Homogenization of elastic–viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes

This paper deals with the prediction of the macroscopic behavior of a multiphase elastic–viscoplastic material. The proposed homogenization schemes are based on an interaction law postulated by Molinari et al. [Molinari, A., Ahzi, S., Kouddane, R. 1997. On the self-consistent modelling of elastic–plastic behavior of polycrystals. Mech. Mater., 26, 43–62]. Self-consistent schemes are developed to describe the behavior of disordered aggregates. The Mori-Tanaka approach is used to capture the behavior of composite materials, where one phase can be clearly identified as the matrix. The proposed schemes are developed within a general framework where compressible elasticity and anisotropy of the materials are taken into account. Inclusions can have various shapes and orientations. Illustrations of the homogenization procedure are given for a two-phase composite materials. Comparisons between results of the literature and predictions based on the interaction law are performed and have demonstrated the efficiency of the proposed homogenization schemes.     

Quasi-steady prediction of coupled bending–torsion flutter under rotating stall

A method is presented in this paper to predict cascade flutter under subsonic stalled flow condition in a quasi-steady manner. The ability to predict the occurrence of aeroelastic flutter is highly important from the compressor design point of view. In the present work, the well known Moore–Greitzer compression system model is used to evaluate the flow under rotating stall and the linearized aerodynamic theory of Whitehead is used to estimate the blade loading. The cascade stability is then predicted by solving the structural model, which is posed as a complex eigenvalue problem. The possibility of occurrence of flutter in both bending and torsional modes is considered and the latter is found to be the dominant one, under subsonic stalled flow, for a large range of frequency ratios examined. It is also shown that the design of compressor blades at frequency ratios close to unity may result in rapid initiation of torsional flutter in the presence of stalled flow. A frequency ratio of 0.9 is primarily emphasized for most part of the study as many interesting features are revealed and the results are physically interpreted. Roughly a pitchfork pattern of energy distribution appears to occur between bending mode and torsional mode which ensures that only one flutter mode is possible at any instant in time. A bifurcation from bending flutter to torsional flutter is shown to occur during which the frequency of the two vibrating modes appear to coalesce for a very short period of time.     

Flutter analysis of a viscoelastic tapered wing under bending–torsion loading

The dynamic stability of a tapered viscoelastic wing subjected to unsteady aerodynamic forces is investigated. The wing is considered as a cantilever tapered Euler–Bernoulli beam. The beam is made of a linear viscoelastic material where Kelvin–Voigt model is assumed to represent the viscoelastic behavior of the material. The governing equations of motion are derived through the extended Hamilton’s principle. The resulting partial differential equations are solved via Galerkin’s method along with the classical flutter investigation approach. The developed model is validated against the well-known Goland wing and HALE wing and good agreement is obtained. Different solution methods, namely; the k method, the p-k method, and the flutter determinant method are compared for the case of elastic wing. However, when the viscoelastic damping is introduced, the k and p-k methods become less effective. The flutter determinant method is modified and employed to carry out non-dimensional parametric study on the Goland wing. The study includes the effects of parameters such as the taper ratio, the density ratio, the viscoelastic damping of wing structure and many other parameters on the flutter speed and flutter frequency. The study reveals that a tapered wing would be more dynamically stable than a uniform wing. It is also observed that the viscoelastic damping provides wider stability region for the wing. The investigation shows that the density ratio, bending-to-torsion frequency ratio, and the radius of gyration have significant effects on the dynamic stability of the wing. Based on the obtained results, a wing with an elastic center and inertial center that are located closer to the mid-chord would be more dynamically stable.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

Determining material true stress–strain curve from tensile specimens with rectangular cross-section

The uniaxial true stress logarithmic strain curve for a thick section can be determined from the load–diameter reduction record of a round tensile specimen. The correction of the true stress for necking can be performed by using the well-known Bridgman equation. For thin sections, it is more practical to use specimens with rectangular cross-section. However, there is no established method to determine the complete true stress–logarithmic strain relation from a rectangular specimen. In this paper, an extensive three-dimensional numerical study has been carried out on the diffuse necking behaviour of tensile specimens made of isotropic materials with rectangular cross-section, and an approximate relation is established between the area reduction of the minimum cross-section and the measured thickness reduction. It is found that the area reduction can be normalized by the uniaxial strain at maximum load which represents the material hardening and also the section aspect ratio. Furthermore, for the same material, specimens with different aspect ratio give exactly the same true average stress–logarithmic strain curve. This finding implies that Bridgmans correction can still be used for necking correction of the true average stress obtained from rectangular specimens. Based on this finding, a method for determining the true stress–logarithmic strain relation from the load–thickness reduction curve of specimens with rectangular cross-section is proposed.     

Principal resonance bifurcation of bending–torsion coupling of aero-engine compressor blade with assembled clearance under lateral displacement excitation of rotor shaft

The principal resonance of bending–torsion coupling of a compressor blade with an assembled clearance and a cubic structural nonlinearity, subjected to the lateral displacement excitation of rotor shaft and aerodynamic loads, is analyzed to explore the topology transformation of amplitude–frequency response along with the changes of the physical parameters. The bifurcation equation of the first-order principal resonance response is derived from using the averaging method. The transition set and the bifurcation figures of the response solution are obtained by the singularity theory. The effects of the main physical parameters of the system on the topology transformation of amplitude–frequency response are discussed.     

Nonlinear aeroelastic stability analysis of wind turbine blade with bending–bending–twist coupling

In this study, the nonlinear aeroelastic stability of wind turbine blade with bending–bending–twist coupling has been investigated for composite thin-walled structure with pretwist angle. The aerodynamic model used here is the differential dynamic stall nonlinear ONERA model. The nonlinear aeroelastic equations are reduced to ordinary equations by Galerkin method, with the aerodynamic force decomposition by strip theory. The nonlinear resulting equations are solved by a time-marching approach, and are linearized by small perturbation about the equilibrium point. The nonlinear aeroelastic stability characteristics are investigated through eigenvalue analysis, nonlinear time domain response, and linearized time domain response.     

Basic solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in?piezoelectric materials

The solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials were given by using the generalized Almansi’s theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack,the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.     

A nongray analysis of radiating–convecting rectangular plate-fin arrays

A complete analysis of heat transfer in rectangular fin arrays has been conducted taking into consideration convection and radiation at all surfaces, as well as radiative exchange between the fins and all neighboring surfaces. The analysis assumes diffuse nongray surfaces and uses the spectral optical properties of stainless steel AISI 430. Three different gray models were proposed in order to assess the effect of the gray assumption on the heat-transfer characteristics. Results are presented for the temperature and radiative-flux distributions along the fin, the radiative flux along the base, the contribution of the radiative component to the overall heat transfer, and the effectiveness of the fin array. Significant deviations in some of these results were found between the nongray model and the gray models. In general, convection was found to be the more effective mode of heat transfer in fin arrays and the effectiveness of the array decreases as the contribution of the radiative component increases.     

Hill’s class of compressible elastic materials and finite bending problems: Exact solutions in unified form

Hill (1978) proposed a natural extension of Hooke’s law to finite deformations. With all Seth-Hill finite strains, Hill’s natural extension presents a broad class of compressible hyperelastic materials over the whole deformation range. We show that a number of known Hookean type finite hyperelasticity models are included as particular cases in Hill’s class and that Bell’s and Ericksen’s constraints may be derived as natural consequences from Hill’s class subjected to internal constraints. Also we present a unified study of finite bending problems for elastic Hill materials. To date exact results are available for certain particular classes of compressible elastic materials, which do not cover Hill’s class. Here, with a novel idea of circumventing the strong nonlinearity we show that it is possible to derive exact solutions in unified form for the whole class of elastic Hill materials. Reduced results are also given for cases subjected to internal constraints.     

Experimental study of non-uniform inlet conditions and three-dimensional effects of vertical air–water two-phase flow in a narrow rectangular duct

To study the three-dimensional interfacial structure development in vertical two-phase flow, air–water upflow experiments were performed in a rectangular duct. Various non-uniform two-phase profiles were created by injecting air from individually controlled spargers at the duct inlet into uniformly injected water flow. A four-sensor conductivity probe was used to measure local void fraction, interfacial area concentration, bubble velocity and Sauter mean diameter at three axial locations to record the development of two-phase parameters. Experimental results showed that the lateral development across the wider dimension of the duct was significant with a non-uniform inlet profile when compared to a uniform inlet profile. It is postulated that lift, wall and turbulent forces are the major contributors to the lateral distribution of the two-phase interfacial structures making this an useful experiment for benchmarking three-dimensional two-fluid models. In examining the interfacial area, the shearing-off of group 1 bubbles (defined as the smaller spherical and distorted bubbles) from the skirt region of group 2 bubbles (defined as the bigger cap and churn bubbles), the coalescence of group 2 bubbles due to wake entrainment, and random collision are the major source and sink mechanisms of interfacial area concentration.     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

Mechanical behaviour of carbon nanotubes under combined twisting–bending

The main objective of this paper is to investigate the mechanical behaviour (strength and stiffness) of carbon nanotubes (CNTs) under combinations of bending and twisting. In order to achieve this goal, molecular dynamics (MD) simulations of bended and twisted CNTs are performed. The LAMMPS code is used, the AIREBO potential is considered for CC bonds, the temperature is kept at 300 K and incremental bending and twisting rotations are imposed to the CNT. Two types of CNTs are analyzed, including zig-zag (8,0) and armchair (5,5) CNTs with similar radius and length. The CNTs are also analyzed for pure bending and pure twisting. The main results are shown in the form of diagrams of energy and moment against imposed rotations. Some relevant conclusions are drawn concerning the influence of loading (bending and twisting) on the stiffness, strength and failure of CNTs: namely, it is concluded that armchair CNTs possess higher strength and fracture toughness under twisting–bending loading than zigzag CNTs; additionally, it is found that both CNTs (armchair and zigzag) still support moderate-to-high bending levels without failure after being extremely twisted and torsionally buckled, even for twisting angles four times those corresponding to torsional buckling; finally, the results prove that CNTs, mostly armchair ones, exhibit very high twisting–bending stiffness and strength and can be used with confidence as torsional spring elements in nanoelectromechanical systems (NEMS).     

Ratchetting under tension–torsion loadings: experiments and modelling

This paper is concerned with the mechanical behaviour of 316 austenitic stainless steel under multiaxial loadings and particular attention is paid to ratchetting under tension–torsion non-proportional loadings. First, a series of uniaxial tests and biaxial tests has been carried out in order to calibrate five different cyclic plasticity models based on an isotropic hardening rule and a non-linear kinematic hardening rule. It is shown that this class of models gives quite good agreement between the experimental and numerical results. Second, another series of ratchetting tests has been carried out under tension–torsion loadings in order to test the prediction capacities of the previous models. It is shown that whereas the models have been calibrated with similar loading paths, four of the five selected models give poor predictions.     

Dynamics of void collapse in shocked energetic materials: physics of void–void interactions

This work presents the response of a porous energetic material subjected to severe transient loading conditions. The porosities, represented by voids, entirely change the response of an otherwise homogeneous material. The variations in terms of energy distribution and maximum temperature reached in the material in the presence of heterogeneities (voids) but in the absence of chemical reactions are studied. This study also accounts for void–void interactions to enhance the understanding of the localization of energy in the material. It is observed that relative position of voids can have important consequence on energy distribution as well as rise in temperature of the energetic material. The relative position of voids further influences the interaction of secondary shock waves generated during the collapse of one void with the downstream voids. This interaction can either enhance or diminish the strength of the shock depending on the location of downstream voids. This work also reveals that the findings from mutual void–void interactions can be used to study systems with multiple voids. This is shown by analyzing systems with 10–25 % void volume fraction. The effect of void–void interactions are connected to the overall response of a chemically inert porous material to imposed transient loads.     

A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A. The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.     

Homogenization of a Hele–Shaw Problem in Periodic and Random Media

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.     

Homogenization of Weakly Coupled Systems of Hamilton–Jacobi Equations with Fast Switching Rates

We consider homogenization for weakly coupled systems of Hamilton–Jacobi equations with fast switching rates. The fast switching rate terms force the solutions to converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and the rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.     

Superelastic behavior in tension–torsion of an initially-textured Ti–Ni shape-memory alloy

A recently-developed crystal-mechanics-based constitutive model for polycrystalline shape-memory alloys [J. Mech. Phys. Solids 49 (2001) 909] is shown to quantitatively predict the superelastic response of an initially-textured Ti–Ni alloy in (i) a proportional-loading, combined tension–torsion experiment, as well as (ii) a path-change, tension–torsion experiment.