We consider the problem of determining an optimal semi-active damping of vibrating systems. For this damping optimization we use a minimization criterion based on the impulse response energy of the system. The optimization approach yields a large number of Lyapunov equations which have to be solved. In this work, we propose an optimization approach that works with reduced systems which are generated using the parametric dominant pole algorithm. This optimization process is accelerated with a modal approach while the initial parameters for the parametric dominant pole algorithm are chosen in advance using residual bounds. Our approach calculates a satisfactory approximation of the impulse response energy while providing a significant acceleration of the optimization process. Numerical results illustrate the effectiveness of the proposed algorithm.

]]>We present in this article a positive finite volume method for diffusion equation on deformed meshes. This method is mainly inspired from , and uses auxiliary unknowns at the nodes of the mesh. The flux is computed so as to be a two-point nonlinear flux, giving rise to a matrix which is the transpose of an M-matrix, which ensures that the scheme is positive. A particular attention is given to the computation of the auxiliary unknowns. We propose a new strategy, which aims at providing a scheme easy to implement in a parallel domain decomposition setting. An analysis of the scheme is provided: existence of a solution for the nonlinear system is proved, and the convergence of a fixed-point strategy is studied.

]]>In this paper, we establish the blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equations in with by and .

]]>We consider an initial and boundary value problem the one dimensional wave equation with damping concentrated at an interior point. We prove a result of a logarithmic decay of the energy of a system with homogeneous Dirichlet boundary conditions. The method used is based on the resolvent estimate approach which derives from the Carleman estimate technique. Under an algebraic assumption describing the right location of the actuator, we prove a logarithmic decay of the energy of solution. We show that this assumption is lower than the one given by and which depends on the diophantine approximations properties of the actuator's location.

]]>In this paper we prove a local result of existence and uniqueness for a free boundary problem for snow avalanche arising from a new model proposed in . The mathematical problem consists of a parabolic free boundary problem with non-standard free boundary conditions (erosion dynamics). The proof is essentially based on a fixed point argument.

]]>A new application of the path-independent M- and L-integrals in linear elastic fracture mechanics is presented for the accurate calculation of loading quantities related to two-cracks problems in engineering structures. Path-independent integrals are used to avoid special requirements concerning crack tip meshing and contour size. The numerical calculation of M- and L-integrals is performed along the external boundary of the model. This global contour includes both crack tips and thus the resulting values represent the sum of loading quantities related to each crack tip. A separation technique is necessary to calculate local values of the J-integral and stress intensity factors. Numerical examples of crack propagation simulations are presented and the resulting crack paths are verified and compared with those from conventional methods.

]]>Momentum and energy equations for vertical flow with viscous dissipation are derived and shown to require that the cross-section mean density is taken as the reference density for calculation of buoyancy forces under the Boussinesq approximation. Solutions are obtained for flow between parallel plane walls, with and without the pressure work as an explicit term in the energy equation. Both walls are at the same temperature, so there is no thermal forcing, but solutions are obtained for all admissible values of dynamic pressure gradient. The passive convection condition, whereby the flow is driven entirely by buoyancy forces resulting from heat generated by the flow's own viscous dissipation, is found on one branch of the dual solutions. However, while theoretically possible, passive convection is not physically realisable with any real fluid.

]]>The paper deals with the homogenization of strongly heterogeneous elastic plates satisfying the Reissner-Mindlin or the Kirchhoff-Love hypotheses. We rigorously justify the limit models obtained by the asymptotic analysis which describe the harmonic waves propagation associated with in-plane displacement and transversal deflection modes in these two classical plate structures. Large contrasts in the coefficients of the elastic material components may result in existence of band gaps for the limit Reissner-Mindlin plates while an analogous property is lost for the deflection of the Kirchhoff-Love model. The different dispersion properties of both the limit plates are related to the changing sign of the limit frequency dependent mass density coefficients.

]]>The behavior of a two-body self-propelling locomotion system in a resistive environment is studied. The motion of the system is excited and sustained by means of a periodic change in the distance between the bodies. A complete analysis of the motion of the system is performed for the case where the resistance forces applied by the environment to the bodies of the system are represented by linear functions of the velocities of these bodies relative to the environment. For the case where the resistance forces are nonlinear functions of the velocities of the bodies, a model based on the averaged equation of motion is used. This model assumes the forces of friction acting in the system to be small in comparison with the excitation force. The motion of the system along a horizontal straight line in an isotropic dry friction environment is investigated in detail for two particular types of excitation modes. The calculated results are compared with the experimental data.

]]>In this paper the fracture behaviors of magnetoelectroelastic cylinder induced by a penny-shaped magnetically dielectric crack are investigated. By employing the Hankel transform technique and introducing three auxiliary functions, the complex question is transformed to solve three coupled nonlinear Fredholm integral equations. The intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) are derived in closed forms. The effects of the radius of the cylinder, applied electric field and magnetic field, dielectric permittivity and magnetic permeability of the crack interior on the COD intensity factor are illustrated numerically. The results corresponding to magnetoelectrically permeable and impermeable boundary conditions are only the special cases of the present model.

]]>We study mathematical properties of quasi-incompressible fluids. These are mixtures in which the density depends on the concentration of one of their components. Assuming that the mixture meets mass and volume additivity constraints, this density-concentration relationship is given explicitly. We show that such a constrained mixture can be written in the form similar to compressible Navier-Stokes equations with a singular relation between the pressure and the density. This feature automatically leads to the density bounded from below and above. After addressing the choice of thermodynamically compatible boundary conditions, we establish the large data existence of weak solution to the relevant initial and boundary value problem. We then investigate one possible limit from the quasi-compressible regime to the incompressible regime.

]]>In this paper we investigate the dynamic behaviour of a thermoelastic diffusion rod clamped at one end and moves freely between two stops at the other. The contact is modelled with the Signorini or normal compliance conditions. The coupled system of equations consists of a hyperbolic equation and two parabolic equations. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. The existence of a weak solution is proved using a penalization method and compensated compactness. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity. We describe the discrete finite element method to our numerical approximations and we show that the given solution converges to the weak solution. Finally, we give an error estimate assuming extra regularity on the solution and we give some results of our numerical experiments.

]]>For the heat equation in a bounded domain we consider the inverse problem of identifying locations and certain properties of the shapes of small heat-conducting inhomogeneities from dynamic boundary measurements on part of the boundary and for finite interval in time. The key ingredient is an asymptotic method based on appropriate averaging of the partial dynamic boundary measurements. Our approach is expected to lead to very effective computational identification algorithms.

]]>In this paper, we study the Gross-Pitaevskii system with trapped dipolar quantum gases. We obtain both the stable regime and the unstable regime. Moreover, via a construction of cross minimization problem, the blow up threshold is established.

]]>Finite element error estimates are derived for the incompressible Stokes equations with variable viscosity. The ratio of the supremum and the infimum of the viscosity appears in the error bounds. Numerical studies show that this ratio can be observed sometimes. However, often the numerical results show a weaker dependency on the viscosity.

]]>The principal focus of this paper is the formulation of a general approach to hyperelastic strain energy functions that does not rely on the use of scalar invariants of tensors. We call this an invariant-free formulation of hyperelasticity. This essentially requires the conversion of the strain energy function from one of scalar products of scalar tensor invariants (all zeroth-order) into one of quadruple contractions between fourth-order tensors, thus preserving directional distinctions through to energy. We begin with an analysis of a range of hyperelastic properties in order to eliminate some non-physical models. In the section after, we are left with the Simo and Pister model and the Compressible neo-Hookean model, and decide on Simo and Pister's model for further study. Presented is a general form of invariant-free hyperelasticity (the so-called generalized strain energy function), and the fitting of the Simo and Pister model into that framework. The novelty of this invariant-free formulation is threefold: first allowing the presentation of strain energy as a fourth-order tensor that explicitly provides the origin of energy contributions from a possible 81 combinations through the simple exchange of the quadruple contraction operator with the Hadamard product; second is a new ability to seamlessly integrate micropolar effects into existing hyperelastic functions (a cursory look); and third is the direction-preserving nature of the formulation, which satisfies the original charter of this work in providing a primer for the natural extension of advanced conventional hyperelastic functions from isotropic materials to anisotropic materials.

]]>Adapted from the parallel and series connections of materials a homogenization method for the calculation of effective material constants of composite materials is derived. A complex structure is divided into small volumes and each volume represents a different material. Thereby the volume fraction of complex-formed inclusions can be approximated. Then, considering the position of volumes to each other (parallel / series connection) the effective material constants for elastic and piezoelectric materials are calculated.

]]>This study is concerned with analysis of the Rayleigh wave field in a 3D isotropic elastic half-space subject to in-plane surface loading. The approach relies on the slow time perturbation of the general representation for the Rayleigh wave eigensolutions in terms of harmonic functions. The resulting hyperbolic-elliptic formulation allows decomposition of the original vector problem of 3D elasticity into a sequence of scalar Dirichlet and Neumann problems for the Laplace equation. The boundary conditions for these are specified through a 2D hyperbolic equation. An example of an impulse tangential load illustrates the efficiency of the derived asymptotic formulation, with the results expressed in terms of elementary functions.

]]>In this paper, we investigated the influence of initial stress on the frequency equation of flexural waves in a transversely isotropic circular cylinder permeated by a magnetic field. The problem is represented by the equations of elasticity taking into account the effect of the magnetic field as given by Maxwell's equations in the quasi-static approximation. The free stress conditions on the inner and outer surfaces of the hollow circular cylinder were used to form a frequency equation in terms of the wavelength, the cylinder radii, the initial stress and the material constants. The frequency equations have been derived in the form of a determinant involving Bessel functions and its roots given the values of the characteristic circular frequency parameters of the first three modes for various geometries. These roots, which correspond to various modes, have been verified numerically and represented graphically in different values for the initial stress. It is recognized that the flexural elastic waves in a solid body propagated under the influence of initial stress can be differentiated in a clear manner from those propagated in the absence of an initial stress. We also observed the initial stress has a great effect on the propagation of magnetoelastic flexural waves. Therefore this research is theoretically useful to convey information on electromagnetic properties of the material: for example through a precise measurement of the surface current induced by the presence of the magnetic field.

]]>The effective solutions for integro-differential equations related to problems of interaction of an elastic thin finite inclusion with a plate, when the inclusion and plate materials possess the creep property are constructed. If the geometric parameter of the inclusion is measured along its length according to the parabolic and linear law we have managed to investigate the obtained boundary value problems of the theory of analytic functions and to get exact solutions and establish behavior of unknown contact stresses at the ends of an elastic inclusion.

]]>Modeling the motion of the Earth's axis, *i.e*., its spin, nutation and its precession, is a prime example of our ongoing effort to simulate the behavior of complex mechanical systems. In fact, models of increasing complexity of this motion have been presented for more than 400 years leading to an increasingly accurate description. The objective of this paper is twofold namely, first, to provide a review of these efforts and, second, to provide an improved analysis, if possible, based on today's numerical possibilities. Newton himself treated the problem of the precession of the Earth, a.k.a. the precession of the equinoxes, in Liber III, Propositio XXXIX of his Principia . He decomposed the duration of the full precession into a part due to the Sun and another part due to the Moon, which would lead to a total duration of 25,918 years. This agrees fairly well with the experimentally observed value. However, Newton does not really provide a concise rational derivation of his result. This task was left to Chandrasekhar in Chapter 26 of his annotations to Newton's book . He follows an approach suggested by Scarborough starting from Euler's equations for the gyroscope and by calculating the torques due to the Sun and to the Moon on a tilted spheroidal Earth. These differential equations can be solved approximately in an analytic fashion, yielding something close to Newton's more or less fortuitous result. However, the equations can also be treated more properly in a numerical fashion by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and discuss the outcome of the numerical solution. A comparison to actual measurements will also be attempted. When solving the Euler equations for the aforementioned case numerically it shows that besides the precessional movement of the Earth's axis there is also a nutational one present. However, as we shall show, if Scarborough's procedure is followed, the period of this nutation turns out to be roughly half a year with a very small amplitude whereas the observed (main) nutational period is much longer, namely roughly nineteen years, and much more intense amplitude-wise. The reason for this discrepancy is based on the assumption that the torques of both the Sun and the Moon are due to gravitational actions within the equinoctial plane. Whilst this is true for the Sun, the revolution of the Moon around the Earth occurs in a plane, which is inclined by roughly 5° w.r.t. the equinoctial. Moreover, this plane rotates such that the ascending and descending nodes of the moon precede with a period of roughly 18 years. If all of this is taken into account the analytically predicted nutation period will be of the order of the observed value . As in the case of the precession we will provide a more stringent analysis based on a numerical solution of the Euler equations, which leads beyond the results presented in Sect.12.10 of .

We develop the elastic constitutive law for the resultant statically and kinematically exact, nonlinear, 6-parameter shell theory. The Cosserat plane stress equations are integrated through-the- thickness under assumption of the Reissner-Mindlin kinematics. The resulting constitutive equations for stress resultant and couple resultants are expressed in terms of two micropolar constants: the micropolar modulus and the micropolar characteristic length *l*. Based on FEM simulations we evaluate their influence on the behaviour of shell models in the geometrically nonlinear range of deformations.

The performance of perturbation method in nonlinear analyses of plates subjected to mechanical, thermal, and thermo-mechanical loadings is investigated. To this end, cylindrical bending of FG plates with clamped and simply-supported edges is considered. The governing equations of Mindlin's first-order shear deformation theory with von Kármán's geometric nonlinearity are solved using one- and two-parameter perturbation methods and the results are compared with the results of an analytical solution. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. It is shown that the accuracy of any-order expansion in perturbation method depends not only on the perturbation parameter, but also on the location chosen for the perturbation parameter and, in general, the solution becomes more accurate when the perturbation parameter is specified at the location where its corresponding response quantity is a maximum. Under thermal loading the possibility of using different parameters as the perturbation parameter for various boundary conditions is investigated. It is observed that, instead of a one-parameter perturbation method, a two-parameter perturbation method must be used in the thermal analysis of FG plates. Also, buckling and post-buckling behavior of FG plates in cylindrical bending is investigated. It is shown that under thermal loading, a bifurcation-type buckling occurs in clamped FG plates. In addition, a snap-through buckling may occur in simply-supported FG plates under thermo-mechanical loading.

A myriad of nondestructive interrogation methods have been developed to assess the initial and progressive integrity of composite materials, with varying degrees of success. Some frontiers, however, have stubbornly resisted this progress; the present paper addresses three of these by discussing our recent progress in the development and application of through-thickness dielectric methods for a wide range of heterogeneous materials. The first topic is the creation of a characterization method and interpretive analysis that can correctly reveal the process of distributed damage initiation and accumulation with non-dilute concentrations in heterogeneous structural composite materials, in contrast to more standard methods that focus on identifying individual flaws. The second topic is development of inverse and adjoint methods to enable the design of heterogeneous materials for specific dielectric responses. And the third topic is the development of a “current state method” for such materials that can provide measurements to predict the subsequent strength and life of individual specimens, in a manner that is somewhat analogous to a single crack analysis in homogeneous materials. The foundation for our discussion of those topics is through-thickness characterization of the low-frequency spectral dielectric response of composite materials to harmonic EMF signals. Successes and remaining challenges are discussed and opportunities for implementation identified.

]]>We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions , we ask whether we can characterize weak limits *u* when as . We assume neither periodicity nor stochasticity for the coefficients, but we demand an abstract averaging property of the homogeneous system on reference volumes. Our conclusion is an effective equation on general domains with general right hand sides. The effective equation uses a causal evolution operator Σ that maps strains to stresses; more precisely, in each spatial point *x*, given the evolution of the strain in the point *x*, the operator Σ provides the evolution of the stress in *x*.

We consider a Neumann problem for the Poisson equation in the periodically perforated Euclidean space. Each periodic perforation has a size proportional to a positive parameter ε. For each positive and small ε, we denote by a suitably normalized solution. Then we are interested to analyze the behavior of when ε is close to the degenerate value , where the holes collapse to points. In particular we prove that if , then can be expanded into a convergent series expansion of powers of ε and that if then can be expanded into a convergent double series expansion of powers of ε and . Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.

]]>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.

]]>In this article the authors continue the discussion in about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincaré operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.

]]>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.

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.

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.

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 .

]]>This paper is concerned with the fundamental solutions, in the framework of thermo-electro-elasticity, for an infinite/half-infinite space of 1D hexagonal quasicrystals (QCs). To this end, three-dimensional static general solutions, in terms of 5 quasi-harmonic functions, are derived with the help of rigorous operator theory and generalized Almansi's theorem. For an infinite/half-infinite space subjected to an external thermal load, corresponding problem is formulated by boundary value problems. Appropriate potential functions are set by a trail-and-error technique. Green functions for the problems in question are obtained explicitly in the closed forms. The present fundamental solutions can be employed to construct 3D analysis for crack, indentation and dislocation problems. Furthermore, these solutions also serve as benchmarks for various numerical simulations.

This paper is concerned with the fundamental solutions, in the framework of thermo-electro-elasticity, for an infinite/halfinfinitespace of 1D hexagonal quasicrystals (QCs). To this end, three-dimensional static general solutions, in terms of 5quasi-harmonic functions, are derived with the help of rigorous operator theory and generalized Almansi's theorem. Foran infinite/half-infinite space subjected to an external thermal load, corresponding problem is formulated by boundaryvalue problems. Appropriate potential functions are set by a trail-and-error technique. Green functions for the problemsin question are obtained explicitly in the closed forms.

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.

]]>A differential geometric description of crystals with continuous distributions of lattice defects and undergoing potentially large deformations is presented. This description is specialized to describe discrete defects, i.e., singular defect distributions. Three isolated defects are considered in detail: the screw dislocation, the wedge disclination, and the point defect. New analytical solutions are obtained for elastic fields of these defects in isotropic solids of finite extent, whereby terms up to second order in strain, involving elastic constants up to third order, are retained in the stress components. The strain measure used in the nonlinear elastic potential – a symmetric function, expressed in material coordinates, of the inverse deformation gradient – differs from that used in previous solutions for crystal defects, and is thought to provide a more realistic depiction of mechanics of large deformation than previous theory involving third-order Lagrangian elastic constants and the Green strain tensor. For the screw dislocation and wedge disclination, effects of core pressure and/or possible contraction along the defect line are considered, and radial displacement contributions arise that are absent in the linear elastic solution, affecting dilatation. Stress components are shown to differ from those of linear elastic solutions near defect cores. Volume change from point defects is strongly affected by elastic nonlinearity.

]]>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 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.

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.

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.