Authors:Leif Kari Pages: 1027 - 1046 Abstract: Abstract The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William–Landel–Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0569-7 Issue No:Vol. 29, No. 5 (2017)

Authors:Leif Kari Pages: 1047 - 1059 Abstract: Abstract The dynamic stiffness of a chemically and physically ageing rubber vibration isolator in the audible frequency range is modelled as a function of ageing temperature, ageing time, actual temperature, time, frequency and isolator dimension. In particular, the dynamic stiffness for an axially symmetric, homogeneously aged rubber vibration isolator is derived by waveguides where the eigenmodes given by the dispersion relation for an infinite cylinder satisfying traction free radial surface boundary condition are matched to satisfy the displacement boundary conditions at the lateral surface ends of the finite rubber cylinder. The constitutive equations are derived in a companion paper (Part 1). The dynamic stiffness is calculated over the whole audible frequency range 20–20,000 Hz at several physical ageing times for a temperature history starting at thermodynamic equilibrium at \(+25\,^{\circ }\hbox {C}\) and exposed by a sudden temperature step down to \(-60\,^{\circ }\hbox {C}\) and at several chemical ageing times at temperature \(+25\,^{\circ }\hbox {C}\) with simultaneous molecular network scission and reformation. The dynamic stiffness results are displaying a strong frequency dependence at a short physical ageing time, showing stiffness magnitude peaks and troughs, and a strong physical ageing time dependence, showing a large stiffness magnitude increase with the increased physical ageing time, while the peaks and troughs are smoothed out. Likewise, stiffness magnitude peaks and troughs are frequency-shifted with increased chemical ageing time. The developed model is possible to apply for dynamic stiffness prediction of rubber vibration isolator over a broad audible frequency range under realistic environmental condition of chemical ageing, mainly attributed to oxygen exposure from outside and of physical ageing, primarily perceived at low-temperature steps. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0573-y Issue No:Vol. 29, No. 5 (2017)

Authors:Alexander Lion; Christoph Mittermeier; Michael Johlitz Pages: 1061 - 1079 Abstract: Abstract A novel approach to represent the glass transition is proposed. It is based on a physically motivated extension of the linear viscoelastic Poynting–Thomson model. In addition to a temperature-dependent damping element and two linear springs, two thermal strain elements are introduced. In order to take the process dependence of the specific heat into account and to model its characteristic behaviour below and above the glass transition, the Helmholtz free energy contains an additional contribution which depends on the temperature history and on the current temperature. The model describes the process-dependent volumetric and caloric behaviour of glass-forming materials, and defines a functional relationship between pressure, volumetric strain, and temperature. If a model for the isochoric part of the material behaviour is already available, for example a model of finite viscoelasticity, the caloric and volumetric behaviour can be represented with the current approach. The proposed model allows computing the isobaric and isochoric heat capacities in closed form. The difference \(c_\mathrm{p} -c_\mathrm{v} \) is process-dependent and tends towards the classical expression in the glassy and equilibrium ranges. Simulations and theoretical studies demonstrate the physical significance of the model. PubDate: 2017-09-01 DOI: 10.1007/s00161-016-0551-9 Issue No:Vol. 29, No. 5 (2017)

Authors:Q. Roirand; D. Missoum-Benziane; A. Thionnet; L. Laiarinandrasana Pages: 1081 - 1092 Abstract: Abstract Textile composites are composed of 3D complex architecture. To assess the durability of such engineering structures, the failure mechanisms must be highlighted. Examinations of the degradation have been carried out thanks to tomography. The present work addresses a numerical damage model dedicated to the simulation of the crack initiation and propagation at the scale of the warp yarns. For the 3D woven composites under study, loadings in tension and combined tension and bending were considered. Based on an erosion procedure of broken elements, the failure mechanisms have been modelled on 3D periodic cells by finite element calculations. The breakage of one element was determined using a failure criterion at the mesoscopic scale based on the yarn stress at failure. The results were found to be in good agreement with the experimental data for the two kinds of macroscopic loadings. The deterministic approach assumed a homogeneously distributed stress at failure all over the integration points in the meshes of woven composites. A stochastic approach was applied to a simple representative elementary periodic cell. The distribution of the Weibull stress at failure was assigned to the integration points using a Monte Carlo simulation. It was shown that this stochastic approach allowed more realistic failure simulations avoiding the idealised symmetry due to the deterministic modelling. In particular, the stochastic simulations performed have shown several variations of the stress as well as strain at failure and the failure modes of the yarn. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0553-2 Issue No:Vol. 29, No. 5 (2017)

Authors:C. Röhrig; T. Scheffer; S. Diebels Pages: 1093 - 1111 Abstract: 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: 2017-09-01 DOI: 10.1007/s00161-017-0560-3 Issue No:Vol. 29, No. 5 (2017)

Authors:Leonell Serrano; Yann Marco; Vincent Le Saux; Gilles Robert; Pierre Charrier Pages: 1113 - 1133 Abstract: Abstract Short-fiber-reinforced thermoplastics components for structural applications are usually very complex parts as stiffeners, ribs and thickness variations are used to compensate the quite low material intrinsic stiffness. These complex geometries induce complex local mechanical fields but also complex microstructures due to the injection process. Accounting for these two aspects is crucial for the design in regard to fatigue of these parts, especially for automotive industry. The aim of this paper is to challenge an energetic approach, defined to evaluate quickly the fatigue lifetime, on three different heterogeneous cases: a classic dog-bone sample with a skin-core microstructure and two structural samples representative of the thickness variations observed for industrial components. First, a method to evaluate dissipated energy fields from thermal measurements is described and is applied to the three samples in order to relate the cyclic loading amplitude to the fields of cyclic dissipated energy. Then, a local analysis is detailed in order to link the energy dissipated at the failure location to the fatigue lifetime and to predict the fatigue curve from the thermomechanical response of one single sample. The predictions obtained for the three cases are compared successfully to the Wöhler curves obtained with classic fatigue tests. Finally, a discussion is proposed to compare results for the three samples in terms of dissipation fields and fatigue lifetime. This comparison illustrates that, if the approach is leading to a very relevant diagnosis on each case, the dissipated energy field is not giving a straightforward access to the lifetime cartography as the relation between fatigue failure and dissipated energy seems to be dependent on the local mechanical and microstructural state. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0561-2 Issue No:Vol. 29, No. 5 (2017)

Authors:Matthias Wunde; Manfred Klüppel Pages: 1135 - 1148 Abstract: Abstract Based on a viscoelastic model, the filler distribution and the amount of interphase of carbon black-filled blends of natural rubber (NR) with styrene-butadiene rubber (SBR) are evaluated. Hereby, the total dissipated energy \(G''\) during dynamical straining is decomposed into the contributions of the different polymer phases and the interphase. For the NR/SBR blends, we find a higher filling of the SBR phase and the interphase and a lower filling of the NR phase. The filler distribution itself depends not only on the affinity of the polymer to the filler but also on the mixing procedure. This is investigated by studying NR/SBR blends prepared by two different mixing procedures. In the standard mixing procedure, the polymers are mixed first, and then, the filler is added. In the batch mixing procedure, the filler is previously mixed in the NR only and then blended with SBR. Batch mixing is resulting in an increase in the filling of the interphase due to filler transfer from NR to SBR. The results for the filler distribution are compared to fatigue crack propagation rates under pulsed excitation. The crack propagation is accelerated when substituting NR with SBR. The batched samples show higher crack propagation rates at higher tearing energies due to a worse dispersion of the carbon black and/or higher filler loading of the interphase. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0562-1 Issue No:Vol. 29, No. 5 (2017)

Authors:A. Herzig; L. Sekerakova; M. Johlitz; A. Lion Pages: 1149 - 1161 Abstract: Abstract The influence of oxygen on elastomers, known as oxidation, is one of the most important ageing processes and becomes more and more important for nowadays applications. The interaction with thermal effects as well as antioxidants makes oxidation of polymers a complex process. Based on the polymer chosen and environmental conditions, the ageing processes may behave completely different. In a lot of cases the influence of oxygen is limited to the surface layer of the samples, commonly referred to as diffusion-limited oxidation. For the lifetime prediction of elastomer components, it is essential to have detailed knowledge about the absorption and diffusion behaviour of oxygen molecules during thermo-oxidative ageing and how they react with the elastomer. Experimental investigations on industrially used elastomeric materials are executed in order to develop and fit models, which shall be capable of predicting the permeation and consumption of oxygen as well as changes in the mechanical properties. The latter are of prime importance for technical applications of rubber components. Oxidation does not occur homogeneously over the entire elastomeric component. Hence, material models which include ageing effects have to be amplified in order to consider heterogeneous ageing, which highly depends on the ageing temperature. The influence of elevated temperatures upon accelerated ageing has to be critically analysed, and influences on the permeation and diffusion coefficient have to be taken into account. This work presents phenomenological models which describe the oxygen uptake and the diffusion into elastomers based on an improved understanding of ongoing chemical processes and diffusion limiting modifications. On the one side, oxygen uptake is modelled by means of Henry’s law in which solubility is a function of the temperature as well as the ageing progress. The latter is an irreversible process and described by an inner differential evolution equation. On the other side, further diffusion of oxygen into the material is described by a model based on Fick’s law, which is modified by a reaction term. The evolved diffusion-reaction equation depends on the ageing temperature as well as on the progress of ageing and is able to describe diffusion-limited oxidation. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0568-8 Issue No:Vol. 29, No. 5 (2017)

Authors:Thomas Glanowski; Vincent Le Saux; Cédric Doudard; Yann Marco; Clément Champy; Pierre Charrier Pages: 1163 - 1179 Abstract: Abstract A methodology is proposed to define an equivalent geometry allowing the use of an uncoupled algorithm to solve thermomechanical problems when cyclic large strain occurs. This methodology is set up on the case of a simple bar and is then challenged on a structural sample for cyclic loadings. It is shown that the definition of the equivalent geometry is dependent on the thermal boundary conditions, which are usually unknowns of the thermal problem. The proposed approach is finally applied to the identification of cyclic dissipation from infrared thermography measurements. PubDate: 2017-09-01 DOI: 10.1007/s00161-017-0572-z Issue No:Vol. 29, No. 5 (2017)

Authors:Maurizio Romeo Abstract: Abstract A micromorphic electroelastic model for polarized liquid crystals is proposed on the basis of a representation of electric multipoles in terms of microdeformation. Nematic liquid crystals are modeled as micropolar continua endowed with intrinsic electric dipole and quadrupole. A nonlinear dimensionless problem for a homogeneous nematic layer is formulated and solved numerically. The existence of a threshold electric potential is discussed, and the corresponding linearized system is also obtained to compare results on small values of deformation and electric field. Differently from common results of the classical continuum approach, asymmetric deformations and electric potentials within the layer are obtained due to the occurrence of non-null intrinsic quadrupole. PubDate: 2017-09-20 DOI: 10.1007/s00161-017-0598-2

Authors:Hicheme Ferdjani; Radhi Abdelmoula Abstract: Abstract This work deals with the propagation of a Dugdale crack at the edge of a half plane. The corresponding singular integral equation is solved semi-analytically. The expressions of the stress intensity factor and of the crack gap are deduced. A propagation criterion deduced from the revisited Griffith theory (Ferdjani and Marigo in Eur J Mech A Solids 53:1–9, 2015) is applied. The length of the process zone is calculated and compared with the literature results. The presented results show the evolution of the applied load with the crack length for different values of the ratio of the critical length of the Dugdale model to the initial crack length. The shape of the crack gap is also presented. Finally, a comparison between the Griffith and Dugdale models is performed. PubDate: 2017-09-16 DOI: 10.1007/s00161-017-0594-6

Authors:Anjanna Matta; Antony A. Hill Abstract: Abstract The thermosolutal instability of double-diffusive convection in an inclined fluid-saturated porous layer with a concentration-based internal heat source is investigated. The linear instability of small-amplitude perturbations to the system is analyzed with respect to transverse and longitudinal rolls. The resultant eigenvalue problem is solved numerically utilizing the Chebyshev tau method. It is shown that an increasing inclination angle causes a strong stabilization in the transverse rolls irrespective of the internal heat source or vertical solutal Rayleigh number. Furthermore, substantial qualitative changes are demonstrated in the linear instability thresholds with variations in the inclination angle and concentration-based heat source. PubDate: 2017-09-14 DOI: 10.1007/s00161-017-0592-8

Authors:Giovanni Romano; Raffaele Barretta; Marina Diaco Abstract: Abstract There are, in each branch of science, statements which, expressed in ambiguous or even incorrect but seemingly friendly manner, were repeated for a long time and eventually became diffusely accepted. Objectivity of physical fields and of their time rates and frame indifference of constitutive relations are among such notions. A geometric reflection on the description of frame changes as spacetime automorphisms, on induced push–pull transformations and on proper physico–mathematical definitions of material, spatial and spacetime tensor fields and of their time-derivatives along the motion, is here carried out with the aim of pointing out essential notions and of unveiling false claims. Theoretical and computational aspects of nonlinear continuum mechanics, and especially those pertaining to constitutive relations, involving material fields and their time rates, gain decisive conceptual and operative improvement from a proper geometric treatment. Outcomes of the geometric analysis are frame covariance of spacetime velocity, material stretching and material spin. A univocal and frame-covariant tool for evaluation of time rates of material fields is provided by the Lie derivative along the motion. The postulate of frame covariance of material fields is assessed to be a natural physical requirement which cannot interfere with the formulation of constitutive laws, with claims of the contrary stemming from an improper imposition of equality in place of equivalence. PubDate: 2017-09-12 DOI: 10.1007/s00161-017-0595-5

Authors:Raimondo Penta; Ariel Ramírez-Torres; José Merodio; Reinaldo Rodríguez-Ramos Abstract: Abstract We derive the new effective governing equations for linear elastic composites subject to a body force that admits a Helmholtz decomposition into inhomogeneous scalar and vector potentials. We assume that the microscale, representing the distance between the inclusions (or fibers) in the composite, and its size (the macroscale) are well separated. We decouple spatial variations and assume microscale periodicity of every field. Microscale variations of the potentials induce a locally unbounded body force. The problem is homogenizable, as the results, obtained via the asymptotic homogenization technique, read as a well-defined linear elastic model for composites subject to a regular effective body force. The latter comprises both macroscale variations of the potentials, and nonstandard contributions which are to be computed solving a well-posed elastic cell problem which is solely driven by microscale variations of the potentials. We compare our approach with an existing model for locally unbounded forces and provide a simplified formulation of the model which serves as a starting point for its numerical implementation. Our formulation is relevant to the study of active composites, such as electrosensitive and magnetosensitive elastomers. PubDate: 2017-08-21 DOI: 10.1007/s00161-017-0590-x

Authors:M. P. Ramos; C. Ribeiro; A. J. Soares Abstract: Abstract In this paper, we study the propagation of sound waves and the dynamics of local wave disturbances induced by spontaneous internal fluctuations in a reactive mixture. We consider a non-diffusive, non-heat conducting and non-viscous mixture described by an Eulerian set of evolution equations. The model is derived from the kinetic theory in a hydrodynamic regime of a fast chemical reaction. The reactive source terms are explicitly computed from the kinetic theory and are built in the model in a proper way. For both time-dependent problems, we first derive the appropriate dispersion relation, which retains the main effects of the chemical process, and then investigate the influence of the chemical reaction on the properties of interest in the problems studied here. We complete our study by developing a rather detailed analysis using the Hydrogen–Chlorine system as reference. Several numerical computations are included illustrating the behavior of the phase velocity and attenuation coefficient in a low-frequency regime and describing the spectrum of the eigenmodes in the small wavenumber limit. PubDate: 2017-08-14 DOI: 10.1007/s00161-017-0591-9

Authors:Babak Ebrahimian; Ali Noorzad; Mustafa I. Alsaleh Abstract: Abstract Recently, the authors have focused on the shear behavior of interface between granular soil body and very rough surface of moving bounding structure. For this purpose, they have used finite element method and a micro-polar elasto-plastic continuum model. They have shown that the boundary conditions assumed along the interface have strong influences on the soil behavior. While in the previous studies, only very rough bounding interfaces have been taken into account, the present investigation focuses on the rough, medium rough and relatively smooth interfaces. In this regard, plane monotonic shearing of an infinite extended narrow granular soil layer is simulated under constant vertical pressure and free dilatancy. The soil layer is located between two parallel rigid boundaries of different surface roughness values. Particular attention is paid to the effect of surface roughness of top and bottom boundaries on the shear behavior of granular soil layer. It is shown that the interaction between roughness of bounding structure surface and the rotation resistance of bounding grains can be modeled in a reasonable manner through considered Cosserat boundary conditions. The influence of surface roughness is investigated on the soil shear strength mobilized along the interface as well as on the location and evolution of shear localization formed within the layer. The obtained numerical results have been qualitatively compared with experimental observations as well as DEM simulations, and acceptable agreement is shown. PubDate: 2017-08-09 DOI: 10.1007/s00161-017-0588-4

Authors:Habib Pouriayevali; Bai-Xiang Xu Abstract: Abstract A comprehensive study on a finite-deformation gradient crystal-plasticity model which has been derived based on Gurtin’s framework (Int J Plast 24:702–725, 2008) is carried out here. This systematic investigation on the different roles of governing components of the model represents the strength of this framework in the prediction of a wide range of hardening behaviors as well as rate-dependent and scale-variation responses in a single crystal. The model is represented in the reference configuration for the purpose of numerical implementation and then implemented in the FEM software ABAQUS via a user-defined subroutine (UEL). Furthermore, a function of accumulation rates of dislocations is employed and viewed as a measure of formation of short-range interactions. Our simulation results reveal that the dissipative gradient strengthening can be identified as a source of isotropic-hardening behavior, which may represent the effect of irrecoverable work introduced by Gurtin and Ohno (J Mech Phys Solids 59:320–343, 2011). Here, the variation of size dependency at different magnitude of a rate-sensitivity parameter is also discussed. Moreover, an observation of effect of a distinctive feature in the model which explains the effect of distortion of crystal lattice in the reference configuration is reported in this study for the first time. In addition, plastic flows in predefined slip systems and expansion of accumulation of GNDs are distinctly observed in varying scales and under different loading conditions. PubDate: 2017-07-22 DOI: 10.1007/s00161-017-0589-3

Authors:Nicolas Charalambakis; George Chatzigeorgiou; Yves Chemisky; Fodil Meraghni Abstract: Abstract In this paper, a review of papers on mathematical homogenization of dissipative composites under small strains and on the interplay between homogenization procedure and dissipation due to mechanical work is presented. Moreover, a critical survey on the links between mathematical homogenization and computational homogenization is attempted. PubDate: 2017-07-13 DOI: 10.1007/s00161-017-0587-5

Authors:Danil A. Kozhevnikov; Mikhail A. Sheremet Abstract: Abstract The effect of surface tension on laminar natural convection in a vertical cylindrical cavity filled with a weak evaporating liquid has been analyzed numerically. The cylindrical enclosure is insulated at the bottom, heated by a constant heat flux from the side, and cooled by a non-uniform evaporative heat flux from the top free surface having temperature-dependent surface tension. Governing equations with corresponding boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The influence of Rayleigh number, Marangoni number, and aspect ratio on the liquid flow and heat transfer has been studied. Obtained results have revealed that the heat transfer rate at free surface decreases with Marangoni number and increases with Rayleigh number, while the average temperature inside the cavity has an opposite behavior; namely, it growths with Marangoni number and reduces with Rayleigh number. PubDate: 2017-07-07 DOI: 10.1007/s00161-017-0586-6