Abstract: On the example of two-phase continua experiencing stress-induced solid–fluid phase transitions, we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type, and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization. PubDate: 2014-10-12

Abstract: Abstract
Exact solutions of elastic Kirchhoff plates are available only for special geometries, loadings and kinematic boundary constraints. An effective solution procedure, based on an analogy between functionally graded orthotropic Saint-Venant beams under torsion and inhomogeneous isotropic Kirchhoff plates, with no kinematic boundary constraints, is proposed. The result extends the one contributed in Barretta (Acta Mech 224(12):2955–2964, 2013) for the special case of homogeneous Saint-Venant beams under torsion. Closed-form solutions for displacement, bending–twisting moment and curvature fields of an elliptic plate, corresponding to a functionally graded orthotropic beam, are evaluated. A new benchmark for computational mechanics is thus provided. PubDate: 2014-10-04

Abstract: Abstract
Bifurcation analysis of a structure constituted by two towers, linked by a viscous device at the tip and subjected to turbulent wind, is carried out. The towers have geometrical and mechanical parameters so that the steady part of the wind, whose contribution is evaluated in the framework of the steady theory, induces a 1:1 resonant double-Hopf bifurcation. The turbulent part of the wind, assumed as composed by two frequencies that are equal and double to the main frequency of the unlinked towers, respectively, induces parametric and external harmonic forces. These forces interact with the self-excitation due to the steady part of the wind, bringing imperfection in the bifurcation scenario. Transitions from resonant to non-resonant cases are analyzed in terms of behavior charts, and post-critical dynamics is studied in the space of bifurcation parameters. PubDate: 2014-09-26

Abstract: A locally phase-shifted Sine–Gordon model well accounts for the phenomenology of unconventional Josephson junctions. The phase dynamics shows resonant modes similar to Fiske modes that appear both in the presence and in the absence of the external magnetic field in standard junctions. In the latter case, they are also in competition with zero field propagation of Sine–Gordon solitons, i.e., fluxons, which give rise to the so-called zero field steps in the current–voltage (I–V) of the junction. We numerically study the I–V characteristics and the resonances magnetic field patterns for some different faceting configurations, in various dissipative regimes, as a function of temperature. The simulated dynamics of the phase is analyzed for lower-order resonances. We give evidence of a nontrivial dynamics due to the interaction of propagating fluxons with localized semifluxons. Numerical results are compared with experimental outcomes obtained on high-quality high-Tc grain boundary YBCO junctions. PubDate: 2014-09-07

Abstract: We design a new mesoscopic thin-film model for shape-memory materials which takes into account thermomechanical effects. Starting from a microscopic thermodynamical bulk model, we guide the reader through a suitable dimension reduction procedure followed by a scale transition valid for specimen large in area up to a limiting model which describes microstructure by means of parametrized measures. All our models obey the second law of thermodynamics and possess suitable weak solutions. This is shown for the resulting thin-film models by making the procedure described above mathematically rigorous. The main emphasis is, thus, put on modeling and mathematical treatment of joint interactions of mechanical and thermal effects accompanying phase transitions and on reduction in specimen dimensions and transition of material scales. PubDate: 2014-09-01

Abstract: Heat transfer in solids is modeled in the framework of kinetic theory of the phonon gas. The microscopic description of the phonon gas relies on the phonon Boltzmann equation and the Callaway model for phonon–phonon interaction. A simple model for phonon interaction with crystal boundaries, similar to the Maxwell boundary conditions in classical kinetic theory, is proposed. Macroscopic transport equation for an arbitrary set of moments is developed and closed by means of Grad’s moment method. Boundary conditions for the macroscopic equations are derived from the microscopic model and the Grad closure. As example, sets with 4, 9, 16, and 25 moments are considered and solved analytically for one-dimensional heat transfer and Poiseuille flow of phonons. The results show the influence of Knudsen number on phonon drag at solid boundaries. The appearance of Knudsen layers reduces the net heat conductivity of solids in rarefied phonon regimes. PubDate: 2014-09-01

Abstract: We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models. PubDate: 2014-09-01

Abstract: By Landau’s approach developed for description of superfluidity of 2He, we derive a mathematical model for a poroelastic medium saturated with a two-phase capillary fluid. The model describes a three-velocity continuum with conservation laws which obey the basic principles of thermodynamics and which are consistent with the Galilean transformations. In contrast to Biot’ linear theory, the equations derived allow for finite deformations. As the acoustic analysis reveals, there is one more longitudinal wave in comparison with the poroelastic medium saturated with a one-phase fluid. We prove that such a result is due to surface tension. PubDate: 2014-09-01

Abstract: The relative merit of lower-order theories, which have been deduced from the three-dimensional theories of continua, is evaluated with respect to the quantified and un-quantified errors in mathematically modeling the physical response of structural elements. Then, the one-dimensional theories are derived with high accuracy, internal consistency and flexibility from the three-dimensional theory of elasticity in order to govern the nonlinear and incremental motions and stability of a functionally graded rod. First, a kinematic-based method of separation of variables is introduced as a method of reduction, which may lead to the lower-order theories with the same order of errors of the three-dimensional theories, and the nonlinear theories of the rod are derived under Leibnitz’s postulate of structural elements by use of Hamilton’s principle. A theorem of uniqueness is proved in solutions of the linear equations of the rod by means of the logarithmic convexity argument. Next, the kinematic basis is expressed by the power series expansion in the cross-sectional coordinates using Weierstrass’s theorem. Mindlin’s method is used so as to derive the equations in an invariant and fully variational form for the small motions superposed on a static finite deformation, the stability analysis and the high-frequency vibrations of the rod. Moreover, the free vibrations of the rod are considered, the basic properties of eigenvalues are examined, and Rayleigh’s quotient is obtained. The invariant equations of the rod, which are expressible in any system of orthogonal coordinates, may provide simultaneous approximations on all the field variables in a direct method of solutions. The equations are indicated to contain some of earlier equations of rods, as special cases, and also, the numerical elasticity solution of a sample application is presented. PubDate: 2014-09-01

Abstract: Abstract
The response of a sandy seabed under wave loading is investigated on the basis of numerical modeling using a multi-scale approach. To that aim, the discrete element method is coupled to a finite volume method specially enhanced to describe compressible fluid flow. Both solid and fluid phase mechanics are upscaled from considerations established at the pore level. Model’s predictions are validated against poroelasticity theory and discussed in comparison with experiments where a sediment analog is subjected to wave action in a flume. Special emphasis is put on the mechanisms leading the seabed to liquefy under wave-induced pressure variation on its surface. Liquefaction is observed in both dilative and compactive regimes. It is shown that the instability can be triggered for a well-identified range of hydraulic conditions. Particularly, the results confirm that the gas content, together with the permeability of the medium are key parameters affecting the transmission of pressure inside the soil. PubDate: 2014-08-29

Abstract: Dynamics of a Timoshenko beam under an influence of mechanical and thermal loadings is analysed in this paper. Nonlinear geometrical terms and a nonuniform heat distribution are taken into account in the considered model. The mathematical model is represented by a set of partial differential equations (PDEs) which takes into account thermal and mechanical loadings. The problem is simplified to two PDEs and then reduced to ordinary differential equations (ODEs) by means of the Galerkin method taking into account three modes of a linear Timoshenko beam. Correctness of the analytical model is verified by a finite element method. Then, the nonlinear model is studied numerically by a continuation method or by a direct numerical integration of ODEs. An effect of the temperature distribution on the resonance near the first natural frequency and on stability of the solutions is presented. The increase of mechanical loading results in hardening of the resonance curve. Thermal loading may stabilise the beam dynamics when the temperature is decreased. The elevated temperature may transit dynamics from regular to chaotic oscillations. PubDate: 2014-08-27

Abstract: In case of the secondary bone fracture healing, four characteristic steps are often distinguished. The
first stage, hematoma and clot formation, which is an object of our study, is important because it prepares
the environment for the following stages. In this work, a new mathematical model describing basic effects
present short after the injury is proposed. The main idea is based on the assumption that blood leaking from
the ruptured blood vessels propagates into a poroelastic saturated tissue close to the fracture and mixes with
the interstitial liquid present in pores. After certain time period from the first contact with surrounding tissue,
the solidification of blood in the fluid mixture starts. This results in clot formation. By assuming the time
necessary to initiate solidification and critical saturation of blood in the mixture, the shape and the structure of
blood clot could be determined. In numerical example, proposed mathematical formulas were used to study
the size of the gap between fractured parts and its effect in blood clot formation. PubDate: 2014-08-23

Abstract: Abstract
The existence theorem of Fichera for the minimum problem of semicoercive quadratic functions in a Hilbert space is extended to a more general class of convex and lower semicontinuous functions. For unbounded domains, the behavior at infinity is controlled by a lemma which states that every unbounded sequence with bounded energy has a subsequence whose directions converge to a direction of recession of the function. Thanks to this result, semicoerciveness plus the assumption that the effective domain is boundedly generated, that is, admits a Motzkin decomposition, become sufficient conditions for existence. In particular, for functions with a smooth quadratic part, a generalization of the existence condition given by Fichera’s theorem is proved. PubDate: 2014-08-21

Abstract: During stress-free thermal analysis with differential scanning calorimetry (DSC), nickel titanium (NiTi) shape memory alloys show a thermal hysteresis which is affected by cooling/heating rates. Moreover, the Ni content of near equiatomic alloys governs the phase transition temperatures. This contribution aims at establishing a constitutive equation which can account for these effects, building on earlier work by Müller, Achenbach and Seelecke (MAS). To be specific, we discuss our new method with a focus on NiTi alloys. As in the original MAS model, our approach is rooted in a non-convex free energy representation and rate equations are utilized to incorporate history dependence during non-equilibrium processes. The relaxation times of these rate equations are determined by characteristic transformation probabilities which in turn are governed by the free energy landscape of our system. We show how the model can be parameterized to rationalize experimental DSC data observed for NiTi samples of variable composition and measured at variable cooling/heating rates. The good agreement between model predictions and experimental results suggests that thermal hystereses are not only related to interfacial strain energy effects but also affected by the transient character of the transformation process incorporating specific thermal relaxation times. Our analysis shows that we observe strong hysteretic effects when the cooling/heating rates exceed these characteristic relaxation rates. PubDate: 2014-08-15

Abstract: This paper gives an overview of wind-induced galloping phenomena, describing its manifold features and the many advances that have taken place in this field. Starting from a quasi-steady model of aeroelastic forces exerted by the wind on a rigid cylinder with three degree-of-freedom, two translations and a rotation in the plane of the model cross section, the fluid–structure interaction forces are described in simple terms, yet suitable with complexity of mechanical systems, both in the linear and in the nonlinear field, thus allowing investigation of a wide range of structural typologies and their dynamic behavior. The paper is driven by some key concerns. A great effort is made in underlying strengths and weaknesses of the classic quasi-steady theory as well as of the simplistic assumptions that are introduced in order to investigate such complex phenomena through simple engineering models. A second aspect, which is crucial to the authors’ approach, is to take into account and harmonize the engineering, physical and mathematical perspectives in an interdisciplinary way—something which does not happen often. The authors underline that the quasi-steady approach is an irreplaceable tool, tough approximate and simple, for performing engineering analyses; at the same time, the study of this phenomenon gives origin to numerous problems that make the application of high-level mathematical solutions particularly attractive. Finally, the paper discusses a wide range of features of the galloping theory and its practical use which deserve further attention and refinements, pointing to the great potential represented by new fields of application and advanced analysis tools. PubDate: 2014-08-14

Abstract: Abstract
This paper examines the class of problems related to the interaction between a finitely deformed incompressible elastic halfspace and contacting elements that include smooth, flat rigid indenters with elliptical and circular shapes and a thick plate of infinite extent. The contact between the finitely deformed elastic halfspace and the contacting elements is assumed to be bilateral. The interaction between both the rigid circular indenter and the finitely deformed halfspace is induced by a Mindlin force that acts at the interior of the halfspace regions and by exterior loads. Similar considerations apply for the contact between the flexible plate of infinite extent and the finitely deformed elastic halfspace. The theory of small deformations superposed on large deformations proposed by Green et al. (Proc R Soc Ser A 211:128–155, 1952) is used as the basis for the formulation of the problem, and results of potential theory and integral transform techniques are used to develop the analytical results. In particular, explicit results are presented for the displacement of the rigid elliptical indenter and the maximum deflection of the flexible plate induced by the Mindlin forces, when the finitely deformed halfspace region has a strain energy function of the Mooney–Rivlin form. PubDate: 2014-08-10

Abstract: Abstract
In this letter, we address the problem of the integrability of a continuum model for granular media at equilibrium. By the means of a formal integrability analysis, we show that the equilibrium limit of such models can be cast into a gradient equation with zero right-hand side. In turn, this implies that the model of interest is inherently Frobenius integrable, in the absence of additional compatibility conditions. Moreover, the quantity inside the gradient is identified with the granular material’s Gibbs free energy. Consequently, the integrability for the model at hand is equivalent to setting the Gibbs free energy of the granular material constant throughout the domain. In other words, integrability is equivalent to the definition of equilibrium employed in statistical physics. PubDate: 2014-07-26

Abstract: The previous paper by the authors (Siriwat and Meleshko in Contin Mech Thermodyn 24:115–148, 2012) was devoted to group analysis of one-dimensional nonisentropic equations of fluids with internal inertia. A direct approach was employed for finding the admitted Lie group. This approach allowed to perform a partial group classification of the considered equations with respect to a potential function. The present paper completes this group classification by an efficient algebraic method. PubDate: 2014-07-25