the influences of the variations in the friction coefficient and the shear modulus upon the contact stresses and the tangential contact force.

]]>A closed form solution is derived for the bonded bimaterial planes at two interfaces. The bonded planes with two interfaces are symmetric with respect to the interface, which is straight. A rational mapping function and complex stress functions are used for the analysis. The problem is reduced to a Riemann-Hilbert problem. Two interfaces problem to derive the general solution is more difficult than one interface problem. As a demonstration of geometry, semi-strips bonded at two parts at the ends of strips are considered. The solution of different geometrical shapes can be obtained by changing the mapping function. Concentrated forces and couples are applied to the each strip. The first derivative of complex stress functions which does not include integral terms with regard to variable of the mapping plane is achieved. Therefore, there is no need of numerical integration to calculate stress components and to determine unknown coefficients in complex stress function. This is very benefit. All elastic constants in complex stress functions are expressed by Dundurs’ parameters. Stress distributions are shown for different lengths of the interface.

]]>In this paper a model describing thermo-elasto-plasticity, phase transitions and transformation-induced plasticity (TRIP) is studied. The main objective is the analysis of a regularization of the corresponding mathematical problem of TRIP and its interaction with classical plasticity under mixed boundary conditions.

]]>The minimal invasive procedure of vertebroplasty is a surgical technique to treat compression fractures of vertebral bodies. During the treatment, liquid bone cement gets injected into the affected vertebral body and therein cures to a solid. In order to investigate the treatment and the impact of injected bone cement, an integrated modelling and simulation framework has been developed. The framework includes (i) the generation of microstructural computer models based on microCT images of human cancellous bone, (ii) computational fluid dynamics (CFD) simulations of bone cement injection into the trabecular structure and (iii) non-linear finite element (FE) simulations of the subsequent bone cement curing. A detailed description of the material behaviour of acrylic bone cements is provided for both simulation stages. A non-linear process-dependent viscosity function is chosen to represent the bone cement behaviour during injection. The bone cements phase change from a highly viscous fluid to a solid is described by a non-linear viscoelastic material model with curing dependent properties. To take into account the distinctive temperature dependence of acrylic bone cements, both material models are formulated in a thermo-mechanically coupled manner. Moreover, the corresponding microstructural CFD- and FE-simulations are performed using thermo-mechanically coupled solvers. An application of the presented modelling and simulation framework to a sample of human cancellous bone demonstrates the capabilities of the presented approach.

]]>We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented.

]]>An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the 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 object at an arbitrary point 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. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.

]]>For a large class of initial data, we prove the existence of classical solutions locally in time to a modified Hele-Shaw problem that takes elastic effects into account. The system arises as sharp interface model of a Cahn-Hilliard system coupled with linearized elasticity. By using the Hanzawa transformation, we can reduce the system to a single evolution equation for the height function. Then short time existence is proven by inverting the linearized operator and applying the contraction mapping principle.

]]>The problem of a spherical shell made of an elastic-plastic second gradient model for plastic materials and subjected to hydrostatic tension is considered. The elastic-plastic second gradient model is a simplified version (porosity neglected) of a second gradient model for plastic porous metals developed, some years ago, by Gologanu, Leblond, Perrin and Devaux, so-called GLPD model. The expressions of the velocity field as well as the ordinary and double stress components are determined for the cases where the spherical shell is modeled by a purely elastic, purely plastic, and elastic-plastic GLPD models. As expected, the solution developed in each case (elastic, ideal-plastic, and elastic-plastic) reduces to that of the first gradient as a special case when the characteristic length scale the GLPD model involves is negligible. Our results allow comparisons between the newly developed solution and the classical elastic-plastic solution for the same model problem; they also provide insights into the influence of the characteristic length scale on the newly developed solution.

]]>The paper deals with numerical realization of discretized, frictionless static contact problems for elastic-perfectly plastic materials and the computational limit analysis. Two numerical methods based on the variational formulation in terms of stresses are analyzed: the semi-smooth Newton method with damping and the alternating direction method of multipliers. These methods are used for tracking the loadings path to determine the discretized limit loading parameter and for solving elastic-perfectly plastic problems.

]]>Dislocation-based analysis of cracked magnetoelectroelastic solid under remotely uniform anti-plane mechanical with in-plane electromagnetic loading is presented. The solution to the generalized dislocation including screw dislocation and electric and magnetic jumps within an incompatible framework are reviewed from the literature. In order to model the system of multiple cracks in the solid, the dislocations are distributed along the crack faces. Then the densities of the dislocations are evaluated by applying the crack-face boundary conditions. Both permeable and impermeable conditions are discussed. The entire field components including shear stress, electric displacements and magnetic inductions are determined for the cracked material, which is an advantage comparing to the methods which only provide crack tip field components. The field intensity factors are also formulated for both permeable and impermeable conditions. Finally examples including horizontal crack, inclined crack and multiple cracks are studied.

]]>This article studies the influence of the nonlocal scale parameter on the deflection of a nonlocal nanobeam and crack growth. Using the Timoshenko hypothesis, a single governing equation is derived and its exact solution can be determined through appropriate end-support conditions. Numerical calculations are carried out for a cantilever microtubule in solution at a given flow speed. The effects of nonlocal scale parameter on the deflection are discussed. Based on the obtained solutions, the double cantilever beam model is utilized to determine energy release rate near a crack tip for an edge crack and a central crack, respectively. It is found that the scale parameter plays different roles in determining stress intensity factors and energy release rates, depending on crack constraints. When neglecting shear deformation, the results for nonlocal Euler-Bernoulli beams can be directly obtained.

]]>In this article we propose a new finite difference scheme for the Falk model system. The Falk model system is a thermoelastic system describing the phase transition occurring on shape memory alloys. Our scheme inherits three important properties: the energy conservative law, the momentum conservation law and the law of increasing entropy. In addition, we show the existence of solution for the scheme and positivity of temperature under some conditions.

]]>This work gives a mathematical account of the propagation of disturbances produced by a sudden draught of air impinging on a cylindrical fibre produced by the melt-spinning process. Accounting for varying tension and aerodynamic drag, the non-dimensional wave-type equation of motion is derived in dimensionless form; the solution of which is sought using the Riemann method of characteristics. Using this procedure, the solutions along the leading forward and reflected characteristics are obtained in closed-form and enable numerical solutions to be obtained via a finite difference routine along the entire computational domain. The analysis shows how such disturbances may be prevented from penetrating beyond the material crystallisation point (where it is extremely susceptible to disturbances) and discusses the application of these findings by optimising the location of a protective shroud.

]]>This paper deals with an energy-entropy-consistent time integration of a thermo-viscoelastic continuum in Poissonian variables. The four differential evolution equations of first-order are transformed by a new G*eneral* E*quation* *for* N*on*-E*quilibrium* R*eversible*-I*rreversible* C*oupling* (GENERIC) format into a matrix-vector notation. Since the entropy is a primary variable, we include thermal constraints to affect the temperatures at the boundary of the body. This enhanced GENERIC format with thermal constraints yields with the related degeneracy conditions structure preservation properties for a system with thermal constraints. The properties of an isolated system are in addition to a constant total linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermo-viscoelastic systems and also for unisolated systems without loads valid. The discretization in time is done with a new TC (*Thermodynamically Consistent*) integrator. This ETC integrator is constructed such, that the algorithmic structural properties after the space-time discretization reflect the underlying enhanced GENERIC format with thermal constraints. As discretization in space the finite element method is used. A projection of the test function of the thermal evolution equation is necessary for an energy-consistent discretization in space. The enhanced GENERIC format with thermal constraints, which is here given in the strong evolution equations, contains the external loads. The consistency properties are discussed for representative numerical examples with different boundary conditions. The coupled mechanical system is solved with a multi-level Newton-Raphson method based on a consistent linearization.

This study focuses on the development of a computationally efficient algorithm for the offline identification of system parameters in nonlinear dynamical systems from noisy response measurements. The proposed methodology is built on the bootstrap particle filter available in the literature for dynamic state estimation. The model and the measurement equations are formulated in terms of the system parameters to be identified - treated as random variables, with all other parameters being considered as internal variables. Subsequently, the problem is transformed into a mathematical subspace spanned by a set of orthogonal basis functions obtained from polynomial chaos expansions of the unknown system parameters. The bootstrap filtering carried out in the transformed space enables identification of system parameters in a computationally efficient manner. The efficiency of the proposed algorithm is demonstrated through two numerical examples - a Duffing oscillator and a fluid structure interaction problem involving an oscillating airfoil in an unsteady flow.

]]>The classic problem of a circular tube or ring buckled by external pressure is investigated by including geometric nonlinearity, material nonlinearity and initial residual stress. Perturbation theory for buckling and immediate postbuckling agree well with numerical integration. It is found that for softening materials the postbuckling may be unstable and catastrophic snap through may occur.

]]>For two types of large-scale bodies, a half-plane and a strip, each of which is weakened by a straight transverse crack, the static problem of elasticity theory is considered. The upper boundary of each body is reinforced by a thin flexible coating. The coating is modeled by special boundary conditions on the upper faces of considered bodies. Three different cases of boundary conditions on the lower face of the strip were studied.

By application of generalized integral transforms to the equilibrium equations in displacements the problems were reduced to the solutions of singular integral equations of first kind with Cauchy kernel to the respect of derivative of the crack opening function. In all considered cases the integral equations consists of a singular term, corresponding to crack behavior in an infinite plate, and a regular term, reflecting the influence of various geometric and physical parameters.

For various sets of model parameters the solutions of the integral equations were built by small parameter and collocation methods; their structure was analyzed. The values of stress intensity factor in the vicinity of the tips of the crack were obtained and analyzed for different coating materials and geometric parameters of the crack.

From the analysis of the numerical results of the problem, it can be concluded that thin flexible coatings significantly reduce stress intensity at a crack tip and therewith significantly increase a reliability of considered elastic bodies.

Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non-mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov-type descent condition and utilizing a Coleman-Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution.

]]>The well-known Hencky solution is only applicable to the problem of deformation of the elastic circular membrane without initial stress under transverse uniformly-distributed loads. The problem considered here is a more general case: an initial tensile or compressive stress has been present in the initially flat circular membrane before the membrane is subjected to the transverse loads. The closed-form solution of the considered problem was presented and all the expressions obtained here for displacements, strains and stresses have the same form as those in the well-known Hencky solution. The initial stress plays an important role in the determination of numerical value of the integral constant controlling membrane equation. The solution obtained here can be regressed into the well-known Hencky solution when the initial stress is equal to zero, and it is therefore called extended Hencky solution.

]]>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.

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.

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.

]]>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.

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.

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 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.

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.

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.

The authors 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 they derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves.

By virtue of the representations of displacements, stresses, and temperature fields in terms of two scalar potential functions and the use of correspondence principle, an analytical derivation of fundamental Green's functions for bi-material half-space composed of a transversely isotropic thermo-elastic layer and an isotropic thermo-visco-elastic half-space affected by finite surface or interfacial sources is presented. With the aid of the potential function relationships, the coupled equations of motion and energy equation in both the half-space and the layer are uncoupled and solved with the aid of Fourier series and Hankel integral transforms. Responses of the medium are derived in the form of improper line integrals related to Hankel inversion transforms. To show the effects of anisotropy and viscoelasticity on the propagation of coupled thermoviscoelastic waves, the derived integrals for displacements, stresses, and temperature Green's functions are evaluated by a numerical scheme.

By virtue of the representations of displacements, stresses, and temperature fields in terms of two scalar potential functions and the use of correspondence principle, an analytical derivation of fundamental Green's functions for bi-material halfspace composed of a transversely isotropic thermo-elastic layer and an isotropic thermo-visco-elastic half-space affected by finite surface or interfacial sources is presented. To show the effects of anisotropy and viscoelasticity on the propagation of coupled thermoviscoelastic waves, the derived integrals for displacements, stresses, and temperature Green's functions are evaluated by a numerical scheme.

The resonant frequency of flexural vibration for a double tapered atomic force microscope (AFM) cantilever has been investigated by considering the damping effect. In this paper the effects of the contact position, contact stiffness, height of the tip, thickness of the beam, the height and breadth taper ratios of cantilever, the angle between the cantilever and the sample surface and damping parameter based on Timoshenko beam theory on the non-dimensional frequency and sensitivity have been studied. The differential Quadrature method (DQM) is employed to solve the nonlinear differential equations of motion. The results show that the resonant frequency decreases when Timoshenko beam parameter or cantilever thickness increases and high order modes are more sensitive to it. The first frequency is sensitive only in the lower range of contact stiffness, but the higher order modes are sensitive to the contact stiffness in a larger range. Increasing the tip height increases the sensitivity of the vibrational modes in a limited range of normal contact stiffness. Increasing the lateral contact stiffness increases the sensitivity to the normal contact stiffness after critical normal contact stiffness, but when the normal contact stiffness is lower than critical normal contact stiffness, the situation is reversed. By increasing the lateral damping parameter, the resonant frequency and sensitivity to the contact stiffness decrease. Furthermore, by increasing the breadth taper ratio, the frequency increases.

The resonant frequency of flexural vibration for a double tapered atomic force microscope (AFM) cantilever has been investigated by considering the damping effect. In this paper the effects of the contact position, contact stiffness, height of the tip, thickness of the beam, the height and breadth taper ratios of cantilever, the angle between the cantilever and the sample surface and damping parameter based on Timoshenko beam theory on the non-dimensional frequency and sensitivity have been studied.

This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface. We define a measurement for the roughness of the cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the laser beam intensity along the free melt surface. Necessary optimality conditions will be deduced. Finally, a numerical solution will be presented and discussed by means of the necessary conditions. physical considerations.

This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface.

The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical and follows Mikhlin [29] approach to the solution of the harmonic problem. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges.

The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges.

The authors would like to correct the proof of Theorem 3(a) in the original paper “On the asymptotic behaviour of the 2D Navier-Stokes equations with Navier friction conditions towards Euler equations.” [ZAMM 89(10), 810-822, 2009].

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