On two models of arbitrarily curved three-dimensional thin interphases in elasticity

In boundary value problems involving thin interphases, it is often desirable to have a model of an interphase which makes possible to solve for the fields in the adjacent media without having to solve for the fields in it. This is usually achieved in the literature by replacing the interphase by a geometrical surface with appropriately designed “imperfect interface” conditions on it. In the present study, carried out in the setting of elasticity, another option is explored: the geometry of the interphase is left intact, and conditions are devised for the displacements and tractions pertaining to the media adjacent to the interphase and evaluated at both sides of it such that they will simulate the presence of the interphase. Those conditions do not involve the fields within the interphase, yet they depend on its material properties and on those of the adjacent media as well, and make possible to solve for the fields in the adjacent media without having to solve for the fields in the interphase. The formulation is given in a parallel orthogonal curvilinear coordinate system suitable for the modeling arbitrarily curved three-dimensional interphases of constant thickness. Both types of the above described interphase models are tested in the setting of a coated infinite fiber embedded in a matrix which is subjected to an anti-plane shear loading and an in-plane transverse shear loading at infinity, and their predictions are compared with the exact solutions for the fields in the three-phase configuration consisting of the interphase and its adjacent media. The model in which the interphase geometry is left intact is observed to perform generally better than the one in which the interphase is replaced by an interface.     

Micromechanical characterizing elastic,thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites

In this work, elastic, thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites are investigated using a 3-dimensional micromechanics-based approach. The main advantage of the proposed micromechanical model is its ability to give closed-form formulation for predicting the effective properties of nanocomposites. In the micromechanical modeling, the interphase formed due to non-boned van der Waals interaction between the continuous CNT and polymer matrix is considered through employing an individual representative volume element. The validity of the model is examined by comparing its results with other theoretical approaches and experimental data available in the literature. The effects of various types of CNTs arrangement in the matrix, i.e. uniform distribution and different functionally graded distributions on the elastic, thermoelastic and viscoelastic properties of polymer nanocomposites are investigated in detail. Furthermore, random arrangement of CNTs in the matrix is modelled. The influences of CNT/polymer matrix interphase and CNT volume fraction on the effective properties of nanocomposites are also studied. Finally, the viscoelastic response of nanocomposites under multiaxial loading is extracted and interpreted.     

含非均匀界面相粒子填充复合材料的有效比热

基于细观力学复合球模型研究了含非均匀界面相粒子填充复合材料的有效热弹性性质,重点讨论了界面相性质的径向分布对有效比热的影响. 首先,将非均匀界面相沿径向离散为多个同心球壳,每个球壳内的材料性质假设是均匀的. 基于上述离散模型,利用含界面相的复合球模型,推导了复合材料的有效体积模量、有效热膨胀系数及有效比热的数值求解表达式;进一步,假设界面相的性质沿径向连续变化,建立了一组微分方程,上述有效性质依赖于该微分方程组的解. 特别地,当界面相杨氏模量为幂次分布时,通过求解该微分方程组得到了有效比热等热弹性性质的解析解. 算例结果表明,应用此方法预测的有效热膨胀系数与实验结果吻合良好;界面相热膨胀系数的径向分布对有效比热和有效热膨胀系数均有显著的影响,而界面相弹性模量的径向分布对有效比热有显著的影响,对有效热膨胀系数的影响相对较小.      

A Cosserat shell model for interphases in elastic media

This study is concerned with the modeling of interphases in elastic media in general, and in composite materials in particular. The aim is to replace a boundary value problem consisting of a three-phase configuration, say that of fiber-interphase-matrix, by a simpler problem which involves the fiber and matrix only, plus certain matching conditions which simulate the interphase. The simplest of such known representations replaces a thin interphase by a “perfect contact interface” (a single surface) across which the displacements and tractions are assumed to be continuous. Another classical model replaces a thin and soft interphase by a “spring-type interface”, across which the tractions are continuous, but the displacement field undergoes a discontinuity. In the present paper, a Cosserat shell model of the interphase is derived which successfully models the original interphase in a unified manner, for the full range of its material parameters relative to those of the neighboring media. The model is derived in the setting of three-dimensional linear elasticity with small deformations and displacements. Comparisons with an existing exact solution of a coated fiber in an infinite matrix show that it performs extremely well even for moderately thick interphases.     

EFFECTIVE SPECIFIC HEATS OF SPHERICAL PARTICULATE COMPOSITES WITH INHOMOGENEOUS INTERPHASES

基于细观力学复合球模型研究了含非均匀界面相粒子填充复合材料的有效热弹性性质,重点讨论了界面相性质的径向分布对有效比热的影响. 首先,将非均匀界面相沿径向离散为多个同心球壳,每个球壳内的材料性质假设是均匀的. 基于上述离散模型,利用含界面相的复合球模型,推导了复合材料的有效体积模量、有效热膨胀系数及有效比热的数值求解表达式;进一步,假设界面相的性质沿径向连续变化,建立了一组微分方程,上述有效性质依赖于该微分方程组的解. 特别地,当界面相杨氏模量为幂次分布时,通过求解该微分方程组得到了有效比热等热弹性性质的解析解. 算例结果表明,应用此方法预测的有效热膨胀系数与实验结果吻合良好;界面相热膨胀系数的径向分布对有效比热和有效热膨胀系数均有显著的影响,而界面相弹性模量的径向分布对有效比热有显著的影响,对有效热膨胀系数的影响相对较小.     

Effective conductivities of thin-interphase composites

A method is presented for approximating the effective conductivity of composite media with thin interphase regions, which is exact to first order in the interphase thickness. The approximations are computationally efficient in the sense the fields need to be computed only in a reference composite in which the interphases have been replaced by perfect interfaces. The results apply whether any two phases of the composite are separated by a single interphase or multiple interphases, whether the conductivities of the composite phases are isotropic or anisotropic, and whether the thickness of an interphase is uniform or varies as a function of position. It is assumed that the conductivities of the interphase materials have intermediate values as opposed to very high or very low conductivities.     

Multi-scale framework for the thermo-viscoelastic analyses of polymer composites

This paper presents a multi-scale framework for analyzing coupled heat conduction and viscoelastic deformation of polymers reinforced with solid spherical particles. The viscoelastic and thermal properties of the polymer constituents are temperature dependent. A simplified micromechanical model for the particle reinforced composite is formulated to obtain the effective thermal properties and viscoelastic responses. The micromechanical model is implemented at material points within elements in the finite element (FE) analyses.     

Rheology of polymer blends

Rheological properties of blends of amorphous and crystalline polymers were studied for a broad range of compositions and temperatures. It was established that below the melting pointT _m the viscoelastic properties of blends of crystalline polymers are similar to those of polymers filled with mineral fillers. In both cases these properties are influenced by the existence, in such systems, of a temporary structural network formed by mineral or polymeric particles and its subsequent breakdown under the action of shear stresses. It was found that an anomalous decrease in the melt viscosity of the main component on addition of a small amount of a second polymer depended on deformation conditions. The comparison of data on viscoelastic properties and thermodynamic interaction between the components in the melt, estimated from the parameter ₂₃ of a new Flory theory, shows that the sharp drop of viscosity takes place in the region of microphase separation due to the appearance of an excess free volume in the interphase region. Calculation of the relaxation spectra for various blends also revealed marked changes when various amounts of a second component were added to the main polymer.     

Structural morphology and relaxation spectra of viscoelastic heterogeneous materials

We analytically derive the relaxation spectra of a two-phase isotropic material whose phases are isotropic Maxwell media, according to the classical and to the generalized self-consistent schemes. Whereas these spectra are continuous in both cases, they exhibit strong differences which can be associated with the different underlying morphology, either symmetrical (polycrystal-type) in the first case or asymmetrical (composite-type) in the second case. The treatment is extended to the (N+1)-phase model which allows us to deal with coated inclusions or with an interphase between the matrix and the inclusions: the interphase is shown to strongly modify the resultant spectrum. More general cases are then considered for different kinds of constitutive behaviour as well as for coated fibre reinforced composites. As a whole, the spectral analysis method appears to be an efficient tool for the investigation of the connection between structural morphology and the overall behaviour of viscoelastic heterogeneous materials.     

Experimental verification of the theory of thixotropy of viscoelastic polymer media

Previous studies [1, 2] were devoted to the formulation of a phenomenological theory of the reversible changes (thixotropy) in the physicomechanical properties of viscoelastic media, in particular, polymer systems. The basis of this theory is the idea that the reversible changes in the structure and properties of viscoelastic thixotropic media under the influence of mechanical action are due to a change in their relaxation spectrum. In this case the behavior of the mechanical properties is entirely and uniquely determined by two material functions which completely characterize the given material: the relaxation time (frequency) distribution function, which describes the behavior of the material in the linear region of deformation, and the thixotropy function, which determines the nature of the changes compression) of the relaxation spectrum on transition to the nonlinear region. The object of this approach is to reflect the effect of the change in supramolecular and molecular structures associated with the flow of polymer system, or in the bonds and orientation of the particles in disperse systems, on the viscoelastic and viscosity properties of the systems concerned. The essential validity of this approach was demonstrated in [3], where it was shown that at sufficiently high strain rates changes occur in the structure of the polymer leading to a change in mechanical properties. In [4] the basic structural functions were found. These, in conjunction with the proposed theory of thixotropy, characterize the behavior of a broad group of polymer systems in both the linear and the nonlinear regions of deformation. In this connection it should be noted that whereas for the linear region the accuracy of the laws obtained lies within the limits ± 100%, on transition to the nonlinear region the error may increase, Therefore, for the purposes of a rigorous quantitative verification of the theory we shall use not the universal functions obtained in [4], but the more precise characteristics of the specific material on which the experiments are performed, since those effects with respect to which it is desired to test the theory usually lie within the limits of 30% of the measured quantities.The authors thank G. V. Vinogradov for organizing and discussing their work.     

A thermodynamic approach to the rheology of highly interactive filler-polymer mixtures. Part II. Comparison with polystyrene/nanosilica mixtures

The dynamic properties as a function of frequency and strain amplitude, steady-state viscosity as a function of shear rate, and transient shear stresses at startup and cessation of shear flow of polystyrene (PS)/fumed silica mixtures of various concentrations were investigated. An abrupt change in the viscoelastic properties was noticed at a concentration above 1% by volume. Observations by means of scanning electron microscopy (SEM) indicate the presence of a three-dimensional network through the bridging of filler particles by the adsorbed polymer. The viscoelastic behavior is simulated utilizing a theory proposed in Part I (Havet and Isayev 2001) based on a double network created by the entangled polymer matrix and the adsorbed polymer with filler concentration taken into account through the bridging density of polymer-filler interactions and a hydrodynamic reinforcement. The steps taken for determining the model parameters required to carry out the simulation are described. The major features of the rheological behavior of highly interactive polymer-filler mixtures are captured qualitatively and in some cases, quantitatively predicted.     

A thermodynamics based damage mechanics model for particulate composites

A micro-mechanical damage model is proposed to predict the overall viscoplastic behavior and damage evolution in a particle filled polymer matrix composite. Particulate composite consists of polymer matrix, particle fillers, and an interfacial transition interphase around the filler particles. Yet the composite is treated as a two distinct phase material, namely the matrix and the equivalent particle-interface assembly. The CTE mismatch between the matrix and the filler particles is introduced into the model. A damage evolution function based on irreversible thermodynamics is also introduced into the constitutive model to describe the degradation of the composite. The efficient general return-mapping algorithm is exploited to implement the proposed unified damage coupled viscoplastic model into finite element formulation. Furthermore, the model predictions for uniaxial loading conditions are compared with the experimental data.     

On the skin friction coefficient in viscoelastic wall-bounded flows

Analysis of the skin friction coefficient for wall bounded viscoelastic flows is performed by utilizing available direct numerical simulation (DNS) results for viscoelastic turbulent channel flow. The Oldroyd-B, FENE-P and Giesekus constitutive models are used. First, we analyze the friction coefficient in viscous, viscoelastic and inertial stress contributions, as these arise from suitable momentum balances, for the flow in channels and pipes. Following Fukagata et al. (Phys. Fluids, 14, p. L73, 2002) and Yu et al. (Int. J. Heat. Fluid Flow, 25, p. 961, 2004) these three contributions are evaluated averaging available numerical results, and presented for selected values of flow and rheological parameters. Second, based on DNS results, we develop a universal function for the relative drag reduction as a function of the friction Weissenberg number. This leads to a closed-form approximate expression for the inverse of the square root of the skin friction coefficient for viscoelastic turbulent pipe flow as a function of the friction Reynolds number involving two primary material parameters, and a secondary one which also depends on the flow. The primary parameters are the zero shear-rate elasticity number, El₀, and the limiting value for the drag reduction at high Weissenberg number, LDR, while the secondary one is the relative wall viscosity, μ_w. The predictions reproduce both types A and B of drag reduction, as first introduced by Virk (Nature, 253, p. 109, 1975), corresponding to partially and fully extended polymer molecules, respectively. Comparison of the results for the skin friction coefficient against experimental data shows good agreement for low and moderate drag reduction which is the region covered by the simulations.     

Small deformation viscoelastic response of gum and highly filled elastomers

Small deformation viscoelastic response has been investigated in a series of five elastomeric binders, both with and without nonreinforcing filler. The filled systems were found to be both nonlinear viscoelastic and thermorheologically complex. These behaviors suggest the existence of a secondary relaxation process. The origin of this secondary process was modeled as an interphase of polymer weakly adsorbed on the filler surface. Decomposition of timetemperature shift factors for filled vs unfilled properties showed that the mechanical response of this interphase followed Arrhenius behavior. Measured activation energies ranged from 24 to 76kJ/mole, depending on the cohesiveenergy density of the elastomeric binder. Finally, these activation energies were related to the strain amplitude dependent nonlinear factors for the polymeric systems which contained no polar groups in their backbone, suggesting that in these systems both the nonlinear and thermorheologically complex nature of the filled materials' viscoelastic response originate from relaxations within this interphase.     

ANALYTICAL SOLUTIONS FOR ELASTOSTATIC PROBLEMS OF PARTICLE-AND FIBER-REINFORCED COMPOSITES WITH INHOMOGENEOUS INTERPHASES

IntroductionRecently ,theinterphaseincompositeshasattractedtheattentionofmanyresearchers[1- 7].Someinterphasesareproducedbydesignandothersbychemicalreactionsinfabricatingcomposites.Fromtheviewpointofmechanicsofcompositematerials,oneofthefundamentalproblemsofcompositeswithinterphasesistopredicttheeffectivemoduli.Therearemanymethodsforpredictingeffectivemoduliofcomposites,suchasgeneralizedself_consistentmethod(GSCM) ,compositesphereandcompositecylindermodels (CCAandCSA)andIDDestimate ,etc …     

Global and local large-deformation response of sub-micron,soft- and hard-particle filled polycarbonate

Since polymers play an increasingly important role in both structural and tribological applications, understanding their intrinsic mechanical response is key. Therefore in the last few decades much effort has been devoted into the development of constitutive models that capture the polymers' intrinsic mechanical response quantitatively. An example is the Eindhoven Glassy Polymer model. In practice most polymers are filled, e.g. with hard particles or fibers, with colorants, or with soft particles that serve as impact modifiers. To characterize the influence of type and amount of filler particles on the intrinsic mechanical response, we designed model systems of polycarbonate with different volume fractions of small, order 100 nm sized, either hard or soft particles, and tested them in lubricated uniaxial compression experiments. To reveal the local effects on interparticle level, three-dimensional representative volume elements (RVEs) were constructed. The matrix material is modeled with the EGP model and the fillers with their individual mechanical properties. It is first shown that (only) 32 particles are sufficient to capture the statistical variations in these systems. Comparing the simulated response of the RVEs with the experiments demonstrates that in the small strain regime the stress is under-predicted since the polymer matrix is modeled by using only one single relaxation time. The yield- and the large strain response is captured well for the soft-particle filled systems while, for the hard-particles at increased filler loadings, the predictions are less accurate. This is likely caused by polymer–filler interactions that result in accelerated physical aging of the polymer matrix close to the surfaces. Modifying the S_a-parameter, that captures the thermodynamic state of the polymer matrix, allows us to correctly predict the macroscopic response after yield. The simulations reveal that all rate-dependencies of the different filled systems originate from that of the polymer matrix. Finally, an onset is presented to predict local and global failure based on critical events on the microlevel, that are likely to cause the over-prediction in the large-strain response of the hard-particle filled systems.     

Imperfect interfaces as asymptotic models of thin curved elastic adhesive interphases

We obtain a limit model for a thin curved anisotropic interphase adherent to two elastic media. Our method is based on asymptotic expansions and energy minimization procedures. The model of perfect interface is obtained at the first order, while an imperfect interface model is obtained at the next order. The conditions of imperfect contact, given in a parallel orthogonal curvilinear coordinate system, involve the interphase material properties, the first order displacement and traction vectors, and their derivatives. An example of implementation of the imperfect interface condition is given for a composite sphere assemblage.     

Effect of Particle Clusters on Carrier Flow Turbulence: A Direct Numerical Simulation Study

Experiments indicate that particle clusters that form in fluidized–bed risers can enhance gas-phase velocity fluctuations. Direct numerical simulations (DNS) of turbulent flow past uniform and clustered configurations of fixed particle assemblies at the same solid volume fraction are performed to gain insight into particle clustering effects on gas-phase turbulence, and to guide model development. The DNS approach is based on a discrete-time, direct-forcing immersed boundary method (IBM) that imposes no-slip and no-penetration boundary conditions on each particle’s surface. Results are reported for mean flow Reynolds number Re _p ?=?50 and the ratio of the particle diameter d _p to Kolmogorov scale is 5.5. The DNS confirm experimental observations that the clustered configurations enhance the level of fluid-phase turbulent kinetic energy (TKE) more than the uniform configurations, and this increase is found to arise from a lower dissipation rate in the clustered particle configuration. The simulations also reveal that the particle-fluid interaction results in significantly anisotropic fluid-phase turbulence, the source of which is traced to the anisotropic nature of the interphase TKE transfer and dissipation tensors. This study indicates that when particles are larger than the Kolmogorov scale (d _p ?>?η), modeling the fluid-phase TKE alone may not be adequate to capture the underlying physics in multiphase turbulence because the Reynolds stress is anisotropic. It also shows that multiphase turbulence models should consider the effect of particle clustering in the dissipation model.     

Suspension-based rheological modeling of crystallizing polymer melts

The applicability of suspension models to polymer crystallization is discussed. Although direct numerical simulations of flowing particle-filled melts are useful for gaining understanding about the rheological phenomena involved, they are computationally expensive. A more coarse-grained suspension model, which can relate the parameters in a constitutive equation for the two-phase material to morphological features, such as the volume fractions of differently shaped crystallites and the rheological properties of both phases, will be more practical in numerical polymer processing simulations. General issues, concerning the modeling of linear and nonlinear viscoelastic phenomena induced by rigid and deformable particles, are discussed. A phenomenological extension of linear viscoelastic suspension models into the nonlinear regime is proposed. A number of linear viscoelastic models for deformable particles are discussed, focusing on their possibilities in the context of polymer crystallization. The predictions of the most suitable model are compared to direct numerical simulation results and experimental data.