Material sensitivity analysis in homogenization of linear elastic composites

Summary  The main goal of the paper is to present theoretical aspects and the finite element method (FEM) implementation of the sensitivity analysis in homogenization of composite materials with linear elastic components, using effective modules approach. The deterministic sensitivity analysis of effective material properties is presented in a general form for an n-components periodic composite, and is illustrated by the examples of 1D as well as of 2D heterogeneous structures. The results of the sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials the method may be applied to computational optimization of engineering composites, to the shape-sensitivity studies and, after some probabilistic extensions, to stochastic sensitivity analysis of random composites. Received 10 November 2000; accepted for publication 24 April 2001     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

A classification of higher-order strain-gradient models – linear analysis

Summary  The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourth-gradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the second-gradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics. Received 13 June 2001; accepted for publication 6 November 2001     

Nonuniform warping torsion of orthotropic composite beams

Summary  The proposed work deals with the behaviour of orthotropic composite beams subjected to torsion, taking into consideration the effects due to nonuniform warping. We start from a one-dimensional approach based on the integral parameter method and consider any complex cross-sectional shape. The basic relations obtained lead to numerical computation based on the finite element method. As a special case of the general results obtained, an example of a sandwich beam is worked out in detail. Received 11 May 1999; accepted for publication 30 November 1999     

Elastic–plastic behavior of a semicircular frame being pressed against a rigid plane

As a simplified structural model, a semicircular frame is used to study the crashworthiness behavior of an aircraft fuselage. The quasi-static large elastic-plastic deformation of a semicircular frame in the process of its being pressed against a rigid ground is analyzed. First, based on the linear elastic assumption, the quasi-static large deformation contact process of the frame can be divided into three phases, i.e., point contact, line contact and post-buckling. By means of a shooting method, the relations between the displacement and contact force as well as the distribution of bending moment in the three phases are obtained. Then, by assuming an elastic, perfectly-plastic moment-curvature relationship for the semi-circular frame, the contact process is analyzed in detail to reveal the plastic collapse mechanism, the traveling of plastic hinge and the force-displacement relationship. In order to verify the analysis, a preliminary experiment was conducted, in which two types of half rings with clamped ends were pressed by a rigid plate. In addition, a numerical simulation is also conducted by employing ABAQUS to analyze both rectangular cross-sectional beam and I-beam. Finally, the theoretical predictions are compared with the experimental results and numerical solutions, showing that the elastic-plastic analysis can predict the contact process very well.     

Thermal instability of a porous medium with relaxation and inertia in the presence of Hall effects

Summary  The thermal instability of a Rivlin–Ericksen fluid in a porous medium is considered in the presence of a uniform vertical magnetic field to include the effect of Hall currents. For the case of stationary convection, the magnetic field has a stabilizing effect on the system, whereas the Hall current has a destabilizing effect on the system. The medium permeability has both stabilizing and destabilizing effects, depending on the Hall parameter M. The kinematic viscoelasticity has no effect on stationary convection. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. The magnetic field (and corresponding Hall currents) introduces oscillatory modes in the system, which would be nonexistent in their absence. The sufficient conditions for the nonexistence of overstability are also obtained. Received 20 May 1999; accepted for publication 8 March 2000     

A new numerical procedure for free vibrations of pretwisted plates

Summary  A numerical procedure is proposed for the analysis of free vibrations of pretwisted thin plates. An accurate strain–displacement relationship based on the thin-shell theory combined with the finite element method using triangular plate elements with three nodes and nine degrees of freedom for each node is utilized. The vibration characteristics of pretwisted thin plates with different twist rates and aspect ratios are studied. The numerical results are compared with the previous results obtained by various types of finite elements and by the Rayleigh–Ritz method. The effect of the twist rate on the vibration characteristics is studied briefly. Received 28 February 2001; accepted for publication 18 December 2001     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

Elastic–plastic contact force history and response characteristics of circular plate subjected to impact by a projectile

A new elastic–plastic impact–contact model is proposed in this paper. By adopting the principle of minimum acceleration for elastic–plastic continue at finite deformation, and with the aid of finite difference method, the proposed model is applied in the problem of dynamic response of a clamped thin circular plate subjected to a projectile impact centrally. The impact force history and response characteristics of the target plate is studied in detail. The theoretical predictions of the impact force and plate deflection are in good agreements with those of LDA experimental data. Linear expressions of the maximum impact force/transverse deflection versus impact velocity are given on the basis of the theoretical results. The project supported by the National Natural Science Foundation of China (10532020).     

Nonsimilar boundary-layer flow over a stationary permeable plate in a rotating fluid with a magnetic field

Summary  The nonsimilar boundary-layer flow and heat transfer over a stationary permeable surface in a rotating fluid in the presence of magnetic field, mass transfer and free stream velocity are studied. The parabolic partial differential equations governing the flow have been solved numerically by using a difference–differential method. For small streamwise distance, these partial differential equations are also solved by a perturbation technique with Shanks transformation. For uniform mass transfer, analytical solutions are obtained. The surface skin friction coefficients and the Nusselt number increase with the magnetic field, suction and streamwise distance from the leading edge of the plate except the skin friction coefficient in the y-direction which decreases with the increasing magnetic field. Received 4 December 2001; accepted for publication 24 September 2002     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

On the modeling and design of piezocomposites with prescribed properties

Summary  Piezoelectric transversely isotropic matrix containing spheroidal piezoelectric inclusions with different properties and of, generally, diverse aspect ratios is considered. A full set of ten effective electrostatic constants is derived, using the method of effective field. The case, when the inclusions are circular cylinders (fibers) is analyzed in detail. The results are compared with those of several earlier works. They constitute the theoretical framework for the design of piezocomposites with prescribed overall properties. Received 3 May 2001; accepted for publication 26 June 2001     

Modelling of a resonant inertial oscillation within a torus

Summary  We consider the air contained in a pneumatic tyre with the purpose of investigating its inertial oscillations. We model the tyre as a torus limited by a membrane in contact with the ground. According to this model, we prove that the flow within this torus may be considered as one at low Mach number and that it is ruled by oscillations of incompressible rotating fluid. Investigating such inertial oscillations, we show that the geostrophic oscillation is resonant, and we study the resonance phenomenon. Received 6 June 2000; accepted for publication 22 November 2000     

Wrinkling of thick orthotropic sandwich plates under general loading conditions

Summary  This contribution presents an efficient analytical model as well as a FE computation of the critical load, which leads to local stability failure (wrinkling) in sandwich structures. The analytical model assumes an orthotropic face layer and a thick transversely isotropic core. In the last section, a more general core material model is considered. Common core materials (foams and honeycombs) can be described with good accuracy within this model. The main advantage of the solution is the consideration of general loading conditions for the orthotropic face layer as well as in-plane deformations of the core. The results of the FE calculations and the analytical model are in good agreement with each other. Received 7 January 1999; accepted for publication 15 June 1999     

Modelling of functionally graded materials by numerical homogenization

Summary  In this contribution, the mechanical behaviour of different ZrO₂/NiCr 80 20 compositions is analysed and compared with experimental findings. The microwave-sintered material is found to possess a slightly dominant ceramic matrix for intermediate volume fractions. Its thermal expansion coefficient deviates from the rule of mixture. The modulus and the stress strain behaviour can be simulated by a numerical homogenization procedure, and the influence of residual stresses is found to be negligible. A newly introduced parameter (matricity) describes the mutual circumvention of the phases and is found to strongly control the stress level of the composite, globally as well as locally. Finally, a graded component and a metal/ceramic bi-material are compared for thermal as well as mechanical loading. Received 23 November 1999; accepted for publication 26 May 2000     

Sliding friction in the presence of ultrasonic oscillations: superposition of longitudinal oscillations

Summary  It was often observed that friction forces can be reduced significantly if ultrasonic oscillations are superposed to the macroscopic sliding velocity. This phenomenon can be used to improve machining processes by addition of ultrasonic vibration to tools or workpieces, and forms the basis for many processes of ultrasonic machining. On the other hand, ultrasonic vibrations can be used to generate motion. The thrusting force of ultrasonic motors is provided to the rotor through friction. In the present paper, a simple theoretical model for friction in the presence of ultrasonic oscillations is derived theoretically and validated experimentally. The model is capable of predicting the reduction of the macroscopic friction force as a function of the ultrasonic vibration frequency and amplitude and the macroscopic sliding velocity. Received 22 November 2000; accepted for publication 6 February 2001     

Influence of unsteady wake on a turbulent separation bubble

 An experimental study was made of turbulent separated and reattaching flow over a blunt body, where unsteady wake was generated by a spoked-wheel type of wake generator with cylindrical rods. The influence of unsteady wake was scrutinized by altering the rotation direction (clockwise and counter-clockwise) and the normalized passing frequency (0 ≤ St _H  ≤ 0.20). The Reynolds number based on the cylindrical rod was Re _d =375. A phase-averaging technique was employed to characterize the unsteady wake. The effect of different rotation directions, which gave a significant reduction of x _R , was examined in detail. The wall pressure fluctuations on the blunt body were analyzed in terms of the spectrum and the coherence. Received: 15 January 2001 / Accepted: 17 July 2001     

Micro/macromechanical plastic limit analyses of composite materials and structures

The load-bearing capacities of ductile composite materials and structures are studied by means of a combined micro/macromechanics approach. Firstly, on the microscopic scale, the aim is to get the macroscopic strength domains by means of the homogenization theory of micromechanics. A representative volume element (RVE) is selected to reflect the microstructures of the composite materials. By introducing the homogenization theory into the kinematic limit theorem of plastic limit analysis, an optimization format to directly calculate the limit loads of the RVE is obtained. And the macroscopic yield criterion can be determined according to the relation between macroscopic and microscopic fields. Secondly, on the macroscopic scale, by introducing the Hill's yield criterion into the kinematic limit theorem, the limit loads of orthotropic structures such as unidirectional fiber-reinforced composite structures are worked out. The finite element modeling of the kinematic limit analysis is deduced into a nonlinear mathematical programming with equality-constraint conditions that can be solved by means of a direct iterative algorithm. Finally, some examples are illustrated to show the application of the present approach. Project supported by the National Natural Science Foundation of China (No. 19902007), the National Foundation for Excellent Doctoral Dissertation of China (No. 200025), the Fund of the Ministry of Education of China for Returned Oversea Scholars and the Basic Research Foundation of Tsinghua University.     

An analysis of the plane-strain compression of a three-layer strip

Summary  The problem considered here is that of the plane-strain compression of a long symmetric strip of a three-layered material between rigid, parallel, rough plates. Two combinations of layers are examined: (a) a viscoplastic material placed between two layers of a rigid/perfectly plastic material, and (b) a rigid/perfectly plastic material placed between two layers of a viscoplastic material. Closed-form solutions are presented for each combination, and qualitative differences between these solutions and solutions obtained for homogeneous materials are discussed. A possible effect of asymptotic behaviour of the solution in the vicinity of maximum-friction surfaces on the general structure of the solution is mentioned. Received 24 July 2000; accepted for publication 6 February 2001     

On estimation methods for effective moduli of microcracked solids

Summary  Most of the conventional methods for estimating the overall elastic moduli of microcracked solids are defined based on the concept of effective medium or effective field. The formal similarity of these methods is examined in this paper. A one-to-one correspondence relation exists between the effective medium methods and the effective field methods in the sense that they yield identical results. In addition to the conventional estimation techniques, any other number of such approaches may be constructed by appropriately specifying the effective matrix compliance (or stiffness) tensor and the effective stress (or strain) field which a microcrack is assumed to be subjected to. To generate continuous spectra of new methods for estimating the effective elastic moduli, two simple and straightforward approaches are proposed, which contain one or two adjustable parameters in order to yield results of good accuracy. The discussion in this paper can be extended to other kinds of heterogeneous materials. Received 4 October 2000; accepted for publication 30 January 2001