A two-dimensional problem for a fibre-reinforced anisotropic thermoelastic half-space with energy dissipation

The theory of thermoelasticity with energy dissipation is employed to study plane waves in a fibre-reinforced anisotropic thermoelastic half-space. We apply a thermal shock on the surface of the half-space which is taken to be traction free. The problem is solved numerically using a finite element method. Moreover, the numerical solutions of the non-dimensional governing partial differential equations of the problem are shown graphically. Comparisons are made with the results predicted by Green–Naghdi theory of the two types (GNII without energy dissipation) and (GNIII with energy dissipation). We found that the reinforcement has great effect on the distribution of field quantities. Results carried out in this paper can be used to design various fibre-reinforced anisotropic thermoelastic elements under thermal load to meet special engineering requirements.     

Numerical reconstruction of one-photon mixed states by means of the degree of linear polarisation

In this paper the authors present the idea for reconstructing one-photon states. Reconstructing a quantum state means measuring the probability distribution P that allows one to write the density operator for the analysed state. The most commonly known approach for the quantum reconstruction is the quantum tomography. Our alternative method assumes that the analysed field is coupled with the reference field which is described by the parameters settled during a measurement. In the proposed gedankenexperiment the degree of linear polarisation (DOLP) of this combined beam is measured using a rotating linear polariser. We state that it is possible to obtain the P-function by changing the parameters of reference beams and by preparing the series of DOLP measurements. This series of data leads to the system of equations. The values of the P-function at chosen points are the unknowns of this system. This article focuses on the numerical algorithm for solving these equations.     

On the symmetry and coaxiality of pertinent stretch and stress tensors during non-proportional loading at finite plastic strain

Summary In this paper we develop agradient theory of internal variables using a variational principle in conjunction with the dissipation inequality. The basic findings are (i), that the internal variables are, non-local in that they obey field equations instead of evolution equations and (ii) they are subject to boundary conditions that are dictated by the applied tractions and displacements as well as the physical structure of the material domain. As a consequence, spatiallyinhomogeneous strain fields exist in the presence ofuniform boundary tractions and/or displacements. This phenomenon is illustrated in the simple case of one dimension.     

A thermodynamic framework for a gradient theory of continuum damage

In this paper, we present a formulation of state variable based gradient theory to model damage evolution and alleviate numerical instability associated within the post-bifurcation regime. This proposed theory is developed using basic microforce balance laws and appropriate state variables within a consistent thermodynamic framework. The proposed theory provides a strong coupling and consistent framework to prescribe energy storage and dissipation associated with internal damage. Moreover, the temporal evolution equation derived here naturally shows the effect of damage—nucleation, growth and coalescence. In addition, the theoretical framework presented here is easily extendable to the addition of other defects (not shown here), and can be generalized to the development of consistent coupled transport equations for species, such as hydrogen (Bammann et al. in JMPS, 2009, submitted), as well as providing a consistent structure for modeling events at diverse length scales.     

Toward an equation of state for solid materials with memory by use of the half-order derivative

Summary The time derivative operator does not depend upon the difference between the current time and the past times; however, the fractional time derivative operator does. Thus, it is reasonable to expect that the fractional derivative would be useful in describing the mathematical theory of the behavior of materials with memory. An equation of state is proposed for solid materials with memory by introducing the half-order fractional calculus derivative in order to relate to the empirical expression used in the fundamental work of Tobolsky and Catsiff. This theory replaces the three empirical functions used by Tobolsky and Catsiff in reducing their experimental data for the low temperature glassy region, the transition region and the quasi-static rubbery plateau region. The square root differential operator with respect to time, D ^1/2, has built in memory since the kernel of this operator depends upon the difference between the current time and the past time. D ^1/2 is a special case of the Abel operator, which is used in the theory of integral equations. The present theory introduces integrals into the standard linear solid resulting in an integral differential equation governing the stress, strain and temperature. It is shown that this proposed linear equation of state for a solid material, which undergoes a second order transition, requires only four phenomenological constants to completely determine the behavior of the solid material. These four phenomenological constants are two relaxation times and two creep times, both of which are functions of the temperature.     

A finite element formulation for thermoelastic damping analysis

We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well‐known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8‐node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three‐dimensional 20‐node iso‐parametric thermoelastic element, which is suitable to model massive structures. For the 8‐node element the dissipation along the plate thickness has been taken into account by introducing a through‐the‐thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled‐field elements. Copyright © 2008 John Wiley & Sons, Ltd.     

Classical plasticity and viscoplasticity models reformulated: theoretical basis and numerical implementation

The well-known phenomenological model of small strain rate-independent plasticity is reformulated in this paper. The main difference from the classical expositions concerns the absence of the plastic strain from the list of state variables. We show that with the proposed choice of state variables, including the total and the elastic strains and strain-like variables which control hardening, we recover all the ingredients of the classical model from a minimum number of hypotheses: instantaneous elastic response and the principle of maximum plastic dissipation. We also show that using a regularized, penalty-like form of the principle of maximum plastic dissipation, we can recover the classical viscoplasticity model. As opposed to the previous schemes used for the finite element implementation of this model (e.g. B-bar method), we propose an approach in which the basic set of equations need not be modified. The operator split method is used to simplify the details of the numerical implementation concerning both the computation of state variables and the incompatible mode based finite element approximations. The latter proves to be indispensable for accommodating the near-incompressible deformation patterns arising in the classical plasticity. An extensive set of numerical simulations is used to illustrate the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

Entanglement and Entropy Squeezing for Moving Two Two-Level Atoms Interaction with a Radiation Field

In this paper, we analyzed squeezing in the information entropy, quantum state fidelity, and qubit-qubit entanglement in a time-dependent system. The proposed model consists of two qubits that interact with a two-mode electromagnetic field under the dissipation effect. An analytical solution is calculated by considering the constants for the equations of motion. The effect of the general form of the time-dependent for qubit-field coupling and the dissipation term on the temporal behavior of the qubit-qubit entanglement, quantum state fidelity, entropy, and variance squeezing are examined. It is shown that the intervals of entanglement caused more squeezing for the case of considering the time-dependent parameters. Additionally, the entanglement between the qubits became more substantial for the case of time dependence. Fidelity and negativity rapidly reached the minimum values by increasing the effect of the dissipation parameter. Moreover, the amount of variance squeezing and the amplitude of the oscillations decreased considerably when the time dependence increased, but the fluctuations increased substantially. We show the relation between entropy and variance squeezing in the presence and absence of the dissipation parameter during the interaction period. This result enables new parameters to control the degree of entanglement and squeezing, especially in quantum communication.     

A creep model with damage based on internal variable theory and its fundamental properties

The creep damage is discussed within Rice irreversible internal state variable (ISV) thermodynamic theory. An ISV small-strain unified creep model with damage is derived by giving the complementary energy density function and kinetic equations of ISVs. The proposed model can describe viscoelasticity and, preferably, three phases of creep deformation. Creep strain results from internal structural adjustment, and different creep stages accompany different thermodynamic properties in terms of flow potential function and energy dissipation rate. During the viscoelastic process, the thermodynamic state of the material system tends to equilibrate spontaneously. The thermodynamic state of the material system without damage tends to equilibrate or achieve steady state after loading. Kinetic equations of ISVs can be derived by one single flow potential function, and the energy dissipation rate decreases monotonically over time. In the entire creep damage process, multiple potentials are needed to characterise evolution of ISVs, rotational fluxes are presented in affinity space, and the thermodynamic state of material system tends to depart from the steady or equilibrium state. The energy dissipation rate can be a measure of the distance between the current thermodynamic state and the equilibrium state. The time derivative of the rate can characterise the development trend of the material, and the integral value in the domain may be regarded as indices to evaluate the long-term stability of the structure.     

形状记忆合金耗能弹簧的力学性能研究

从Helmholt自由能出发,建立了SMA材料在复杂应力状态下的热力学本构关系,并由此推导出SMA耗能弹簧的热力学本构关系;在Brinson相变发展方程的基础上,引入等效应力的概念后,建立了内变量马氏体百分数的发展方程。利用上述理论,结合普通金属弹簧的设计方法,建立了SMA耗能弹簧的设计方法;利用MATLAB编写的计算机程序进行了仿真计算。研究结果表明SMA耗能弹簧具有较好的耗能能力。本文建立的计算理论较好地吻合了其他学者的研究结果,适用于实际工程问题的理论分析,具有较大的适用价值。     

Decoherence of the two-mode squeezed vacuum state in a diffusion process

We have studied the case in which one mode of the light field in the two-mode squeezed vacuum state evolves in a diffusion channel. By virtue of thermo-entangled state representation and the technique of integration within an ordered product, the evolution formula of the field density operator is given. Its non-classical properties, such as squeezing effect, antibunching effect, the violation of Cauchy–Schwartze inequality and the entanglement property between two modes, are studied. The influences of the squeezing parameter and the dissipation time on the non-classical properties are discussed. The results obtained by the numerical method show that its non-classical properties are all weakened with the dissipation. On the other hand, its squeezing effect and the entanglement property between two modes are strengthened, but its antibunching effect and the violation of Cauchy–Schwartze inequality are weakened with the increase of the squeezing parameter.     

On finite element modelling of surface tension: Variational formulation and applications – Part II: Dynamic problems

In Part I, a finite element model of surface tension has been discussed and used to solve some quasi-static problems. The quasi-static analysis is often required to find not only the initial shape of the liquid but also the static equilibrium state of a liquid body before a dynamic analysis can be carried out. In general, natural and industrial processes in which surface tension force is dominant are of dynamic nature. In this second part of this work, the dynamic effects will be included in the finite element model described in Part I.A fully Lagrangian finite element method is used to solve the free surface flow problem and Newtonian constitutive equations describing the fluid behaviour are approximated over a finite time interval. As a result the momentum equations are function of nodal position instead of velocities. The resulting ordinary differential equation is integrated using Newmark algorithm. To avoid overly distorted elements an adaptive remeshing strategy is adopted. The adaptive strategy employs a remeshing indicator based on viscous dissipation functional and incorporates an appropriate transfer operator.The validation of the model is performed by comparing the finite element solutions to available analytical solutions of a droplet oscillations and experimental results pertaining to stretching of a liquid bridge.     

Condensate theory versus pairing theory for degenerate Bose systems

The statistical variational principle for the thermodynamic potential is used to investigate the quantum statistically degenerate phase of an interacting Bose system. Starting with a general mean-field trial Hamiltonian, we find a set of integral equations that has two different types of solution: (a) the condensate sssssstheory with a finite c-number part for the lowest momentum state operator; (b) the pairing theory with zero c-number part. The condensate theory is identical to the self-consistent Hartree-Fock-Bogoliubov approximation of the Beliaev equations, whereas the pairing theory is identical to the theory proposed by Evans and Imry. We solve the resulting integral equations for both theories numerically and show that—independent of the special form of interacting potentials—the condensate theory gives the lower free energy and the higher transition temperature. Therefore we conclude that the condensate theory describes the thermodynamically stable state rather than the pairing theory.Project of the Sonderforschungsbereich Frankfurt/Darmstadt financed by special funds of the Deutsche Forschungsgemeinschaft.     

A viscoplastic theory with thermodynamic considerations

Summary A thermodynamic foundation using the concept of internal state variables is given for a general theory of viscoplasticity for initially isotropic materials. Three, fundamental, internal, state variables are admitted; they are: a tensorial back stress for kinematic effects, and scalar drag and yield strengths for isotropic effects. All three are considered to evolve phenomenologically according to competitive processes between strain hardening, deformation induced dynamic recovery, and thermally induced static recovery. Within this phenomenological framework, a thermodynamically admissible set of evolution equations is proposed. The theory allows each of the three internal variables to be composed as a sum of independently evolving constituents. The evolution of internal state can also include terms that vary linearly with the external variable rates, whose presence affects the energy dissipation properties of a material.     

Magneto-thermoelastic Response in a Functionally Graded Isotropic Unbounded Medium Under a Periodically Varying Heat Source

This paper deals with the problem of magneto-thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of linear theory of generalized thermoelasticity with energy dissipation (TEWED) and without energy dissipation (TEWOED) having a finite conductivity. The governing equations of generalized thermoelasticity (GN model) for a functionally graded material (FGM) under the influence of a magnetic field are established. The Laplace–Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Leguerre’s method and the inversion of the Laplace transformation is done numerically using a method based on a Fourier series expansion technique. Numerical estimates of the displacement, temperature, stress, and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic materials is obtained by taking a suitable non-homogeneous parameter. Finally, the results obtained are presented graphically to show the effect of a non-homogeneous, magnetic field and damping coefficient on displacement, temperature, stress, and strain.     

A mathematical model for an upper convected Maxwell fluid with an elastic core: Study of a limiting case

In this paper we formulate a mathematical model for a continuum which behaves like an upper convected visco-elastic Maxwell fluid if the stress is above a certain threshold and like a neo-Hookean elastic solid if the stress is below that threshold. The constitutive equations for each phase are derived within the context of the theory of natural configurations and by means of the criterion of the maximization of the rate of dissipation [11]. We then focus on a limiting case in which the continuum becomes an elastic-rigid body. In this limiting case the constitutive relation of the material becomes implicit and, although there is no energy dissipation, it cannot be included in the class of hyperelastic (or Green) bodies. The stress indeed cannot be expressed as a function of the strain. This class of materials was first introduced by Rajagopal in [15] and is the subject of the forthcoming papers [3] and [4].     

Magneto-thermo-elastic response in a perfectly conducting medium with three-phase-lag effect

This paper deals with the problem of magneto-thermo-elastic interactions in an unbounded, perfectly conducting elastic medium due to the presence of periodically varying heat sources in the context of linear theory of generalized thermo-elasticity with energy dissipation (TEWED or GN-III model), without energy dissipation (TEWOED or GN-II model) and three-phase-lag model (3P model). The governing equations of generalized thermo-elasticity of the above models under the influence of a magnetic field are established. The Laplace-Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Laguerre’s method, and the inversion of the Laplace transformation is done numerically using a method based on Fourier series expansion technique. Displacement, temperature, stress and strain distributions have been computed numerically and presented graphically in numbers of figures. A comparison of the results for different theories (GN-II, GN-III and 3P model) and the effect of magnetic field and damping coefficient on the physical quantities has been discussed.     

Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics

This paper presents a new class of assumed strain finite elements to use in combination with general energy‐momentum‐conserving time‐stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time. The case of interest corresponds to nearly incompressible material responses, in the fully non‐linear finite strain elastic and elastoplastic ranges. The new elements consider the classical scaling of the deformation gradient with an assumed Jacobian (its determinant) defined locally through a weighted averaging procedure at the element level. The key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or B‐bar operator. The developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta, and the conservation/dissipation law of energy for elastic and inelastic problems, respectively. Care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws, and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration. With these developments at hand, a new general B‐bar operator is introduced that satisfies these conditions. The new operator not only accounts for the spatial interpolations (e.g. bilinear displacements with piece‐wise constant volume) but also depends on the discrete structure of the equations in time. The aforementioned conservation/dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes, unconditionally of the time step size and the material model (elastic or elastoplastic). Different finite elements are considered in this framework, including quadrilateral and triangular elements for plane problems and brick elements for three‐dimensional problems. Several representative numerical simulations are presented involving, in particular, the use of energy‐dissipating momentum‐conserving time‐stepping schemes recently developed by the author and co‐workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements, including these conservation/dissipation properties in time and their locking‐free response in the quasi‐incompressible case. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved unsteady,unstructured, artificial compressibility,finite volume scheme for viscous incompressible flows: Part I. Theory and implementation

A robust, artificial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible flows over a wide range of Reynolds and Rayleigh numbers. Artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations. A locally generalized preconditioner is introduced, designed to be robust and offer good convergence rates. Free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme. Memory efficiency is achieved through a multistage, pseudo‐time‐explicit time‐marching solution procedure. A node‐centred dual‐cell edge‐based finite volume discretization technique, suitable for unstructured grids, is used due to its computational efficiency and high‐resolution spatial accuracy. In the interest of computational efficiency and ease of implementation, stabilization is achieved via a scalar‐valued artificial dissipation scheme. Temporal accuracy is facilitated by employing a second‐order accurate, dual‐time‐stepping method. In this part of the paper the theory and implementation details are discussed. In Part II, the scheme will be applied to a number of example problems to solve flows over a wide range of Reynolds and Rayleigh numbers. Copyright © 2002 John Wiley & Sons, Ltd.     

A boundary element formulation for analysis of elastoplastic plates with geometrical nonlinearity

In this paper a new boundary element method formulation for elastoplastic analysis of plates with geometrical nonlinearities is presented. The von Mises criterion with linear isotropic hardening is considered to evaluate the plastic zone. Large deflections are assumed but within the context of small strain. To derive the boundary integral equations the von Kármán’s hypothesis is taken into account. An initial stress field is applied to correct the true stresses according to the adopted criterion. Isoparametric linear elements are used to approximate the boundary unknown values while triangular internal cells with linear shape function are adopted to evaluate the domain value influences. The nonlinear system of equations is solved by using an implicit scheme together with the consistent tangent operator derived along the paper. Numerical examples are presented to demonstrate the accuracy and the validity of the proposed formulation.