Generalizations of the Navier–Stokes fluid from a new perspective

In this paper we study incompressible fluids described by constitutive equations from a different perspective, than that usually adopted, namely that of expressing kinematical quantities in terms of the stress. Such a representation is the appropriate way to express fluids like the classical Bingham fluid or fluids whose material moduli depend on the pressure. We consider models wherein the symmetric part of the velocity gradient is given by a “power-law” of the stress. This stress power-law model automatically satisfies the constraint of incompressibility without our having to introduce a Lagrange multiplier to enforce the constraint. The model also includes the classical incompressible Navier–Stokes model as a special subclass. We compare the stress power-law model with the classical power-law models and we show that the stress power-law model can, for certain parameter values, exhibit qualitatively different response characteristics than the classical power-law models and—on the other hand—it can be, for certain parameter values, used as a substitute for the classical power-law models. Using a stress power-law model we study several steady flow problems and obtain exact analytical solutions, and we argue that the possibility to obtain an exact analytical solution suggests, among others, that using these models provides an interesting alternative to the classical power-law models for which reasonable exact analytical solutions cannot be obtained. Finally, we discuss the issue of the choice of boundary conditions, and we show that the choice of boundary conditions has, at least for one of the problems that we study, a profound impact on the solvability of the boundary value problem.     

Pressure sensitivity and strength-differential effect of fiber-reinforced polymer matrix composites

Based on an energy approach, a simple nonlinear theory for the fiber-reinforced polymer matrix composite is developed. The yield stress and elastic moduli of the polymer matrix are taken to be pressure-dependent, and the ensuring pressure sensitivity and the strength-differential effect of the overall nonlinear response are then investigated. The stress-strain curves of the transversely isotropic composite are calculated at several levels of superimposed hydrostatic pressure, and it is found that the flow stress, except for a pure tension or compression along the fiber direction, can increase markedly under a high pressure. The strength-differential effect, which characterizes the different responses of the material between tension and compression, also exhibits a strong dependence on the loading mode and the applied pressure. The simple theory is shown to be accurate enough to compare favorably with an exact solution; it also yields results which are in close agreement with the experimental data of a grap hite/epoxy system at several selected pressures.     

A higher-order shear and normal deformation functionally graded plate model: some recent results

A higher-order shear and normal deformations plate theory is employed for stress analysis and free vibration of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. This idealization is required in order to obtain the closed-form solutions of some fundamental solid mechanics problems and also simplify the evaluation and development of numerical models of the structures made of FGMs. The material properties of FG plates such as Young’s moduli and material density are considered in this case to vary continuously in the thickness direction according to the volume fraction of constituents and mathematically modeled as exponential and power law functions. Poisson’s ratio is assumed to be constant. The effect of variation of material properties in terms of material grading index on the deformations, stresses, and natural frequency of FG plates is studied. The accuracy of the presented numerical solutions has been established with the solutions available of other models and the exact three-dimensional (3D) elasticity solutions.     

A class of generalized mid-point algorithms for the Gurson–Tvergaard material model

We investigate the generalized mid-point algorithms for the integration of elastoplastic constitutive equations for the pressure-dependent Gurson–Tvergaard yield model. By exact linearization of the algorithms and decomposition of the stresses into hydrostatic and deviatoric parts, a formula for explicitly calculating the consistent tangent moduli with the generalized mid-point algorithms is derived for the Gurson–Tvergaard model. The generalized mid-point algorithms, together with the consistent tangent moduli, have been implemented into ABAQUS via the user material subroutine. An analytical solution of the Gurson–Tvergaard model for the plane strain tension case is given and the performances of the generalized mid-point algorithms have been assessed for plane strain tension and hydrostatic tension problems and compared with the exact solutions. We find that, in the two problems considered, the generalized mid-point algorithms give reasonably good accuracy even for the case using very large time increment steps, with the true mid-point algorithm (α = 0·5) the most accurate one. Considering the extra non-symmetrical property of the consistent tangent moduli of the algorithms with α < 1, the Euler backward algorithm (α = 1) is, perhaps, the best choice.     

On the numerical integration of a class of pressure-dependent plasticity models

An unconditionally stable algorithm for the numerical integration of elastoplastic pressure-dependent constitutive relations is analysed in detail in this paper. The application of the method to plane stress problems, in which the out-of-plane strain component is not defined kinematically, is discussed. The tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations. The algorithm is applied to Gurson's constitutive model, some one-dimensional problems are solved, and comparisons with exact solutions are made. The paper closes with a numerical study of the necking of an axi-symmetric specimen using Gurson's plasticity model to describe the constitutive behaviour of the material.     

Effects of the orientational distribution of cracks in isotropic solids

The paper deals with the elastic characterisation of microcracked solids: we analyse dispersions of cracks with arbitrary non-random orientational distributions. Particular cases of angular distributions are given by cracks all oriented in a given direction or cracks uniformly random oriented in the space. A unified theory covers all the orientational distributions between the random and the parallel ones. The micromechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent elastic moduli of the overall material. The analysis has been performed in two-dimensional (2D) elasticity (plane stress and plane strain) with slit like cracks and in three-dimensional (3D) elasticity with planar circular cracks. The elastic behaviour of the microcracked solid depends upon the density of cracks and upon their orientational distribution. In particular, this study allows us to state that in two-dimensions the elastic behaviour of such a microcracked material is completely defined by one order parameter, which depends on the given angular distribution while the elastic characterisation in three-dimensions depends on two order parameters. The particular cases of isotropic orientations of cracks (both in 2D and in 3D) have been generalised to higher values of the cracks density by means of the method of the iterated homogenisation, which leads to some differential equations. Their solutions show that the equivalent elastic moduli depend exponentially on the cracks density.     

Numerical simulation of the Split Hopkinson Pressure Bar test technique for concrete under compression

The Split Hopkinson Pressure Bar [SHPB] technique has been commonly used to investigate concrete compressive response under high strain rate. However, there appears to be a lack of agreement in the literature about a number of critical issues pertaining to this test method. In this paper, computational simulation models are employed to critique the technique and obtain a better understanding of it. Influential parameters are identified and attempts are made to shed light on some controversial issues surrounding the interpretation of high strain rate test data. The results show that significantly different strain rates can be obtained from the same SHPB test depending on the method used to estimate the strain rate value. Furthermore, comparing the results of simulations with pressure-independent and pressure-dependent constitutive material models show that strength increases associated with strain rate are strongly, but not totally, reliant upon the confinement introduced by lateral inertial effects and the frictional condition at the interface between the pressure bars and the specimen. Based on these observations, it is argued that the so-called ‘rate-enhanced’ models that explicitly account for strength increases as a function of strain rate should not be used in numerical simulations that already account for the effects of lateral confinement, since such models would tend to double-count the strain rate effect.     

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli

Two real-domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency are investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher-order and higher floating-point precision numerical approximation, the latter being novel in this context. A general formula for higher-order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating-point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher-order approximations in standard double precision and increased quadruple precision. It is found that, as the order of the approximation increases, the elasticity moduli tend toward the analytical solution. Using higher floating-point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.     

Analytical elastic solutions for pressurized hollow cylinders with internal functionally graded coatings

The main objective of this work is to obtain analytical solutions for thick-walled cylinders subjected to internal and external pressure in which the entire wall is made of functionally graded material or of only a thin functionally graded coating present on the internal homogeneous wall. We assume that the materials are isotropic with constant Poisson’s ratio; as far as the Young modulus is concerned, we consider a power and an exponential. The proposed analytical solutions show the effects of the different profiles describing the graded properties of the materials on the stress and displacement fields; in addition, comparisons between graded coating and conventional homogeneous coating highlight the advantage of the graded material on the interface stress reduction. Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties. This investigation permits us to optimize the elastic response of cylinders under pressure by tailoring the thickness variation of the elastic properties and to reduce manufacturing costs given by the technological limitations that occur to produce entire functionally graded walls.     

阻尼材料复模量测试中的动力学分析

流变振动仪利用等截面杆的纵向简谐振动来测定材料的阻尼，十分方便和有效。本文把振动杆件作为一个连续体，指出响应（一端的应力）和激励（另一端的应变）之比（作者称之为试件复模量）不仅反映了试件的材料性质，而且与试件的动特性参数（如质量密度）和几何参数密切相关；提出了由实验得到的试件复模量得出材料复模量的计算方法。剪切简谐振动试验也有类似情况。     

On two micromechanics theories for determining micro–macro relations in heterogeneous solids

The average-field theory and the homogenization theory are briefly reviewed and compared. These theories are often used to determine the effective moduli of heterogeneous materials from their microscopic structure in such a manner that boundary-value problems for the macroscopic response can be formulated. While these two theories are based on different modeling concepts, it is shown that they can yield essentially the same effective moduli and boundary-value problems. A hybrid micromechanics theory is proposed in view of this correspondence. This theory leads to a more accurate computation of the effective moduli, and applies to a broader class of microstructural models. Hence, the resulting macroscopic boundary-value problem gives better estimates of the macroscopic response of the material. In particular, the hybrid theory can account for the effects of the macrostrain gradient on the macrostress in a natural manner.     

Relaxation behavior of neat and particulate filled polypropylene in uniaxial and multiaxial compression

The recently developed confined compression test was used to measure the viscoelastic bulk and shear relaxation moduli of neat, glass bead and talc filled polypropylene. In this paper further modifications of the test are introduced and a criterion for the assessment of the quality of experimental data is suggested. As expected, shear as well as the bulk relaxation moduli were found to increase with the addition of particles. In order to determine the pressure sensitivity of the material, unconfined compression tests were also performed and compared with the confined tests through interconversion of the measured moduli. In agreement with earlier results on other polymers, it turned out that the relaxation response is significantly retarded at higher confinement levels. It is shown that the effect of filler particles on the long-term behavior depends on the specific uniaxial or multiaxial stress state. Poisson’s ratio was calculated by interconversion from the bulk and shear relaxation modulus; these results show that with a single test in the confined configuration, a complete viscoelastic characterization of the material can be obtained.     

Creep response of GFRP–concrete hybrid structures: Application to a footbridge prototype

Glass fibre reinforced polymer (GFRP) pultruded profiles have been increasingly used in civil engineering structural applications in the past few decades owing to their high strength, low weight and corrosion resistance. Nevertheless, the low material moduli, which makes design most often governed by deformability and instability phenomena, the brittle failure mechanisms and the high initial costs, have been delaying their widespread use. Hybrid GFRP–concrete structural solutions have been proposed to overcome the aforementioned limitations, namely the low material moduli. Furthermore, GFRP material creep models suggest that such hybrid structures may reduce the creep deformations when compared to full GFRP structures. In this context, this paper presents experimental and analytical investigations about the creep behaviour of a hybrid GFRP–concrete footbridge comprising two I-shaped GFRP pultruded profiles and a thin deck made of steel fibre reinforced self-compacting concrete (SFRSCC). The experiments comprised flexural creep tests on a 6.0 m long footbridge prototype subjected to a uniformly distributed load for up to 2642 h, during which deflections and axial deformations were monitored. In order to assess the influence of loading and environmental conditions on the creep behaviour of the structural system, the prototype was tested for three different combinations of load levels and seasons. Experimental results showed that (i) GFRP–concrete hybrid structures lead to a considerable decrease of the creep deformations of GFRP structures and that (ii) environmental conditions significantly influence the viscoelastic response of these hybrid structures. The models proposed, based on the creep response of the constituent materials, were able to predict the observed structural response for the different load levels and environmental conditions with very good accuracy. Therefore, they are proposed to predict the long-term response of GFRP–concrete structures instead of empirical models based on short-term experimental data.     

Nonlinear large deflection buckling analysis of compression rod with different moduli

Many novel materials exhibit a property of different elastic moduli in tension and compression. One such material is graphene, a wonder material, which has the highest strength yet measured. Investigations on buckling problems for structures with different moduli are scarce. To address this new problem, first, the nondimensional expression of the relation between offset of neutral axis and deflection curve is derived based on the phased integration method, and then using the energy method, load–deflection relation of the rod is determined; second, based on the improved constitutive model for different moduli, large deformation finite element formulations are developed, and combined with the arc-length method, finite element iterative program for rods with different moduli is established to obtain buckling critical loads; third, material mechanical properties testing of graphite, which is the raw material of graphene, is performed to measure the tensile and compressive elastic moduli; moreover, buckling tests are also conducted to investigate the buckling behavior of this kind of graphite rod. By comparing the calculation results of the energy method and finite element method with those of laboratory tests, the analytical model and finite element numerical model are demonstrated to be accurate and reliable. The results show that it may lead to unsafe results if the classic theory was still adopted to determine the buckling loads of those rods composed of a material having different moduli. The proposed models could provide a novel approach for further investigation of nonlinear mechanical behavior for other structures with different moduli.     

A simple viscoelastic model for soft tissues the frequency range 6-20 MHz

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

Nanorheology of regenerated silk fibroin solution

We have investigated the rheological properties of regenerated silk fibroin (RSF), a viscoelastic material at micro and nano length scales, by video microscopy. We describe here the principles and technique of video microscopy as a tool in such investigations. In this work, polystyrene beads were dispersed in the matrix of RSF polymer and the positions of the embedded beads diffusing were tracked using video microscopy. An optical tweezer was used to transport and locate the bead at any desired site within the micro-volume of the sample, to facilitate the subsequent free-bead video analysis. The position information of the beads was used to obtain the time dependant mean squared displacement (MSD) of the beads in the medium and hence to calculate the dynamic moduli of the medium. We present here the results of rheological measurements of the silk polymer network in solution over a frequency range, whose upper limit is the frame capture rate of our camera at full resolution. The technique is complementary to other microrheological techniques to characterize the material, but additionally enables one to characterize local inhomogeneities in the medium, features that get averaged out in bulk characterization procedures.     

A simple viscoelastic model for soft tissues in the frequency range 6-20 MHz.

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

On the use of a spectrum-based model for linear viscoelastic materials

A new spectrum-based model for describing the behavior of time-dependent materials is presented. In this paper, unlike most prior modeling techniques, the time-dependent response of viscoelastic materials is not expressed through the use of series. Instead, certain criteria have been imposed to select a spectrum function that has the potential of describing a wide range of material behavior. Another consequence of choosing the spectrum function of the type used in this paper is to have a few closed form analytic solutions in the theory of linear viscoelasticity. The Laplace transform technique is used to obtain the necessary formulae for viscoelastic Lame' functions, relaxation and bulk moduli, creep bulk and shear compliance, as well as Poisson's ratio. By using the Elastic–Viscoelastic Correspondence Principle (EVCP), material constants appearing in the proposed model are obtained by comparing the experimental data with the solution of the integral equation for a simple tensile test. The resulting viscoelastic functions describe the material properties which can then be used to express the behavior of a material in other loading configurations. The model's potential is demonstrated and its limitations are discussed.     

Process design for the production of a ceramic-like body from recycled waste glass

The influence of fabrication variables on the sintering behaviour of a ceramic-like body containing 90% recycled waste glass was inferred from measurements of some of the fired properties (moduli of rupture and elasticity, firing shrinkage, bulk density and porosity). Interpretation of these results in terms of viscous sintering theory indicates the relative influence on the sintering behaviour of factors such as particle size and distribution, clay binder content and plasticity, pressing pressure, heating/cooling rate, firing temperature and time, thus enabling the fabrication variables to be optimized. Comparison of the physical properties of the resulting glass-based bodies with those of commercial ceramic tile bodies indicates that the glass-based bodies are very comparable with the best ceramic tiles tested, and considerably better than several commercially-produced clay-based bodies.