Abstract: Abstract
Since the pioneering work by Treloar, many models based on polymer chain statistics have been proposed to describe rubber elasticity. Recently, Alicandro, Cicalese, and the first author rigorously derived a continuum theory of rubber elasticity from a discrete model by variational convergence. The aim of this paper is twofold. First, we further physically motivate this model and complete the analysis by numerical simulations. Second, in order to compare this model to the literature, we present in a common language two other representative types of models, specify their underlying assumptions, check their mathematical properties, and compare them to Treloar’s experiments. PubDate: 2014-01-01

Abstract: Abstract
This work presents the development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions
${\phi_\alpha}$
using principles of continuum mechanics: conservation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for constituents are decoupled. In this case, one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures p
α of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose
${p_\alpha=\phi_\alpha p}$
and
${\sum_\alpha^\nu p_\alpha = p}$
which implies
${\sum_\alpha^\nu \phi_\alpha = 1}$
which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifications of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in
${H^{k,p}\left(\bar{\Omega}^e\right)}$
scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents. PubDate: 2014-01-01

Abstract: Abstract
The covariance principle of differential geometry within a four-dimensional (4D) space-time ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space-time and the use of 4D tensors, operations and operators. This enables to separate covariance (i.e. frame-indifference) and material objectivity (i.e. material-indifference). We propose here a method to build a constitutive relation for thermo-elastic materials using such a 4D formalism. A 4D generalization of the classical variational approach is assumed leading to a model for a general thermo-elastic material. The isotropy of the relation can be ensured by the use of the invariants of variables, which offers new possibilities for the construction of constitutive relations. It is then possible to build a general frame-indifferent but not necessarily material-indifferent constitutive relation. It encompasses both the 3D Eulerian and Lagrangian thermo-elastic isotropic relations for finite transformations. PubDate: 2014-01-01

Abstract: Abstract
A continuum model for a graphene sheet undergoing infinitesimal in-plane deformations is derived by applying the arguments of homogenization theory. The model turns out to coincide with that found by various authors with different methods, but it avoids anticipations on the validity of any properly adjusted or generalized Cauchy–Born rule. The constitutive equation for stress and the effective Young’s modulus and Poisson ratio are explicitly given in terms of the bond constants. PubDate: 2014-01-01

Abstract: Abstract
In this paper, we study mass flow rate of rarefied gas flow through micro/nanoscale channels under simultaneous thermal and pressure gradients using the direct simulation Monte Carlo (DSMC) method. We first compare our DSMC solutions for mass flow rate of pure temperature-driven flow with those of Boltzmann-Krook-Walender equation and Bhatnagar-Gross-Krook solutions. Then, we focus on pressure–temperature-driven flows. The effects of different parameters such as flow rarefaction, channel pressure ratio, wall temperature gradient and flow bulk temperature on the thermal mass flow rate of the pressure–temperature-driven flow are examined. Based on our analysis, we propose a correlated relation that expresses normalized mass flow rate increment due to thermal creep as a function of flow rarefaction, normalized wall temperature gradient and pressure ratio over a wide range of Knudsen number. We examine our predictive relation by simulation of pressure-driven flows under uniform wall heat flux (UWH) boundary condition. Walls under UWH condition have non-uniform temperature distribution, that is, thermal creep effects exist. Our investigation shows that developed analytical relation could predict mass flow rate of rarefied pressure-driven gas flows under UWH condition at early transition regime, that is, up to Knudsen numbers of 0.5. PubDate: 2014-01-01

Abstract: Abstract
An analytical solution of the problem of the propagation of a Lüders band in an isotropic strain gradient plasticity medium is provided based on a softening–hardening constitutive law. A detailed description is given of the plastic strain distribution in the finite size band front. The solution is shown to be harmonic in the band front and exponential in the band tail. Particular attention is paid to the conditions to be applied at the interface between both regions. This solution is then used to validate finite element simulations of the Lüders band formation and propagation in a plate in tension. The approach is shown to suppress the spurious mesh dependence exhibited by conventional finite element simulations of the Lüders behavior and to provide a finite width band front in agreement with the experimental observations from strain field measurements. PubDate: 2013-12-29

Abstract: Abstract
Instability and stress–strain behavior were investigated for 2D regular assemblies of cylindrical particles. Biaxial shear experiments were performed on three sets of assemblies with regular, albeit increasingly defective structures. These experiments revealed unique instability behavior of these assemblies. Continuum models for the assemblies were then constructed using the granular micromechanics approach. In this approach, the constitutive equations governing the behavior of inter-particle contacts are written in local or microscopic level. The behavior of the RVE is then retrieved by using either kinematic constraint or least squares (static constraint) along with the principle of virtual work to equate the work done by microscopic force–displacement conjugates to that of the macroscopic stress and strain tensor conjugates. The ability of the two continuum approaches to describe the measured stress–strain behavior was evaluated. The continuum models and the local constitutive laws were used to perform instability analyses. The onset of instability and orientation of shear band was found to be well predicted by the instability analyses with the continuum models. Further, macro-scale instability was found to correlate with the instability of inter-particle contacts, although with some variations for the two modeling approaches. PubDate: 2013-12-28

Abstract: Abstract
In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013), Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus μ
c
, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology. PubDate: 2013-12-19

Abstract: Abstract
Parameters identification of nonstationary systems is a very challenging topic that has only recently received critical attention from the research community. Aim of the paper is the structural health monitoring of bridge-like structures excited by a massive moving load, whose characteristics, such as the mass and speed, are unknown, in the presence of a localized damage along the structure. A novel method for the simultaneous identification of both the load characteristics and damage parameters from vibration measurements is proposed: the data processing relies on the ensemble empirical mode decomposition and the normalized Hilbert transform. Neither a priori information about the response of the undamaged structure nor the free decay of the damaged system is required, only a single-point measurement is needed. The empirical instantaneous frequency is firstly employed to estimate the load characteristics; secondly, the effect of the moving mass is filtered from the instantaneous frequency, and then, the damage position is identified. The performance of the technique are studied varying the load characteristics, damage locations, and crack depths. The effect of ambient noise is also taken into account. Numerical experiments show the identification is rather accurate, results are not very sensitive to the crack location and depth, while they are sensibly affected by the speed of the moving load. PubDate: 2013-12-15

Abstract: 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: 2013-11-30

Abstract: Abstract
This paper deals with modelling of landslide propagation. Its purpose is to present a methodology of analysis based on mathematical, constitutive and numerical modelling, which includes both well-established theories together with some improvements which are proposed herein. Concerning the mathematical model, it is based on Biot–Zienkiewicz equations, from where a depth-integrated model is developed. The main contribution here is to combine a depth-integrated description of the soil–pore fluid mixture together with a set of 1D models dealing with pore pressure evolution within the soil mass. In this way, pore pressure changes caused by vertical consolidation, changes of total stresses resulting from height variations and changes of basal surface permeability can be taken into account with more precision. Most of rheological models used in depth-integrated models are derived either heuristically (the case of Voellmy model, for instance), or from general 3D rheological models. Here, we will propose an alternative way, based on Perzyna’s viscoplasticity. The approach followed for numerical modelling is the SPH method, which we have enriched adding a 1D finite difference grid to each SPH node, in order to improve the description of pore water profiles in the avalanching soil. This paper intends to be a homage to Professor Felix Darve, who has very much contributed to the field of modern geomechanics. PubDate: 2013-11-26

Abstract: Abstract
This paper is devoted to a micromechanical study of mechanical properties of cement-based materials by taking into account effects of water saturation degree and carbonation process. To this end, the cement-based materials are considered as a composite material constituted with a cement matrix and aggregates (inclusions). Further, the cement matrix is seen as a porous medium with a solid phase (CSH) and pores. Using a two-step homogenization procedure, a closed-form micromechanical model is first formulated to describe the basic mechanical behavior of materials. This model is then extended to partially saturated materials in order to account for the effects of water saturation degree on the mechanical properties. Finally, considering the solid phase change and porosity variation related to the carbonation process, the micromechanical model is coupled with the chemical reaction and is able to describe the consequences of carbonation on the macroscopic mechanical properties of material. Some comparisons between numerical results and experimental data are presented. PubDate: 2013-11-23

Abstract: 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: 2013-11-22

Abstract: Abstract
In the present paper, the simplest model of strain-gradient elasticity will be considered, that is, the isotropy in a bidimensional space. Paralleling the definition of the classic elastic moduli, our aim is to introduce second-order isotropic moduli having a mechanical interpretation. A general construction process of these moduli will be proposed. As a result, it appears that many sets can be defined, each of them constituted of 4 moduli: 3 associated with 2 distinct mechanisms and the last one coupling these mechanisms. We hope that these moduli (and the construction process) might be useful for forthcoming investigations on generalized continuum mechanics. PubDate: 2013-11-20

Abstract: 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: 2013-11-07

Abstract: Abstract
The paper determines the forms of equations of equilibrium for stable coherent phase interfaces in isotropic solids. If the first phase satisfies the Baker Ericksen inequalities strictly and the principal stretches of the second phase differ from those of the first phase, one obtains the equality of three specific generalized scalar forces and of a generalized Gibbs function. The forms of these quantities depend on the signs of the increments of the principal stretches across the interface. The proof uses the rank 1 convexity condition for isotropic materials (Šilhavý in Proc. R. Soc. Edinb 129A:1081–1105, 1999) and is available only if the two phases are not too far from each other or if one of the two phases is a fluid (liquid). The result does not follow from the representation theorems for isotropic solids. PubDate: 2013-11-01

Abstract: Abstract
A continuum description of multiphase flows, in which excess physical quantities associated with phase interfaces and the three-phase contact line are incorporated, is briefly presented. A thermodynamic analysis, based on the Müller–Liu thermodynamic approach of the second law of thermodynamics, is performed to derive the expressions of the constitutive variables in thermodynamic equilibrium. Non-equilibrium responses are proposed by use of a quasi-linear theory. A set of constitutive equations for the surface and line constitutive quantities is postulated. Some restrictions for the emerging material parameters are derived by means of the minimum conditions of the surface and line entropy productions in thermodynamic equilibrium. Hence, a complete continuum mechanical model to describe excess surface and line physical quantities is formulated. PubDate: 2013-11-01

Abstract: Abstract
We study the dispersion relation for sound in rarefied polyatomic gases (hydrogen, deuterium and hydrogen deuteride gases) basing on the recently developed theory of extended thermodynamics (ET) of dense gases. We compare the relation with those obtained in experiments and by the classical Navier–Stokes Fourier (NSF) theory. The applicable frequency range of the ET theory is proved to be much wider than that of the NSF theory. We evaluate the values of the bulk viscosity and the relaxation times involved in nonequilibrium processes. The relaxation time related to the dynamic pressure has a possibility to become much larger than the other relaxation times related to the shear stress and the heat flux. PubDate: 2013-11-01

Abstract: Abstract
In this work, the structural and transport properties of Mg-doped Sn-based alloys have been investigated. The temperature-dependent transport and structural properties of Sn–Mg alloys were investigated for five different samples (Pure Sn, Sn-1.0 wt% Mg, Sn-2.0 wt% Mg, Sn-6.0 wt% Mg and Pure Mg). Scanning electron microscopy (SEM), X-ray diffraction and energy dispersive X-ray analysis measurements were carried out in order to clarify the structural properties of the samples. It was found that the samples had tetragonal crystal symmetry, except for pure Mg which had hexagonal crystal symmetry. We also found that the cell parameters changed slightly with the addition of Mg element. The SEM micrographs of the samples showed that they had smooth surfaces with a clear grain boundary. The electrical and thermal conductivity of the samples were measured by four-point probe and the radial heat flow method, respectively. The electrical resistivity of the samples increased almost linearly with the increasing temperature. The thermal conductivity values ranged between 0.60 and 1.00 W/Km, while they decreased slightly with temperature and increased with Mg composition. The thermal conductivity values of the alloys were in between the values of pure Sn and Mg. The thermal conductivity results of the alloys were compared with other available results, and a good agreement was seen between the results. In addition, the temperature coefficients of electrical resistivity and thermal conductivity were determined; these were independent of the composition of the alloying elements. PubDate: 2013-11-01