ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Cover image for Vol. 95 Issue 11

Early View (Online Version of Record published before inclusion in an issue)

Editor: Holm Altenbach

Impact Factor: 1.162

ISI Journal Citation Reports © Ranking: 2014: 68/257 (Mathematics Applied); 77/137 (Mechanics)

Online ISSN: 1521-4001

Associated Title(s): GAMM-Mitteilungen, Mathematische Nachrichten, Mathematical Logic Quarterly, PAMM


  1. 1 - 76
  1. Original Articles

  2. Original Manuscripts

    1. On the time decay of solutions for non-simple elasticity with voids

      Zhuangyi Liu, Antonio Magaña and Ramón Quintanilla

      Article first published online: 30 OCT 2015 | DOI: 10.1002/zamm.201400290

  3. Original Articles

    1. Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane

      Maya Bassam, Denis Mercier, Serge Nicaise and Ali Wehbe

      Article first published online: 7 OCT 2015 | DOI: 10.1002/zamm.201500172

      In , Belkacem and Kasimov studied the stability of an one-dimensional Timoshenko system in inline image 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 inline image and its analogous system in inline image. 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 inline image. 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.

    2. Low velocity impact analysis of composite laminated beams subjected to multiple impacts in thermal field

      Mostafa Sabzikar Boroujerdy and Yasser Kiani

      Article first published online: 30 SEP 2015 | DOI: 10.1002/zamm.201500132

      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.

  4. Original Manuscripts

    1. Debonding fracture of bonded bimaterial semi-strips subjected to concentrated forces and couples

      Norio Hasebe and Seiji Kato

      Article first published online: 16 SEP 2015 | DOI: 10.1002/zamm.201500125

      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.

  5. Original Articles

    1. Torsional frequency analyses of microtubules with end attachments

      Khameel B. Mustapha and Basil T. Wong

      Article first published online: 10 SEP 2015 | DOI: 10.1002/zamm.201500007

    2. Comparison of continuous and discontinuous Galerkin approaches for variable-viscosity Stokes flow

      Ragnar S. Lehmann, Mária Lukáčová-Medvid'ová, Boris J. P. Kaus and Anton A. Popov

      Article first published online: 7 SEP 2015 | DOI: 10.1002/zamm.201400274

    3. Magnetohydrodynamic unsteady separated stagnation-point flow of a viscous fluid over a moving plate

      S. Dholey

      Article first published online: 11 AUG 2015 | DOI: 10.1002/zamm.201400218

      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 inline image 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 inline image, 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.

    4. Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

      N. P. Lazarev

      Article first published online: 6 AUG 2015 | DOI: 10.1002/zamm.201500128

      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.

    5. Modeling the effects of material properties on the pull-in instability of nonlocal functionally graded nano-actuators

      Hamid M. Sedighi, Farhang Daneshmand and Mohamadreza Abadyan

      Article first published online: 20 JUL 2015 | DOI: 10.1002/zamm.201400160

      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.

    6. A generalization of Noether's theorem for a non-material volume

      Leonardo Casetta, Hans Irschik and Celso Pupo Pesce

      Article first published online: 20 JUL 2015 | DOI: 10.1002/zamm.201400196

      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.

    7. The Riemann problem for the Chaplygin gas equations with a source term

      Chun Shen

      Article first published online: 20 JUL 2015 | DOI: 10.1002/zamm.201500015

      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.

    8. The motion induced by a radially stretching membrane in a rotating fluid system

      Patrick D. Weidman

      Article first published online: 16 JUL 2015 | DOI: 10.1002/zamm.201500094

      The flow induced above an impermeable sheet undergoing linear radial stretching in a system rotating at angular velocity Ω is investigated. The problem is governed by the single parameter inline image, where a is the stretching rate of the membrane. A similarity reduction of the Navier-Stokes equations leads to a pair of nonlinearly-coupled ordinary differential equations which are numerically solved over the range inline image. Coriolis forces acting on the radially stretching flow in the vicinity of the wall induces azimuthal flow in the boundary layer. A large-σ analysis of the system leads to the Ekman equations describing the balance of Coriolis and viscous forces for which an exact solution is presented. Numerical results for the radial and azimuthal shear stresses, along with the magnitude of the velocity induced into the boundary layer, are shown to be in good agreement with the large-σ asymptotic results.

    9. Generation of shocks by the Biermann battery

      Manuel Núñez

      Article first published online: 16 JUL 2015 | DOI: 10.1002/zamm.201500034

      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.

    10. The study of fluid flow and heat transfer of a viscous incompressible fluid between a rotating solid disk and a stationary permeable disk using the Brinkman-Darcy model

      Dayle C. Jogie and Balswaroop Bhatt

      Article first published online: 2 JUL 2015 | DOI: 10.1002/zamm.201400089

      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.

    11. Energy conservation and dissipation properties of time-integration methods for nonsmooth elastodynamics with contact

      Vincent Acary

      Article first published online: 1 JUL 2015 | DOI: 10.1002/zamm.201400231

      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.

    12. Deformations near an elliptical hole with surface effect in a laminated anisotropic thin plate

      Xu Wang and Peter Schiavone

      Article first published online: 25 JUN 2015 | DOI: 10.1002/zamm.201400271

      This work is concerned with the coupled stretching and bending deformation around an elliptical hole with surface energy in a laminated and inhomogeneous anisotropic elastic thin plate within the context of the Kirchhoff theory. A closed-form full-field solution is derived by using the octet formalism recently developed by Cheng and Reddy (2002, 2003, 2004, 2005) and by incorporating a simplified version of the surface elasticity model. In particular, explicit real-form expressions of the hoop membrane stress resultant, hoop bending moment, in-plane displacements and slopes on the mid-plane along the edge of the elliptical hole are obtained.

    13. On mathematical problems for viscoelastic multi-mechanism models in the isothermal case

      Nils Hendrik Kröger, Michael Wolff and Michael Böhm

      Article first published online: 12 JUN 2015 | DOI: 10.1002/zamm.201400171

      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.

    14. An attraction-repulsion chemotaxis system with logistic source

      Qingshan Zhang and Yuxiang Li

      Article first published online: 8 JUN 2015 | DOI: 10.1002/zamm.201400311

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

      • display math

      under homogeneous Neumann boundary conditions in a smooth bounded domain inline image inline image. 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 inline image.

    15. Analysis of a contact problem with normal damped response and unilateral constraint

      Mikaël Barboteu, David Danan and Mircea Sofonea

      Article first published online: 3 JUN 2015 | DOI: 10.1002/zamm.201400304

      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.

    16. An analytical method for predicting the anti-plane effective magnetoelectroelastic coefficients of composites containing doubly periodic multicoated fibers

      Y. L. Xu and J. H. Xiao

      Article first published online: 11 MAY 2015 | DOI: 10.1002/zamm.201500003

      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.

    17. Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm

      Peter Benner, Patrick Kürschner, Zoran Tomljanović and Ninoslav Truhar

      Article first published online: 3 MAY 2015 | DOI: 10.1002/zamm.201400158

      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.

    18. A positive scheme for diffusion problems on deformed meshes

      Xavier Blanc and Emmanuel Labourasse

      Article first published online: 3 MAY 2015 | DOI: 10.1002/zamm.201400234

      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.

    19. Blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equations

      Qian Zhang

      Article first published online: 29 APR 2015 | DOI: 10.1002/zamm.201500040

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

    20. Remark on the pointwise stabilization of an elastic string equation

      Fathi Hassine

      Article first published online: 29 APR 2015 | DOI: 10.1002/zamm.201400260

      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.

    21. On a free boundary problem arising in snow avalanche dynamics

      Benedetta Calusi, Lorenzo Fusi and Angiolo Farina

      Article first published online: 29 APR 2015 | DOI: 10.1002/zamm.201400250

      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.

    22. A new application of M- and L-integrals for the numerical loading analysis of two interacting cracks

      Paul O. Judt and Andreas Ricoeur

      Article first published online: 21 APR 2015 | DOI: 10.1002/zamm.201500012

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

    23. When is natural convection completely passive?

      Anthony Kay

      Article first published online: 16 APR 2015 | DOI: 10.1002/zamm.201400177

      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.

    24. Elastodynamics of strongly heterogeneous periodic plates using Reissner-Mindlin and Kirchhoff-Love models

      Eduard Rohan and Bernadette Miara

      Article first published online: 16 APR 2015 | DOI: 10.1002/zamm.201400145

      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.

    25. The motion of a two-body limbless locomotor along a straight line in a resistive medium

      N. Bolotnik, M. Pivovarov, I. Zeidis and K. Zimmermann

      Article first published online: 14 APR 2015 | DOI: 10.1002/zamm.201400302

      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.

    26. A penny-shaped magnetically dielectric crack in a magnetoelectroelastic cylinder under magnetoelectromechanical loads

      L. L. Liu, W. J. Feng and P. Ma

      Article first published online: 13 APR 2015 | DOI: 10.1002/zamm.201500049

      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.

    27. On PDE analysis of flows of quasi-incompressible fluids

      Eduard Feireisl, Yong Lu and Josef Málek

      Article first published online: 11 APR 2015 | DOI: 10.1002/zamm.201400229

      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.

    28. Analytical and numerical results for a dynamic contact problem with two stops in thermoelastic diffusion theory

      Moncef Aouadi and Maria I. M. Copetti

      Article first published online: 7 APR 2015 | DOI: 10.1002/zamm.201400285

      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.

    29. On an inverse boundary problem for the heat equation when small heat conductivity defects are present in a material

      M. Bouraoui, L. El Asmi and A. Khelifi

      Article first published online: 2 APR 2015 | DOI: 10.1002/zamm.201300265

      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.

    30. Blow up threshold for the Gross-Pitaevskii system with trapped dipolar quantum gases

      Baiyu Liu, Li Ma and Jing Wang

      Article first published online: 2 APR 2015 | DOI: 10.1002/zamm.201400189

      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.

    31. Finite element methods for the incompressible Stokes equations with variable viscosity

      Volker John, Kristine Kaiser and Julia Novo

      Article first published online: 1 APR 2015 | DOI: 10.1002/zamm.201400291

      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.

    32. An invariant-free formulation of neo-Hookean hyperelasticity

      David C. Kellermann and Mario M. Attard

      Article first published online: 1 APR 2015 | DOI: 10.1002/zamm.201400210

      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.

    33. Simplified homogenization technique for engineering applications

      Dennis Bäcker

      Article first published online: 1 APR 2015 | DOI: 10.1002/zamm.201400188

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

    34. On the 3D Rayleigh wave field on an elastic half-space subject to tangential surface loads

      Nihal Ege, Barış Erbaş and Danila A. Prikazchikov

      Article first published online: 1 APR 2015 | DOI: 10.1002/zamm.201400211

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

    35. The influence of hydrostatic stress on the frequency equation of flexural waves in a magnetoelastic transversly isotropic circular cylinder

      Abo-el-nour N. Abd-alla, Aishah Raizah and Luca Placidi

      Article first published online: 21 MAR 2015 | DOI: 10.1002/zamm.201400059

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

    36. The effective solution of two-dimensional integro-differential equations and their applications in the theory of viscoelasticity

      Nugzar Shavlakadze

      Article first published online: 21 MAR 2015 | DOI: 10.1002/zamm.201400091

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

    37. Geometrically nonlinear FEM analysis of 6-parameter resultant shell theory based on 2-D Cosserat constitutive model

      S. Burzyński, J. Chróścielewski and W. Witkowski

      Article first published online: 12 MAR 2015 | DOI: 10.1002/zamm.201400092

      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 inline image 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.

    38. On perturbation method in mechanical, thermal and thermo-mechanical loadings of plates: cylindrical bending of FG plates

      F. Fallah, A. Nosier, M. Sharifi and F. Ghezelbash

      Article first published online: 4 MAR 2015 | DOI: 10.1002/zamm.201400136

      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.

    39. Non-periodic homogenization of infinitesimal strain plasticity equations

      Martin Heida and Ben Schweizer

      Article first published online: 4 MAR 2015 | DOI: 10.1002/zamm.201400112

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

    40. A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach

      Massimo Lanza de Cristoforis and Paolo Musolino

      Article first published online: 4 MAR 2015 | DOI: 10.1002/zamm.201400035

      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 inline image a suitably normalized solution. Then we are interested to analyze the behavior of inline image when ε is close to the degenerate value inline image, where the holes collapse to points. In particular we prove that if inline image, then inline image can be expanded into a convergent series expansion of powers of ε and that if inline image then inline image can be expanded into a convergent double series expansion of powers of ε and inline image. Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.

    41. Numerical solution of stochastic partial differential equations using a collocation method

      Minoo Kamrani

      Article first published online: 23 FEB 2015 | DOI: 10.1002/zamm.201400080

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

    42. On the inverse problem of the two-velocity tree-like graph

      Sergei Avdonin, Choque Rivero Abdon, Günter Leugering and Victor Mikhaylov

      Article first published online: 23 FEB 2015 | DOI: 10.1002/zamm.201400126

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

    43. A representation theorem for the circular inclusion problem

      Nkem Ogbonna

      Article first published online: 23 FEB 2015 | DOI: 10.1002/zamm.201300147

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

    44. A strain-softening bar revisited

      Serge N. Gavrilov and Ekaterina V. Shishkina

      Article first published online: 23 FEB 2015 | DOI: 10.1002/zamm.201400155

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

    45. An improved displacement boundary condition of piezoelectric cantilever beams

      Lian-Zhi Yang, Liangliang Zhang, Lianying Yu, Minzhong Wang and Yang Gao

      Article first published online: 23 FEB 2015 | DOI: 10.1002/zamm.201400169

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

    46. Consideration of spatial variation of the friction coefficient in contact mechanics analysis of laterally graded materials

      Serkan Dag

      Article first published online: 3 FEB 2015 | DOI: 10.1002/zamm.201400116

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

    47. Solution of bonded bimaterial problem of two interfaces subjected to concentrated forces and couples

      Norio Hasebe and Seiji Kato

      Article first published online: 3 FEB 2015 | DOI: 10.1002/zamm.201400179

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

    48. Well-posedness of a thermo-elasto-plastic problem with phase transitions in TRIP steels under mixed boundary conditions

      Sören Boettcher, Michael Böhm and Michael Wolff

      Article first published online: 3 FEB 2015 | DOI: 10.1002/zamm.201300287

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

  6. Articles

    1. Modelling and simulation of acrylic bone cement injection and curing within the framework of vertebroplasty

      Ralf Landgraf, Jörn Ihlemann, Sebastian Kolmeder, Alexander Lion, Helena Lebsack and Cornelia Kober

      Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400064

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

    2. Existence and uniqueness for frictional incremental and rate problems – sharp critical bounds

      L.-E. Andersson, A. Pinto da Costa and M. A. Agwa

      Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400143

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

  7. Original Manuscripts

    1. On a non-stationary load on the surface of a semiplane with mixed boundary conditions

      Veniamin D. Kubenko

      Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400202

      An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the object at an arbitrary point of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.

    2. Classical solutions for a modified Hele-Shaw model with elasticity

      Helmut Abels and Stefan Schaubeck

      Article first published online: 22 JAN 2015 | DOI: 10.1002/zamm.201400099

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

  8. Articles

    1. Analytical elastic-plastic analyses of a spherical shell subjected to hydrostatic tension based on a strain gradient model for plastic metals

      Koffi Enakoutsa

      Article first published online: 21 JAN 2015 | DOI: 10.1002/zamm.201400131

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

  9. Original Articles

    1. Dislocation-based fracture analysis of functionally graded magnetoelectroelastic solids

      S. Mahmoud Mousavi

      Article first published online: 21 JAN 2015 | DOI: 10.1002/zamm.201400197

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

  10. Original Papers

    1. Effect of scale parameter on the deflection of a nonlocal beam and application to energy release rate of a crack

      X.-L. Peng, X.-F. Li, G.-J. Tang and Z.-B. Shen

      Article first published online: 20 JAN 2015 | DOI: 10.1002/zamm.201400132

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

    2. A conservative finite difference scheme for the Falk model system of shape memory alloys

      Shuji Yoshikawa

      Article first published online: 20 JAN 2015 | DOI: 10.1002/zamm.201300177

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

    3. A mathematical account of draught-induced disturbances in the fibre spinning process

      L. J. Crane and A. G. McVeigh

      Article first published online: 20 JAN 2015 | DOI: 10.1002/zamm.201400029

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


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