The simulation of oscillatory pipe flow on an analogue computer

Summary A formulation of equations governing oscillatory flow of a fluid in a circular pipe is given. The equations are then cast in a form suitable for analogue computation and a corresponding programming diagram suitable for use with an EAI 680 Analogue Computer is provided. Results have been obtained forNewtonian flow for a range ofReynolds numbers and for combinations of the realReynolds number, and the ratio of elastic to viscous forces in the case of linear viscoelastic fluids.The displacement profiles given in this work do not depend upon specific rheological models and are therefore of universal validity for linear fluids.     

A stability investigation for an incompressible simple fluid with fading memory

The nonlinear equations of motion for an incompressible simple fluid, occupying a fixed bounded container, are formulated on the basis of the finitelinear viscoelasticity theory for materials with fading memory; this formal boundary-initial value problem is then viewed as a nonlinear abstract evolution equation on a certain Hilbert space. It is shown that a linearized version of this evolution equation is associated with a linear dynamical system on this Hilbert space, and several results for stability and asymptotic behavior for this linearized problem are proved through the use of Liapunov stability methods. On the assumption that the original nonlinear evolution equation also is associated with some dynamical system on the same space, it is shown that the rest condition of the fluid is stable and all motions are bounded. The Liapunov function employed for this purpose can be interpreted as a mechanical energy function for the fluid.E. F. Infante's work was supported in part by the U.S. Office of Naval Research (grant N0014-76-C-0278P002), the U.S. National Science Foundation (grant MCS-76-07247 A03), and the U.S. Army Research Office (grant AROD 31-124-73-G-130); that of J. A. WALKER was supported in part by the U.S. National Science Foundation (grant ENG76-81570) and the U.S. Air Force (grant AFOSR-76-3063A).     

The influence of mechanical constraint upon the switching of a ferroelectric memory capacitor

The role of mechanical constraint upon the switching response of a ferroelectric thin film memory capacitor is explored. The memory capacitor is represented by a two dimensional ferroelectric island whose non-linear behaviour is modelled by a crystal plasticity constitutive law within the finite element method. The switching response of the device, in terms of remnant charge storage, is determined as a function of geometry and constraint. Various types of constraint on the ferroelectric capacitor are considered, including the presence of a silicon dioxide passivation layer, a silicon substrate and metallic electrodes. The effect of the relative resistance to 90 degree switching and 180 degree switching is also explored in a tetragonal ferroelectric device. Throughout the study, the finite element calculations are compared with the behaviour of a material element subjected to various degrees of mechanical constraint.     

A computer method for simulating service loads

A recently presented study addressed the problem of analyzing field data that are best characterized as nonstationary stochastic signais. The analysis method hypothesizes that the nonstationary signal consists of two stationary signals, which belong to different populations, occurring consecutively according to a suitable probabilistic model. The analysis procedure involves the following: segmenting the time history, estimating the population of each segment, estimating the power spectrum of each segment, averaging the power spectra which belong to each population, presenting the power spectra via parameters of digital filters (which shape white noise sequences into sequences with the measured power spectra), and measuring the parameters of the probabilistic model. In this paper a simulation method is presented that uses the results of the analysis method mentioned above to create a sequence that simulates the statistical characteristics of the nonstationary field data. This simulation method is designed to be efficiently implemented on a general-purpose computer of any size, including micros. First, a review of the stochastic model is given. Then the steps of simulation are presented: generating a white sequence on the digital computer, generating the probabilistic model, and developing an algorithm for using digital filters in shaping the power spectra. Sample results are shown to reflect the soundness of the procedure. This simulation method can prove useful in computer studies of the fatigue of mechanical components under field loading. Since it is exactly reproducible in different laboratories, this method can also serve in comparison studies of fatigue-life prediction procedures. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

A note on the crack analogue model for fretting fatigue

The contact of a flat punch over a half-plane under constant normal loads, and oscillating tangential and bulk loads is studied, with the aim to improve the crack analogue (CA) model for fretting fatigue (FF) (Acta Mater. 46(9) (1998) 2955). New analytical results are found for a range of conditions, finding the effect of bulk loads and of partial slip which were not considered in the original CA model. Implications for the FF life assessment methodology are found to be significant.     

Model of electrothermal convection of an ideal dielectric in a horizontal capacitor

The behavior of a nonuniformly heated dielectric in a variable electric field of a horizontal capacitor is studied. The action of the dielectrophoretic mechanism of instability associated with the temperature dependence of the dielectric permeability is considered. The model of electroconvection of the ideal dielectric is obtained for the boundary conditions corresponding to the no-flow condition on the rigid surface. Nonlinear flow regimes are investigated in the case of microgravity. A map of the regimes is constructed.     

A Preisach type model for temperature driven hysteresis memory erasure in shape memory materials

We establish the well-posedness and thermodynamic consistency of a variational inequality modeling temperature-induced memory erasure in shape memory materials. It is shown that the input–output operator is continuous with respect to uniform convergence.     

A three-dimensional constitutive model for shape memory alloys

Shape memory alloys (SMAs) are materials that, among other characteristics, have the ability to present high deformation levels when subjected to mechanical loading, returning to their original form after a temperature change. Literature presents numerous constitutive models that describe the phenomenological features of the thermomechanical behavior of SMAs. The present paper introduces a novel three-dimensional constitutive model that describes the martensitic phase transformations within the scope of standard generalized materials. The model is capable of describing the main features of the thermomechanical behavior of SMAs by considering four macroscopic phases associated with austenitic phase and three variants of martensite. A numerical procedure is proposed to deal with the nonlinearities of the model. Numerical simulations are carried out dealing with uniaxial and multiaxial single-point tests showing the capability of the introduced model to describe the general behavior of SMAs. Specifically, uniaxial tests show pseudoelasticity, shape memory effect, phase transformation due to temperature change and internal subloops due to incomplete phase transformations. Concerning multiaxial tests, the pure shear stress and hydrostatic tests are discussed showing qualitatively coherent results. Moreover, other tensile–shear tests are conducted modeling the general three-dimensional behavior of SMAs. It is shown that the multiaxial results are qualitative coherent with the related data presented in the literature.     

A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.     

A finite element method for liquid with memory

Steady, two-dimensional flows of viscoelastic liquid with memory are analyzed by means of a nonlinear integral constitutive equation, novel streamlined finite elements, Galerkin's method of weighted residuals, and Newton iteration. By their relation to streamlines the new elements allow, for the first time, full Newton iteration of the algebraic equation set to which the governing integrodifferential system reduces. Streamlines are computed simultaneously with velocity components and pressure, the primary unknowns; to track the history of liquid particles the deformation equations are solved analytically in a Protean system of coordinates that conforms to the streamlines. In the cases studied the Newton iteration converges quadratically to a creeping flow state of Weissenberg number (product of upstream wall shear rate and relaxation time of the liquid) up to about 20 in channel flow, 10 in film flow, and 2 in die-swell flow. Shear-thinning shrinks the domain of convergence; it also reduces die-swell. A slip boundary condition in the vicinity of the contact line extends the domain of convergence.     

An analogue study for flame flickering

An analogue experiment is proposed to simulate flame flickering comprising a free ascending column fed on its side with a light gas (helium) emerging from a vertical slot in ambient air. The convective motion of the helium jet is considered to represent the motion of burnt gases of buoyant jet flames. The helium jet is accelerated by buoyancy effects and the flow field is similar to that of burnt gases observed for real buoyant flames. The vertical velocity profile of the steady helium jet is measured at different vertical distances. The unsteady helium jet is also studied by measuring the instability frequency as a function of ambient pressure at different injection flow rates, and by analyzing the tomography images of the helium jet. The instability morphology is the same as that observed on real buoyant flames. We conclude that this type of instability can be approximately characterized by the maximum vertical velocityu _max, and the distance betweenu _max in the helium ascending column andu = o in the ambient air. For this type of instability the local vorticity is proportional to which can be influenced by gravity and ambient pressure. Theoretical prediction of the instability frequency as a function of gravity and ambient pressure has been obtained, and is in good agreement with the experimental results.List of symbols C ₁,C ₂ constants - F instability frequency - F _c critical frequency - F _m the most amplified frequency - F (K, ) function defined in (11) - g gravitational acceleration - g reduced gravity acceleration g(₀-^*)/^* - k real wave number of the disturbance - K reduced wave numberK=2k - K _c reduced wave number of the critical instability mode - K _m nondimensional wavenumber of the most amplified mode - L vertical characteristic length (in x direction) - P ambient pressure - u local vertical buoyant velocity (inx direction) - u _max local maximum vertical velocity - v local velocity component iny direction (horizontal) - V ₀ injection velocity of helium (iny direction) - x vertical distance measured from the leading edge of boundary layer - y horizontal distance measured from the exit plane of the vertical slot - Z(K, ) function defined in equation (11) Greek symbols distance betweenu _max in the helium ascending column andu = o in the ambient air - - wavelength of instability - _c critical wavelength - _m the most amplified wavelength - ^* helium density at slot exit - ₀ ambient air density - ^* helium dynamic viscosity at slot exit - v ^* helium kinematic viscosity at slot exit - complex number presented in disturbancee ^i(kx+t) - _i imaginary part of , representing the amplification rate of disturbance - _r real part of , where ( _r /k) represents the group velocity - reduced complex number of , defined      

A thermo-mechanically coupled field model for shape memory alloys

The impressive properties of shape memory alloys are produced by means of solid-to-solid phase transformations where thermal effects play an important role. In this paper, we present a model for polycrystalline shape memory alloys which takes full thermo-mechanical coupling into account. Starting from the equations of the first and the second law of thermodynamics, we derive evolution equations for the volume fractions of the different martensitic variants and a related equation for heat conduction. A thermodynamic analysis allows to formulate a complete expression for the dissipation caused by phase transformation and heat flux. This allows to model the experimentally well-documented transformation fronts in tension tests by a finite element scheme without further assumptions. Additionally, the number of required model parameters is very small in comparison with phenomenological approaches. Numerical examples are presented which show a good agreement with experimental observations.     

A thermodynamic finite-strain model for pseudoelastic shape memory alloys

A thermodynamic finite-strain model describing the pseudoelastic response of shape memory alloys is proposed. The model is based on a self-consistent Eulerian theory of finite deformations using the logarithmic rate. Purely elastic material response is derived from a hyperelastic potential. The mass fraction of martensite is introduced as internal state variable to indicate the thermomechanical state of the phase transforming material. The evolution of martensite is governed by a kinetic law which is derived from the Helmholtz free energy of the two-phase solid and takes the heat generated during phase transition into account. The material model is implemented into a finite element code in an updated Lagrangian scheme and calibrated to experimental data. Simulations under different loading conditions illustrate the characteristics of the model.     

形状记忆合金结构拓扑优化设计方法研究

形状记忆合金由于其优良的力学特性得到了广泛关注,并形成了一系列具有变革性的创新应用.为了充分提升形状记忆合金结构的力学性能,提出了一种基于实体各向同性材料惩罚模型SIMP (Solid Isotropic Materi-al with Penalization)的形状记忆合金结构拓扑优化方法.基于ZM宏观唯象本构模型,考虑形状记忆合金材料特性,对拓扑优化过程中引入的中间密度材料的奥氏体和马氏体弹性模量以及相变转变应力进行插值.同时,考虑形状记忆合金本身的材料非线性和结构在大变形下的几何非线性效应,以获得准确的力学响应.采用三密度场法来避免最终设计结果出现的棋盘格现象、网格依赖性和大量中间密度单元.利用超单元法来改善由于低密度单元引起的非线性有限元分析过程的数值不稳定问题.利用伴随法对优化模型中的响应函数进行灵敏度分析.最后,通过二维和三维的数值算例验证了本文的优化设计方法,结果表明本文提出的拓扑优化框架能够对预期性能的形状记忆合金结构方案进行求解.     

A constitutive model for shape memory alloys accounting for thermomechanical coupling

This paper presents a generalized Zaki-Moumni (ZM) model for shape memory alloys (SMAs) [cf. Zaki, W., Moumni, Z., 2007a. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 55, 2455-2490 accounting for thermomechanical coupling. To this end, the expression of the Helmholtz free energy is modified in order to derive the heat equation in accordance with the principles of thermodynamics. An algorithm is proposed to implement the coupled ZM model into a finite element code, which is then used to solve a thermomechanical boundary value problem involving a superelastic SMA structure. The model is validated against experimental data available in the literature. Strain rate dependence of the mechanical pseudoelastic response is taken into account with good qualitative as well as quantitative accuracy in the case of moderate strain rates and for mechanical results in the case of high strain rates. However, only qualitative agreement is achieved for thermal results at high strain rates. It is shown that this discrepancy is mainly due to localization effects which are note taken into account in our model. Analyzing the influence of the heat sources on the material response shows that the mechanical hysteresis is mainly due to intrinsic dissipation, whereas the thermal response is governed by latent heat. In addition, the variation of the area of the hysteresis loop with respect to the strain rate is discussed. It is found that this variation is not monotonic and reaches a maximum value for a certain value of strain rate.     

A computer model of the two-roll mill for generalised Newtonian flow

We investigate the flow of a generalised Newtonian liquid between two contra-rotating cylinders of equal size — the so called two-roll mill problem. A finite element method is used to obtain solutions for the case of a Carreau model and for a power law model. Consideration is given to the influence of shearthinning on the flow pattern around the cylinders. Results are presented for different speeds of rotation of the cylinders and for various values of fluid parameters. A comparison is made with the analytical solution of Jeffrey.