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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

]]>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.

]]>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.

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.

]]>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.

]]>The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles we have addressed the problem of the optimal design of sensors for purely conservative wave models. We analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. In this way, using fine asymptotic spectral properties of the Laplacian, we disclosed the links between the geometric properties of the domain where waves propagate and the existence of optimal locations for the sensors or, by the contrary, the emergence of relaxation phenomena. Here we show that spectral randomised observability is equivalent to the property of spectral controllability by means of a discrete set of lumped controls acting everywhere on the domain, and distributed according to the shape of the eigenfunctions. Our results on optimal observation then find natural equivalents on the problem of optimal spectral control. We also give an interpretation of these results in terms of a feedback stabilization property, ensuring the exponential decay of the energy of solutions as time tends to infinity.

]]>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.

The authors consider the problem of determining an optimal semi-active damping of vibrating systems. For this damping optimization they 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, they 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. Their 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.

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 .

]]>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 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.

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.

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.

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.

]]>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.

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