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

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

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

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

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

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

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

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

]]>We propose a hysteretic model for electromechanical coupling in piezoelectric materials, with the strain and the electric field as inputs and the stress and the polarization as outputs. This constitutive law satisfies the thermodynamic principles and exhibits good agreement with experimental measurements. Moreover, when it is coupled with the mechanical and electromagnetic balance equations, the resulting PDE system is well-posed under the hypothesis that hysteretic effects take place only in one preferred direction. We prove the existence and uniqueness of its global weak solutions for each initial data with prescribed regularity. One of the tools is a new Lipschitz continuity theorem for the inverse Preisach operator with time dependent coefficients.

]]>This paper considers the crack problem for a semi-infinite plate subjected to a thermal shock by using hyperbolic heat conduction equations and equations of motion. The Laplace transform and Fourier transform are used to reduce the problem to a system of singular integral equations which are solved numerically. Numerical results are presented illustrating the influence of non-Fourier effect and inertia effect on the stress intensity factor and normalized crack opening displacements. The results show that the stress intensity factors have higher amplitude and an oscillating feature comparing to those obtained under the conventional Fourier thermal conduction condition and quasi-static hypothesis. These results can help understand the crack behaviors of advanced materials under thermal impact loading.

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

]]>Basic relations of the resultant linear six-field theory of shells are established by consistent linearization of the resultant 2D non-linear theory of shells. As compared with the classical linear shell models of Kirchhoff-Love and Timoshenko-Reissner type, the six-field linear shell model contains the drilling rotation as an independent kinematic variable as well as two surface drilling couples with two work-conjugate surface drilling bending measures are present in description of the shell stress-strain state. Among new results obtained here within the six-field linear theory of elastic shells there are: 1) formulation of the extended static-geometric analogy; 2) derivation of complex BVP for complex independent variables; 3) description of deformation of the shell boundary element; 4) the Cesáro type formulas and expressions for the vectors of stress functions along the shell boundary contour; 5) discussion on explicit appearance of gradients of 2D stress and strain measures in the resultant stress working.

]]>The present paper is devoted to the study of the effective properties of 2D unbounded composite materials with temperature dependent conductivities. We consider a special case of nonlinear composites, when the conductivity coefficients of the matrix and the composite constituencies are proportional. This allows us to transform the problem for the nonlinear composite to a problem for an equivalent linear composite and then to find a solution of the nonlinear type. Analyzing the effective properties of the composites we derive relationships between their average properties. We show that, when computing the effective properties of the representative cell of the nonlinear composite, the result may depend not only on the temperature but also on its gradient.

]]>The method of dimensionality reduction (MDR) is extended to the axisymmetric frictionless unilateral Hertz-type contact problem for a viscoelastic half-space and an arbitrary axisymmetric rigid indenter under the assumption that the circular contact area (arbitrarily evolving in time) remains singly connected during the whole process of indentation. In particular, the MDR is applied to study in detail the so-called rebound indentation problem, where the contact radius has a single maximum. It is shown that the obtained closed-form analytical solution for the rebound indentation displacement (recorded in the recovery phase, when the contact force vanishes) does not depend on the indenter shape.

]]>The analysis of the numerical stability of co-simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well-known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force-, force/displacement- and displacement/displacement-decomposition. The stability analysis of co-simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single-mass oscillator, a linear two-mass oscillator is used here for analyzing the stability of co-simulation methods. The two-mass co-simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co-simulation test model with a linear co-simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis.

]]>In this work we consider the non-simple theory of elastic material with voids and we investigate how the coupling of these two aspects of the material affects the behavior of the solutions. We analyze only two kind of different behavior, slow or exponential decay. We introduce four different dissipation mechanisms in the system and we study, in each case, the effect of this mechanism in the behavior of the solutions.

]]>This classical Korteweg capillarity system is here encapsulated in a quintic derivative nonlinear Schrödinger equation for a model Kármán-Tsien type capillarity law. An integrable subsystem is isolated and invariants of motion are use to construct novel exact solutions. The latter involve parameters introduced via a gauge and reciprocal transformation.

]]>This paper has been inspired by ideas presented by V. V. Kozlov in his works . In the present work the main goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational, i.e., vakonomic ones. So far, we can see that they give quite different results. The vakonomic model from the mathematical point of view seems to be more elegant. The similar problems were examined by M. Jóźwikowski and W. Respondek in their paper .

]]>A 3D beam model, i.e. a beam that may deform in space and experience longitudinal and torsional deformations, is developed considering Timoshenko's theory for bending and assuming that the cross section rotates as a rigid body but may deform in longitudinal direction due to warping. The cross sectional properties are firstly calculated and then inserted at the equation of motion. The beam is assumed to be with an arbitrary cross section, with linearly varying thickness and width, and with an initial twist. The model is appropriate for open and closed thin-walled cross sections, and also for solid cross sections. The objective of the current research is to demonstrate that complex beam structures can be modeled accurately with reduced number of degrees of freedom.

]]>In , Belkacem and Kasimov studied the stability of an one-dimensional Timoshenko system in with one distributed temperature or Cattaneo dissipation damping. They proved that the heat dissipation alone is sufficient to stabilize the system. But there is a difference between the Timoshenko system in and its analogous system in . For this reason, the stability results are no longer the same and of intrinsic difference. In this paper, we consider the stability of some distributed systems involving Mindlin-Timoshenko plate in the plane. If the plate is subject to two internal distributed damping then, using a direct approach based on the Fourier transform, we establish a polynomial energy decay rate for initial data in . In the case of indirect internal stability, when only one among the two equations is effectively damped while the second is indirectly damped through the coupling, we have two different situations. To be more precise, if the equation of the displacement in the vertical direction of the plate is only damped then, the system is unstable. Next, when the control is acting on the equation of the angles of rotation of a filament of the plate, no decay can be proved but our conjecture is a polynomial stability. Finally, unlike the one-dimensional case, we show that, under a heat conduction (by Fourier or Cattaneo law), the plate is unstable.

]]>In this research, dynamic behaviour of a composite laminated beam subjected to multiple projectiles is analysed. Temperature elevation is also taken into account. Hertz law of contact is used to model the impact phenomenon between the projectiles and the target. Beam obeys the first order shear deformation theory assumptions. Governing motion equations of the beam and projectiles are obtained using the Hamilton principle. Conventional Ritz method suitable for arbitrary in-plane and out-of-plane boundary conditions is implemented to reduce the partial differential equations into time-dependent ordinary differential equations. Time domain solution of such equations is extracted by means of the well-known fourth-order Runge-Kutta method. After validating the proposed model with the available numerical data, parametric studies are conducted to investigate the influences of multiple impactors, beam characteristics, boundary conditions and thermal environment. It is shown that, temperature elevation decreases the contact force and increases the contact time.

]]>Debonding fracture of a bimaterial strip with two interfaces is investigated subjected to concentrated forces and couples. In the previous paper (ZAMM, see below), closed form stress functions were derived for the bonded bimaterial planes with two interfaces. As a demonstration of geometry, semi-strips bonded at two places of the ends of strips subjected to concentrated forces and couples were analyzed. Using the stress function, the stress intensities of debonding (SID) are obtained. To investigate the accuracy of SID calculated by the stress function, a comparison with the results obtained by a boundary element analysis is carried out and it is confirmed that they agree well each other. It is stated that SID is the square root of the strain energy release rate and the same as the strain energy release rate for evaluating the strength of the fracture. Then the debonding extension behaviors are investigated for some initial debonding states and three loading conditions, concentrated forces, couples and both combined loadings, using SID. Expressions to calculate SID for arbitrary loading magnitudes are derived. Fatigue growth of debonding under cyclic loading is also investigated, using Paris law regarding fatigue.

]]>The motion of self-propelling limbless locomotion systems in a linear viscous environment is considered. The resistance (friction) force acting on an element of the systems is assumed to be proportional to the velocity of this element relative to the environment, the coefficient of proportionality (coefficient of friction) being constant. Two models of interaction of the locomotor with the environment are distinguished. In the first model, the coefficient of friction is constant for a mass element, whereas in the second model, this coefficient is constant for a length element. It is shown that progressive locomotion is impossible for the first model and is possible for the second model. This is explained by the fact that in the second model, the coefficient of friction for a mass element is in fact controlled by changing the length of this element due to deformation of the locomotor's body. The first model applies for lumped mass systems, while the second model is adequate for distributed mass limbless locomotors, like worms. For both models, the equations of motion of the system's center of mass are derived and analyzed.

]]>This study is concerned with the mathematical modeling of microtubules for torsional vibration analysis. Microtubules (MTs) are prone to mechanical torsion, and hence conductance oscillation, when they act as molecular pathways for cargo transport. The period of this torsional oscillation has an intimate, but yet-to-be-understood relationship with the MTs’ stiffness and end conditions. In the spirit of the foregoing concern, this study embarks on the characterization of the shift in the torsional resonance of an isolated MT in the presence of an end attachment in the form of either outer kinetochores (treated as a torsional spring) or a linker molecule (treated as a concentrated mass). The MT is itself idealized as a strain gradient micro-shaft, with its governing equation augmented by enriched boundary conditions due to the attachments. The solution of the model is then sought through the differential transformation method (DTM). Validation of the reduced form of the model is evaluated with benchmark results in the literature. Under a rapidly increasing magnitude of an attached rotary mass, the analyses indicate the existence of a drift of the frequency towards zero, a scenario that could induce a transition to rigid-body motion. Conversely, in the presence of spring-like elastic anchors, the analyses reveal a softening effect that increases the torsional resonant frequencies of the MT. Sensitivity assessments through Pareto analyses predict an interaction effect between the size-effect and the magnitude of an attached mass.

]]>We describe a Discontinuous Galerkin (DG) scheme for variable-viscosity Stokes flow which is a crucial aspect of many geophysical modelling applications and conduct numerical experiments with different elements comparing the DG approach to the standard Finite Element Method (FEM). We compare the divergence-conforming lowest-order Raviart-Thomas (RT_{0}P_{0}) and Brezzi-Douglas-Marini (BDM_{1}P_{0}) element in the DG scheme with the bilinear Q_{1}P_{0} and biquadratic Q_{2}P_{1} elements for velocity and their matching piecewise constant/linear elements for pressure in the standard continuous Galerkin (CG) scheme with respect to accuracy and memory usage in 2D benchmark setups.
We find that for the chosen geodynamic benchmark setups the DG scheme with the BDM_{1}P_{0} element gives the expected convergence rates and accuracy but has (for fixed mesh) higher memory requirements than the CG scheme with the Q_{1}P_{0} element without yielding significantly higher accuracy. The DG scheme with the RT_{0}P_{0} element is cheaper than the other first-order elements and yields almost the same accuracy in simple cases but does not converge for setups with non-zero shear stress. The known instability modes of the Q_{1}P_{0} element did not play a role in the tested setups leading to the BDM_{1}P_{0} and Q_{1}P_{0} elements being equally reliable. Not only for a fixed mesh resolution, but also for fixed memory limitations, using a second-order element like Q_{2}P_{1} gives higher accuracy than the considered first-order elements.

In this paper we prove the existence of weak solutions for the inclusion problem in anti-plane Cosserat elasticity in Sobolev space setting, and for the corresponding systems of boundary integral equations.

]]>A theoretical study on the problem of piezoelectric materials with a nano elliptic cavity considering surface effect subjected to far-field antiplane mechanical load and inplane electric load is reported. Surface effect is introduced based on the theory of Gurtin-Murdoch surface/interface model. A rigorous whole-field solution is presented by using the complex variable elasticity theory and the conformal mapping method, in terms of which closed-form solution of the stress/electric displacement fields and the stress/electric displacement intensity factors is obtained. The results reveal that the stress/electric displacement fields and the stress/electric displacement intensity factors are size dependent when the size of the cavity/crack is on the order of nanometer. The present solution approaches to the classical electroelastic results with the increase of the size of the nano elliptic cavity and crack. The influences of the elliptic cavity shape ratio on the stress and electric displacement concentration factors are discussed.

]]>An analysis has been made for the unsteady separated stagnation-point (USSP) flow of an incompressible viscous and electrically conducting fluid over a moving surface in the presence of a transverse magnetic field. The unsteadiness in the flow field is caused by the velocity and the magnetic field, both varying continuously with time *t*. The effects of Hartmann number *M* and unsteadiness parameter β on the flow characteristics are explored numerically. Following the method of similarity transformation, we show that there exists a definite range of for a given *M*, in which the solution to the governing nonlinear ordinary differential equation divulges two different kinds of solutions: one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). We also show that below a certain negative value of β dependent on *M*, only the RFS occurs and is continued up to a certain critical value of β. Beyond this critical value no solution exists. Here, emphasis is given on the point as how long would be the existence of RFS flow for a given value of *M*. An interesting finding emerges from this analysis is that, after a certain value of *M* dependent on , only the AFS exists and the solution becomes unique. Indeed, the magnetic field itself delays the boundary layer separation and finally stabilizes the flow since the reverse flow can be prevented by applying the suitable amount of magnetic field. Further, for a given positive value of β and for any value of *M*, the governing differential equation yields only the attached flow solution.

The equilibrium problems for two-dimensional elastic body with a rigid delaminated inclusion are considered. In this case, there is a crack between the rigid inclusion and the elastic body. Non-penetration conditions on the crack faces are given in the form of inequalities. We analyze the dependence of solutions and derivatives of the energy functionals on the thickness of rigid inclusion. The existence of the solution to the optimal control problem is proved. For that problem the cost functional is defined by derivatives of the energy functional with respect to a crack perturbation parameter while the thickness parameter of rigid inclusion is chosen as the control function.

]]>Dynamic pull-in behavior of nonlocal functionally graded nano-actuators by considering Casimir attraction is investigated in this paper. It is assumed that the nano-bridge is initially at rest and the fundamental frequency of nano-structure as a function of system parameters is obtained asymptotically by Iteration Perturbation Method (IPM). The effects of actuation voltage, nonlocal parameter, properties of FGM materials and intermolecular force on the dynamic pull-in behavior are studied. It is exhibited that two terms in series expansions are adequate to achieve the acceptable approximations for fundamental frequency as well as the analytic solution. Comparison between the obtained results based on the asymptotic analysis and the reported experimental and numerical results in the literature, verify the effectiveness of the asymptotic analysis.

]]>Variable-mass conditions can occur in a variety of practical problems of engineering. Investigations on problems of this type have been figuring as a particular research field of mechanics and applied mathematics. The fundamental issue is that the basic equations of classical mechanics were originally formulated for the case of an invariant mass contained in a material volume. Therefore, appropriate formulations are required when dealing with variable-mass problems. The scope of the present article is devoted to arbitrarily moving control volumes formulated within the framework of Ritz's method, that is, to non-material volumes in the sense discussed by Irschik and Holl . We aim at demonstrating a generalized version of Noether's theorem such that it can be grounded on the generalized Hamilton's principle for a non-material volume in the form derived by Casetta and Pesce . This will consistently allow the consideration of conservation laws, written from a Noetherian approach, in this particular context of non-material volumes. To test the proposed formulation, the problem of a rotating drum uncoiling a strip will be addressed.

]]>The Riemann solutions for the one-dimensional Chaplygin gas equations with a Coulomb-like friction term are constructed explicitly. It is shown that the delta shock wave appears in the Riemann solutions in some certain situations. The generalized Rankine-Hugoniot conditions of delta shock wave are established and the position, propagation speed and strength of delta shock wave are given, which enables us to see the influence of Coulomb-like friction term on the Riemann solutions for the Chaplygin gas equations clearly. In addition, the relations connected with the area transportation are derived which include mass and momentum transportation.

]]>The generation of magnetic field in shock surfaces separating regions of different electron density is a well known phenomenon. We study how this generation will affect the original structure of ionic flow. In a one-dimensional geometry, it turns out that the leading magnetosonic wavefront produced by the seed field may be compressional, ultimately evolving into a shock in a finite time. The time where this shock occurs depends on few parameters: sound velocity, Alfvén velocity and the variation of the magnetic field at the original surface at time zero. The alternative is that the magnetosonic wave may stabilize or damp out, which always happens if we start from a null magnetic field.

]]>The study of momentum and heat transfer has been carried out for the case of a viscous incompressible fluid between a rotating solid and a stationary permeable disk, whose depth is equal to that of the free fluid. Navier-Stokes equations govern the flow in the free fluid, while the flow in the porous region is governed by a combination of Brinkman and Darcy equations, respectively. Energy equations in the free fluid region and the porous region have been considered. A two step numerical process is employed; series expansions are first created to give analytical approximations of momentum and energy equations in MAPLE, while a Runge-Kutta algorithm bvp4c is then employed in MATLAB to numerically evaluate the velocity and temperature distributions in the flow fields. Velocity profiles, temperature profiles and relevant streamlines are sketched for various models involving variations in parameters such as Reynolds number, Brinkman number, and Prandtl number. It is observed that various parameters have differing effects on associated profiles which are subsequently discussed in the paper.

]]>This article is devoted to the study of the conservation and the dissipation properties of the mechanical energy of several time–integration methods dedicated to the elasto–dynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized–α schemes leads to energy blow-up, we study two schemes dedicated to the time–integration of nonsmooth systems with contact: the Moreau–Jean scheme and the nonsmooth generalized–α scheme. The energy conservation and dissipation properties of the Moreau–Jean is firstly shown. In a second step, the nonsmooth generalized–α scheme is studied by adapting the previous works of Krenk and Høgsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber–Hughes–Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact.

]]>We deal with special kinds of viscoelastic multi-mechanism models (MM models) in series connection. The MM models under consideration consist of a finite number of rheological Kelvin-Voigt elements and, possibly, a thermoelastic element. An important new item is the possible coupling between the KV elements leading to a new quality. After dealing in short with the modeling, we investigate two resulting three-dimensional mathematical problems in the isothermal case. In particular, we show existence and uniqueness of weak solutions for the corresponding initial-boundary value problems for displacements, stresses and partial strains.

]]>This paper deals with the attraction-repulsion chemotaxis system with logistic source

under homogeneous Neumann boundary conditions in a smooth bounded domain . Under a growth restriction on logistic source and suitable assumptions on the positive parameters χ, ξ, α, β, γ and δ, we show the existence of global bounded classical solutions. The global weak solution is also constructed if the logistic damping effect is rather mild. Furthermore, we obtain the asymptotic behavior of solutions for the logistic source .

]]>We consider a mathematical model which describes the equilibrium of a viscoelastic body in frictional contact with an obstacle. The contact is modeled with normal damped response and unilateral constraint for the velocity field, associated to a version of Coulomb's law of dry friction. We present a weak formulation of the problem, then we state and prove an existence and uniqueness result of the solution. The proof is based on arguments of history-dependent quasivariational inequalities. We also study the dependence of the solution with respect to the data and prove a convergence result. Further, we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate of the discretization. Finally, we provide numerical simulations which illustrate the behavior of the solution with respect to the frictional contact conditions and validate the theoretical convergence results.

]]>The present paper deals with the magnetoelectroelastic composites containing a doubly periodic array of multicoated fibers under anti-plane shear loads and in-plane electromagnetic loads. By introducing the generalized eigenstrain, the heterogeneous magnetoelectroelastic medium is equivalent to a homogeneous magnetoelectroelastic medium with the periodically distributed generalized eigenstrains. Then the homogeneous magnetoelectroelastic medium with the generalized eigenstrain is solved analytically under the applied load conditions, the generalized stresses and strains in the fibers, coatings and matrix are derived. Based on the average-field theory, the solutions of the generalized stresses and strains are applied to determine the anti-plane effective magnetoelectroelastic properties of the composites. Two-phase (fiber/matrix) and three-phase (fiber/coating/matrix) magnetoelectroelastic composites are examined, and the comparison between the obtained results and the existing results shows the accuracy of the proposed method. Several four-phase magnetoelectroelastic composites with epoxy matrix are studied, and the influences of the composites microstructures on the effective magnetoelectric coefficient are discussed.

]]>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.

]]>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.

]]>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.

]]>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 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, the authors 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.

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 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 authors 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 *Gc* and the micropolar characteristic length *l*. Based on FEM simulations they evaluate their influence on the behaviour of shell models in the geometrically nonlinear range of deformations.

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.

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 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.

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 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.

]]>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.

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