Authors:Sergey G. Fedosin Pages: 361 - 371 Abstract: The virial theorem is considered for a system of randomly moving particles that are tightly bound to each other by the gravitational and electromagnetic fields, acceleration field and pressure field. The kinetic energy of the particles of this system is estimated by three methods, and the ratio of the kinetic energy to the absolute value of the energy of forces, binding the particles, is determined, which is approximately equal to 0.6. For simple systems in classical mechanics, this ratio equals 0.5. The difference between these ratios arises by the consideration of the pressure field and acceleration field inside the bodies, which make additional contribution to the acceleration of the particles. It is found that the total time derivative of the system’s virial is not equal to zero, as is assumed in classical mechanics for systems with potential fields. This is due to the fact that although the partial time derivative of the virial for stationary systems tends to zero, but in real bodies the virial also depends on the coordinates and the convective derivative of the virial, as part of the total time derivative inside the body, is not equal to zero. It is shown that the convective derivative is also necessary for correct description of the equations of motion of particles. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0536-8 Issue No:Vol. 29, No. 2 (2017)

Authors:M. Vaz; M. R. Lange Pages: 373 - 383 Abstract: Modeling strategies aimed at thermo-mechanical coupled problems has been developed for a wide range of engineering applications. Staggered-type coupling procedures have been largely used in materials processing operations, especially in commercial codes, owing to their simplicity and flexibility. The present work shows that, in thermo-plastic problems, the classical implementation of the most common coupling procedure may present accuracy issues and time-stepping dependency. Numerical experiments indicate that an iterative coupling scheme constitutes a viable and simple approach to this class of problems. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0537-7 Issue No:Vol. 29, No. 2 (2017)

Authors:Charles Demay; Jean-Marc Hérard Pages: 385 - 410 Abstract: This work is dedicated to the modeling of gas–liquid flows in pipes. As a first step, a new two-layer model is proposed to deal with the stratified regime. The starting point is the isentropic Euler set of equations for each phase where the classical hydrostatic assumption is made for the liquid. The main difference with the models issued from the classical literature is that the liquid as well as the gas is assumed compressible. In that framework, an averaging process results in a five-equation system where the hydrostatic constraint has been used to define the interfacial pressure. Closure laws for the interfacial velocity and source terms such as mass and momentum transfer are provided following an entropy inequality. The resulting model is hyperbolic with non-conservative terms. Therefore, regarding the homogeneous part of the system, the definition and uniqueness of jump conditions is studied carefully and acquired. The nature of characteristic fields and the corresponding Riemann invariants are also detailed. Thus, one may build analytical solutions for the Riemann problem. In addition, positivity is obtained for heights and densities. The overall derivation deals with gas–liquid flows through rectangular channels, circular pipes with variable cross section and includes vapor–liquid flows. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0531-0 Issue No:Vol. 29, No. 2 (2017)

Authors:A. Sellitto; V. Tibullo; Y. Dong Pages: 411 - 428 Abstract: By means of a nonlinear generalization of the Maxwell–Cattaneo–Vernotte equation, on theoretical grounds we investigate how nonlinear effects may influence the propagation of heat waves in isotropic thin layers which are not laterally isolated from the external environment. A comparison with the approach of the Thermomass Theory is made as well. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0538-6 Issue No:Vol. 29, No. 2 (2017)

Authors:J. F. Ganghoffer; M. B. Boubaker Pages: 429 - 455 Abstract: We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0539-5 Issue No:Vol. 29, No. 2 (2017)

Authors:Evgeniy Yu. Vitokhin; Mikhail B. Babenkov Pages: 457 - 475 Abstract: We consider a series of problems with a short laser impact on a thin metal layer accounting various boundary conditions of the first and second kind. The behavior of the material is modeled by the hyperbolic thermoelasticity of Lord–Shulman type. We obtain analytical solutions of the problems in the semi-coupled formulation and numerical solutions in the coupled formulation. Numerical solutions are compared with the analytical ones. The analytical solutions of the semi-coupled problems and numerical solutions of the coupled problems show qualitative match. The solutions of hyperbolic thermoelasticity problems are compared with those obtained in the frame of the classical thermoelasticity. It was determined that the most prominent difference between the classical and hyperbolic solutions arises in the problem with fixed boundaries and constant temperature on them. The smallest differences were observed in the problem with unconstrained, thermally insulated edges. It was shown that a cooling zone is observed if the boundary conditions of the first kind are given for the temperature. Analytical expressions for the velocities of the quasiacoustic and quasithermal fronts as well as the critical value for the attenuation coefficient of the excitation impulse are verified numerically. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0540-z Issue No:Vol. 29, No. 2 (2017)

Authors:Mohammad Reza Salimpour; Rasool Kalbasi; Giulio Lorenzini Pages: 477 - 491 Abstract: This paper inquires the effectiveness of a PCM-based heat sink as a reliable solution to portable electronic devices. This sink is composed of a PCM with low thermal conductivity and fins to boost its conductivity. The optimization is subjected to fixed heat sink volume filled with PCM between vertical equidistant fins. New fins are installed in the unheated space existing in each enclosure which is not involved in thermal distribution from vertical fins to the PCM. Based on the same principle, new fins generations are augmented stepwise to the multi-scale structure. The steps of adding fins will continue up to the point that the objective function reaches its maximal value, i.e., maximizing the longest safe operation time without allowing the electronics to reach the critical temperature. The results indicate that in each length of the enclosure, the optimum volume fraction and the best fins distance values exist in which the heat sink performance becomes maximum, and adding more fins lowers the performance of the heat sink. Increasing the enclosure’s length by \(2^{n}\) does not change them. For an enclosure with constant length, the optimal number of steps for adding fins within the enclosure is a function of the fin thickness. The results indicate that increasing the thickness changes the optimal number of adding fins inside the enclosure (normally a decrease). As the fin thickness is lowered, there will be a higher effect by adding vertical fins in the enclosure. Numerical simulations cover the Rayleigh number range \(2\times 10^{5}\le \hbox {Ra}_{\mathrm{H}} \le 2.7\times 10^{8}\) , where H is the heat sink height. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0541-y Issue No:Vol. 29, No. 2 (2017)

Authors:Alireza Beheshti Pages: 493 - 507 Abstract: This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0542-x Issue No:Vol. 29, No. 2 (2017)

Authors:Martina Costa Reis; Yongqi Wang Pages: 509 - 534 Abstract: Based on the Eulerian spatial averaging theory and the Müller–Liu entropy principle, a two-fluid model for reactive dilute solid–liquid mixtures is presented. Initially, some averaging theorems and properties of average quantities are discussed and, then, averaged balance equations including interfacial source terms are postulated. Moreover, constitutive equations are proposed for a reactive dilute solid–liquid mixture, where the formation of the solid phase is due to a precipitation chemical reaction that involves ions dissolved in the liquid phase. To this end, principles of constitutive theory are used to propose linearized constitutive equations that account for diffusion, heat conduction, viscous and drag effects, and interfacial deformations. A particularity of the model is that the mass interfacial source term is regarded as an independent constitutive variable. The obtained results show that the inclusion of the mass interfacial source term into the set of independent constitutive variables permits to easily describe the phase changes associated with precipitation chemical reactions. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0546-6 Issue No:Vol. 29, No. 2 (2017)

Authors:Felix A. Reich; Wilhelm Rickert; Oliver Stahn; Wolfgang H. Müller Pages: 535 - 557 Abstract: Based on the principles of rational continuum mechanics and electrodynamics (see Truesdell and Toupin in Handbuch der Physik, Springer, Berlin, 1960 or Kovetz in Electromagnetic theory, Oxford University Press, Oxford, 2000), we present closed-form solutions for the mechanical displacements and stresses of two different magnets. Both magnets are initially of spherical shape. The first (hard) magnet is uniformly magnetized and deforms due to the field induced by the magnetization. In the second problem of a (soft) linear-magnetic sphere, the deformation is caused by an applied external field, giving rise to magnetization. Both problems can be used for modeling parts of general magnetization processes. We will address the similarities between both settings in context with the solutions for the stresses and displacements. In both problems, the volumetric Lorentz force density vanishes. However, a Lorentz surface traction is present. This traction is determined from the magnetic flux density. Since the obtained displacements and stresses are small in magnitude, we may use Hooke’s law with a small-strain approximation, resulting in the Lamé-Navier equations of linear elasticity theory. If gravity is neglected and azimuthal symmetry is assumed, these equations can be solved in terms of a series. This has been done by Hiramatsu and Oka (Int J Rock Mech Min Sci Geomech Abstr 3(2):89–90, 1966) before. We make use of their series solution for the displacements and the stresses and expand the Lorentz tractions of the analyzed problems suitably in order to find the expansion coefficients. The resulting algebraic system yields finite numbers of nonvanishing coefficients. Finally, the resulting stresses, displacements, principal strains and the Lorentz tractions are illustrated and discussed. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0544-8 Issue No:Vol. 29, No. 2 (2017)

Authors:U. S. Mahabaleshwar; K. R. Nagaraju; P. N. Vinay Kumar; Dumitru Baleanu; Giulio Lorenzini Pages: 559 - 567 Abstract: In this paper, we investigate the theoretical analysis for the unsteady magnetohydrodynamic laminar boundary layer flow due to impulsively stretching sheet. The third-order highly nonlinear partial differential equation modeling the unsteady boundary layer flow brought on by an impulsively stretching flat sheet was solved by applying Adomian decomposition method and Pade approximants. The exact analytical solution so obtained is in terms of rapidly converging power series and each of the variants are easily computable. Variations in parameters such as mass transfer (suction/injection) and Chandrasekhar number on the velocity are observed by plotting the graphs. This particular problem is technically sound and has got applications in expulsion process and related process in fluid dynamics problems. PubDate: 2017-03-01 DOI: 10.1007/s00161-016-0543-9 Issue No:Vol. 29, No. 2 (2017)

Authors:C. Röhrig; T. Scheffer; S. Diebels Abstract: Composite materials are of great interest for industrial applications because of their outstanding properties. Each composite material has its own characteristics due to the large number of possible combinations of matrix and filler. As a result of their compounding, composites usually show a complex material behavior. This work is focused on the experimental testing of a short fiber-reinforced thermoplastic composite at room temperature. The characteristic behavior of this material class is often based on a superposition of typical material effects. The predicted characteristic material properties such as elasto-plasticity, damage and anisotropy of the investigated material are obtained from results of cyclic uniaxial tensile tests at constant strain rate. Concerning the manufacturing process as well as industrial applications, the experimental investigations are extended to multiaxial loading situations. Therefore, the composite material is examined with a setup close to a deep-drawing process, the Nakajima test (Nakazima et al. in Study on the formability of steel sheets. Yawate Technical Report No. 264, pp 8517–8530, 1968). The evaluation of the experimental investigations is provided by an optical analysis system using a digital image correlation software. Finally, based on the results of the uniaxial tensile tests, a one-dimensional macroscopic model is introduced and first results of the simulation are provided. PubDate: 2017-02-27 DOI: 10.1007/s00161-017-0560-3

Authors:Hans-Dieter Alber Abstract: We study how the propagation speed of interfaces in the Allen–Cahn phase field model for phase transformations in solids consisting of the elasticity equations and the Allen–Cahn equation depends on two parameters of the model. The two parameters control the interface energy and the interface width, but change also the interface speed. To this end, we derive an asymptotic expansion of second order for the interface speed, called the kinetic relation, and prove that it is uniformly valid in both parameters. As a consequence, we show that the model error is proportional to the interface width divided by the interface energy. We conclude that simulations of interfaces with low interface energy based on this model require a very small interface width, implying a large numerical effort. Effective simulations thus need adaptive mesh refinement or other advanced techniques. PubDate: 2017-02-18 DOI: 10.1007/s00161-017-0558-x

Authors:Q. Yang; Q. C. Lv; Y. R. Liu Abstract: This paper is concerned with Hamilton’s principle for inelastic bodies with conservative external forces. Inelasticity is described by internal variable theory by Rice (J Mech Phys Solids 19:433–455, 1971), and the influence of strain change on the temperature field is ignored. Unlike Hamilton’s principle for elastic bodies which has an explicit Lagrangian, Hamilton’s principle for inelastic bodies generally has no an explicit Lagrangian. Based on the entropy inequality, a quasi Hamilton’s principle for inelastic bodies is established in the form of inequality and with an explicit Lagrangian, which is just the Lagrangian for elastic bodies by replacing the strain energy with free energy. The quasi Hamilton’s principle for inelastic bodies states that the actual motion is distinguished by making the action an maximum. The evolution equations of internal variables can not be recovered from the quasi Hamilton’s principle. PubDate: 2017-02-11 DOI: 10.1007/s00161-017-0557-y

Authors:K. S. Surana; A. D. Joy; J. N. Reddy Abstract: This paper presents a non-classical continuum theory in Lagrangian description for solids in which the conservation and the balance laws are derived by incorporating both the internal rotations arising from the Jacobian of deformation and the rotations of Cosserat theories at a material point. In particular, in this non-classical continuum theory, we have (i) the usual displacements \(( \pmb {\varvec{u }})\) and (ii) three internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\) about the axes of a triad whose axes are parallel to the x-frame arising from the Jacobian of deformation (which are completely defined by the skew-symmetric part of the Jacobian of deformation), and (iii) three additional rotations \(({}_e \pmb {\varvec{{\varTheta } }})\) about the axes of the same triad located at each material point as additional three degrees of freedom referred to as Cosserat rotations. This gives rise to \( \pmb {\varvec{u }}\) and \({}_e \pmb {\varvec{{\varTheta } }}\) as six degrees of freedom at a material point. The internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\) , often neglected in classical continuum mechanics, exist in all deforming solid continua as these are due to Jacobian of deformation. When the internal rotations \({}_i \pmb {\varvec{{\varTheta } }}\) are resisted by the deforming matter, conjugate moment tensor arises that together with \({}_i \pmb {\varvec{{\varTheta } }}\) may result in energy storage and/or dissipation, which must be accounted for in the conservation and the balance laws. The Cosserat rotations \({}_e \pmb {\varvec{{\varTheta } }}\) also result in conjugate moment tensor which, together with \({}_e \pmb {\varvec{{\varTheta } }}\) , may also result in energy storage and/or dissipation. The main focus of the paper is a consistent derivation of conservation and balance laws that incorporate aforementioned physics and associated constitutive theories for thermoelastic solids. The mathematical model derived here has closure, and the constitutive theories derived using two alternate approaches are in agreement with each other as well as with the condition resulting from the entropy inequality. Material coefficients introduced in the constitutive theories are clearly defined and discussed. PubDate: 2017-01-25 DOI: 10.1007/s00161-017-0554-1

Authors:Mykhailo Semkiv; Patrick D. Anderson; Markus Hütter Abstract: The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case. PubDate: 2017-01-20 DOI: 10.1007/s00161-016-0550-x

Authors:Tomáš Roubíček Abstract: A rather general model for fluid and heat transport in poro-elastic continua undergoing possibly also plastic-like deformation and damage is developed with the goal to cover various specific models of rock rheology used in geophysics of Earth’s crust. Nonconvex free energy at small elastic strains, gradient theories (in particular the concept of second-grade nonsimple continua), and Biot poro-elastic model are employed, together with possible large displacement due to large plastic-like strains evolving during long time periods. Also the additive splitting is justified in stratified situations which are of interest in modelling of lithospheric crust faults. Thermodynamically based formulation includes entropy balance (in particular the Clausius–Duhem inequality) and an explicit global energy balance. It is further outlined that the energy balance can be used to ensure, under suitable data qualification, existence of a weak solution and stability and convergence of suitable approximation schemes at least in some particular situations. PubDate: 2017-01-13 DOI: 10.1007/s00161-016-0547-5

Authors:A. Paglietti Abstract: In laminar flow, viscous fluids must exert appropriate elastic shear stresses normal to the flow direction. This is a direct consequence of the balance of angular momentum. There is a limit, however, to the maximum elastic shear stress that a fluid can exert. This is the ultimate shear stress, \(\tau _\mathrm{y}\) , of the fluid. If this limit is exceeded, laminar flow becomes dynamically incompatible. The ultimate shear stress of a fluid can be determined from experiments on plane Couette flow. For water at \(20\,^{\circ }\hbox {C}\) , the data available in the literature indicate a value of \(\tau _\mathrm{y}\) of about \(14.4\times 10^{-3}\, \hbox {Pa}\) . This study applies this value to determine the Reynolds numbers at which flowing water reaches its ultimate shear stress in the case of Taylor–Couette flow and circular pipe flow. The Reynolds numbers thus obtained turn out to be reasonably close to those corresponding to the onset of turbulence in the considered flows. This suggests a connection between the limit to laminar flow, on the one hand, and the occurrence of turbulence, on the other. PubDate: 2017-01-10 DOI: 10.1007/s00161-016-0549-3

Authors:P. Ván Abstract: Single-component nonrelativistic dissipative fluids are treated independently of reference frames and flow-frames. First the basic fields and their balances are derived, then the related thermodynamic relations and the entropy production are calculated and the linear constitutive relations are given. The usual basic fields of mass, momentum, energy and their current densities, the heat flux, pressure tensor and diffusion flux are the time- and spacelike components of the third-order mass–momentum–energy density-flux four-tensor. The corresponding Galilean transformation rules of the physical quantities are derived. It is proved that the non-equilibrium thermodynamic frame theory, including the thermostatic Gibbs relation and extensivity condition and also the entropy production, is independent of the reference frame and also the flow-frame of the fluid. The continuity-Fourier–Navier–Stokes equations are obtained almost in the traditional form if the flow of the fluid is fixed to the temperature. This choice of the flow-frame is the thermo-flow. A simple consequence of the theory is that the relation between the total, kinetic and internal energies is a Galilean transformation rule. PubDate: 2017-01-09 DOI: 10.1007/s00161-016-0545-7

Authors:Teodor M. Atanacković; Marko Janev; Sanja Konjik; Stevan Pilipović Abstract: We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed. PubDate: 2017-01-06 DOI: 10.1007/s00161-016-0548-4