Basic Constitutive Equations Research Articles

On temperature in thermoelastoplasticity

Nowadays, the determination of temperature needs often to be treated in the framework of thermoplasticity or thermoviscoplasticity, for example, during metal forming, crashes, … There are still many great difficulties. Some concern the numerical integration of the equations involved; we can say that they are solved reasonably well. Others are more fundamental, and concern the basic constitutive equations and the writing of the thermodynamical equations; they still belong to unsolved problems. Recently, many papers have been published in this area (for ex. [P. Germain, E.H. Lee, in: A. Sawczuk (Ed.), Problems of Plasticity, Noorthoff Int. Publ., 1974. [14]] and [D. Rittel, Z.G. Wang, M. Merzer, Adiabatic shear failure and dynamic stored energy of cold work, Phys. Rev. Lett. 96 (7) (2006) 075502. [18]]); we have thought that it would be useful to discuss our old paper that appeared, in French, in the Journal Archives of Mechanics 26 (4) (1974) 701–715, and to underline what our contributions were at that time: (i) on a possible definition of the homogenized temperature of an aggregate (polycrystals or composites); (ii) on a possible interpretation of the occurrence of ‘adiabatic bands’ by considering only the influence of temperature, for ex. [A. Baltov, T. Vinh, C. R. Acad. Sci. Paris Sér. A 291 (1972) 275. [13]] and [D. Rittel, Z.G. Wang, M. Merzer, Adiabatic shear failure and dynamic stored energy of cold work, Phys. Rev. Lett. 96 (7) (2006) 075502. [18]]. For this, with the help of a physical microscopical model, we have explained where some energy could be stored in the aggregate, and we have shown that only in the case of elastic and thermal isotropy, were we able to defined its temperature T, as the mean of the local temperature θ, of its constituents. Then we wrote the thermodynamical equations (assuming small perturbations) for various hypotheses: isothermic, adiabatic or general heat condition. Finally, we analyzed a simple rod that is subjected to a quasi-static loading at a constant strain rate and then, when the rod is dynamically loaded as during the Taylor test i.e. during the impact of the rod on a rigid wall. We have shown that there is a non-constant ratio between the plastic power and the stored power and that there was no concentration (adiabatic band), while taking into account the influence of the temperature in the equations. To cite this article: W.K. Nowacki, J. Zarka, C. R. Mecanique 336 (2008).

Numerical modelling of hot forming processes

In this paper, first, the state of the art of numerical modelling in hot forming processes will be overviewed. Then the paper briefly reviews the theoretical background of metal forming simulation, including the basic constitutive equations, and the information flow in process modelling. The important process variables and the main characteristics of various hot forming processes will also be discussed. Finally, some industrial examples will be shown. An integrated approach to forging and heat treatment will also be illustrated.

Modelling of Bingham-like fluids with deformable core

We present here two extensions of the usual Bingham model in which we account for deformations in the region where the shear is below the threshold τ o . In particular, we present two models that consider the material in the above region as a neo-Hookean solid or as a visco-elastic upper convected fluid. Both models are developed in the general 3-D framework of the natural configurations theory. After formulating the basic constitutive equations we focus on two 1-D problems: Shear flow driven by a constant stress and flow in a channel driven by a constant pressure gradient. For both the models, the resulting mathematical problems are free boundary problems involving parabolic and hyperbolic equations.

Universal relations in isotropic nonlinear magnetoelasticity

In this paper we first summarize the basic constitutive equations for (nonlinear) magnetoelastic solids capable of large deformations. Equivalent formulations are given using either the magnetic induction vector or the magnetic field vector as the independent magnetic variable in addition to the deformation gradient. The constitutive equations are then specialized to incompressible, isotropic magnetoelastic materials in order to determine universal relations. A universal relation, in this context, is an equation that relates the components of the stress tensor and the components of the magnetic field and/or the components of the magnetic induction that holds independently of the specific choice of constitutive law for the considered class or subclass of materials. As has been shown previously for the case in which the magnetic induction is the independent magnetic variable, in the general case there exists only one possible universal relation. We show that this is also the case if the magnetic field is taken as the independent variable and that the universal relations resulting from the two cases are equivalent. A number of special cases are found for certain specializations of the constitutive equations. These include some connections between the deformation, the magnetic field and magnetic induction that do not involve the components of the stress tensor. Universal relations are then examined for some representative homogeneous and inhomogeneous universal solutions.

Asperity Model of an Earthquake: Dynamic Problem

A previous study that presented the static solution for an asperity model of an earthquake is extended to solve the dynamic problem that develops when failure occurs on the boundary of an asperity patch and then spreads over the surrounding displacement shadow region. The boundary integral equation method is coupled with basic constitutive equations for failure and friction to solve the dynamic problem, with different parameters used for the strong asperity patch and weak shadow region. No friction, displacement-weakening friction, and velocity-strengthening friction are all investigated. Depending on the type and amount of friction that is present, the dynamic solutions for slip on the fault exhibit a range of different features, including overshoot of the static solution and oscillation, rupture front velocities that may be greater than or less than the S velocity and change with position, and either total or partial release of the static moment. Common characteristics of the solutions are that failure on the asperity patch is almost independent of failure on the shadow region and that the displacement deficit on the shadow region is released by propagating slip pulses. The stress concentrations of the asperity model are sufficient to produce nonlinear elastic effects in a region extending outward from the fault to distances comparable with the dimensions of the shadow region. Beginning with the solutions for slip on the fault, waveforms are simulated for an earthquake of magnitude M w 1.44 and compared with data recorded at a distance of 8.65 km. Simulations that contain both source and propagation effects are capable of explaining most of the basic features of the observational data, including general agreement with the shape of the waveforms in the time domain, the levels and slopes of the spectra at low frequencies (less than 10 Hz) and at high frequencies (greater than 100 Hz), and some of the interference effects present in both the time and frequency domains.

Basic Equations and Constitutive Equations of Micropolar Magnetic Fluids with E-B Analogy and the Abraham Expression of Electromagnetic Momentum

A complete set of basic equations for magnetic fluids with internal rotation is proposed in this paper. The basic equations are derived from the conservation laws of mass, linear momentum, angular momentum and energy, while the constitutive equations are obtained by the thermodynamical method that is based on the free energy and the dissipation function. The concrete expression of constitutive equations are determined by the principle of material frame indifference and the principle of maximal dissipation rate. The Abraham expression of the electromagnetic momentum and E-B analogy are adopted in this theory. It is shown that the difference in the basic equations in case of adopting the Abraham expression and the Minkowski expression, respectively. The constitutive equation of magnetization which includes the Shliomis relaxation equation (1972) is proposed.

Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics

11R1. Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics. - L Gaul, M Ko¨gl, and M Wagner. Springer-Verlag, New York. 2003. 488 pp. ISBN 3-540-00463-7. $79.95. Reviewed by JT Chen (Dept of Harbor and River Eng, Natl Taiwan Ocean Univ, PO Box 7-59, Keelung 202, Taiwan, ROC).As the title of this book emphasizes, an introductory course to the boundary element method (BEM) and advanced formulations is presented. The book contains four parts: Part I The Direct Boundary Element Method, Part II Dual Reciprocity Method (DRM), Part III Hybrid Boundary Element Methods, and Part IV Appendix. Part I can be seen as an introductory course, while Parts II and III cover advanced topics that contain the authors’ research material. The appendices in Part VI contain some fundamental solutions and particular solutions for the DRM. Exercises and programs are not provided. The reviewer found that an on-line book of BEM by the first author is available on the web site of http://www.bem.uni-stuttgart.de/. This site provides a more friendly and suitable textbook for beginners since exercises are given. As it is common with other BEM books, this text begins with an introduction and mathematically preliminaries. A special chapter on continuum physics is added in Chapter 3: the basic laws and constitutive equations for elastodynamics, heat conduction, electrodynamics, thermoelasticity, acoustics, and piezoelectricity are covered to provide a complete overview on the physical modeling. Chapters 4 and 5 introduce the direct BEM for potential problems of the Laplace and Navier equations with emphases on the issues of anisotropy and piezoelectricity. No indirect formulations in terms of single-layer or double-layer representations are developed. Numerical integration schemes for regular and singular integrals are addressed in Chapter 6. Parts II and III on advanced topics include the dual reciprocity BEM and the hybrid BEM. Chapter 7 basically follows the DRM book by Partridge et al. for a general introduction to the method. Chapter 8 focus on the solution of the DRM equation of motion and Chapters 9 and 10 present the application of the DRM to piezoelectricity and thermoelasticity. The hybrid BEM is derived from variational principles of mechanics that are reviewed in Chapter 11. Hybrid displacement and hybrid stress methods are both addressed in Chapters 12 and 13, respectively. Since the presented hybrid BEM uses the same source of variational principles as for hybrid FEM, one obtains symmetric system matrices. In contrast to symmetric Galerkin BEM, the hybrid BEM does not require a double integration over the boundary. Although this book can be used as a text in a BEM course, it contains some original results regarding the dual reciprocity BEM and the variational formulation of the hybrid BEM. The book is thus recommended to graduate students and engineers. The authors have succeeded in fulfilling their aim of a dual-purpose textbook. In Part I, students as well as practitioners find a clear introduction to the method, whereas Parts II and III can serve as a valuable reference to researchers and engineers. The main distinction of the book in comparison to available works on the BEM may be its focus on the application of the method to anisotropy, piezoelectricity and thermoelasticity, as well as the presentation of the hybrid BEM. The book contains 488 pages with 135 figures. The quality of print and figures is adequate. In general, this is a well-written book and is recommended to individuals and libraries.

Basic Equations of Micropolar Magnetic Fluids

A complete set of the basic equations for magnetic fluids with internal rotation is proposed in this paper. The basic equations and constitutive equations are derived by the thermodynamical method in which free energy and dissipation function are used on basis of the conservation laws. The balance equations are derived from the conservation laws of mass, momentum, angular momentum and energy. The constitutive equations such as the stress tensor, the coupled stress tensor, the magnetization are determined by the principle of maximal dissipation rate and the principle of material frame indifference. The constitutive equation of magnetization which includes Shliomis's relaxation equation is proposed.

Asperity Model of an Earthquake: Static Problem

A previous study that presented the static solution for an asperity model of an earthquake is extended to solve the dynamic problem that develops when failure occurs on the boundary of an asperity patch and then spreads over the surrounding displacement shadow region. The boundary integral equation method is coupled with basic constitutive equations for failure and friction to solve the dynamic problem, with different parameters used for the strong asperity patch and weak shadow region. No friction, displacement-weakening friction, and velocity-strengthening friction are all investigated. Depending on the type and amount of friction that is present, the dynamic solutions for slip on the fault exhibit a range of different features, including overshoot of the static solution and oscillation, rupture front velocities that may be greater than or less than the S velocity and change with position, and either total or partial release of the static moment. Common characteristics of the solutions are that failure on the asperity patch is almost independent of failure on the shadow region and that the displacement deficit on the shadow region is released by propagating slip pulses. The stress concentrations of the asperity model are sufficient to produce nonlinear elastic effects in a region extending outward from the fault to distances comparable with the dimensions of the shadow region. Beginning with the solutions for slip on the fault, waveforms are simulated for an earthquake of magnitude M w 1.44 and compared with data recorded at a distance of 8.65 km. Simulations that contain both source and propagation effects are capable of explaining most of the basic features of the observational data, including general agreement with the shape of the waveforms in the time domain, the levels and slopes of the spectra at low frequencies (less than 10 Hz) and at high frequencies (greater than 100 Hz), and some of the interference effects present in both the time and frequency domains.

A strain-gradient thermodynamic theory of plasticity based on dislocation density and incompatibility tensors

In this work, we discuss a thermodynamic theory of plasticity for self-organization of collective dislocations in FCC metals. The theory is described by geometrical tensor quantities of crystal defect fields such as dislocation density tensor, representing net mobile dislocation density and geometrically necessary boundaries, and the incompatibility tensor representing immobile dislocation density. Conservation laws for the two kinds of dislocation density are formulated with dislocation products and interactions terms. Based on the second law of thermodynamics, we drive basic constitutive equations for the dislocation flux, production and interaction terms of dislocations. We also derive a set of reaction-diffusion equations for the dislocation density tensor and incompatibility tensor which describes the vein and persistent slip band (PSB) ladder structures. These equations are analyzed using linear stability and bifurcation analysis. An intrinsic mesoscopic length scale is determined which provides an estimate for the wavelength of the PSBs. The basic aspects of the model are motivated and substantiated by analyzing the stress fields of various possible dislocation configurations using discrete dislocation dynamics.

気液二相流のモデリングと基礎式・構成式

Basic equations and constitutive equations of gas-liquid two-phase flow were explained based on the various modeling. The local instantaneous formulation and averaged formulation of basic equations were described based on typical models such as homogeneous model, drift flux model and two-fluid model. The constitutive equations for these averaged basic equations were discussed. The modeling of turbulence in gas-liquid two-phase flow were also explained and basic equations and constitutive equations of turbulence were described.

Creep Hardening Rule under Multiaxial Repeated Stress Changes

Anisotropic creep behavior of polycrystalline metals under multiaxial nonproportional repeated loading conditions is modeled from phenomenological points of view. The creep model consists of basic constitutive equations (BCEs) and an auxiliary hardening rule (AUX) to enhance the predictive capability of the BCEs. The BCEs are specified on the basis of a modification of the conventional kinematic hardening model, and they are characterized by a new kinematic hardening variable which is defined as the sum of two component variables ; one represents the back stress in the conventional sense and the other a flow resistance in the opposite direction of the deviatoric stress. The AUX is governed by a memory region in which only the evolution of the back stress takes place. Two different formulations of the AUX are presented. The applicability of the creep model is discussed on the basis of simulations for multiaxial nonproportional repeated creep of type 304 stainless steel at high temperature.

A Constitutive Model for Anisotropic Creep Deformation

Anisotropic creep behavior of polycrystalline metals under repeated stress changes is modeled from a phenomenological point of view. The creep model consists of basic constitutive equations (BCE) and an auxiliary hardening rule (AUX) to enhance the predictive capability of the BCE. The BCE is characterized by a kinematic hardening variable which is defined as the sum of two component variables; one represents the back stress and the other a flow resistance in the opposite direction of the stress deviator. The AUX is governed by a memory region in which only the evolution of the back stress takes place. The validity of the creep model is discussed on the basis of simulations for multiaxial nonproportional repeated creep of type 304 stainless steel at 650°C.

Thermostatics of an infinite mixture of an elastic solid and a fluid containing a spherical thermal inclusion

The basic constitutive equations for a mixture are given by Green and Naghdi [1]. In the present paper, we follow [2], and we obtain the thermoelastic equations for a mixture of an elastic solid and a fluid. We observe that the thermoelastic equations of a mixture and the Lamé's equations of the thermoelasticity are formally the same. Throughout this paper, we find a more generalized form of Betti's reciprocal theorem and Maysel's formula for a mixture. Finally, a solution of a spherical thermal inclusion of radius a in an infinite mixture is obtained by using generalized Maysel's formula and by direct integration of the governing differential equations.

Basic researches on thermo-hydraulic non-equilibrium phenomena related to nuclear reactor safety.

A review was made of recent developments of fundamental researches on thermo-hydraulic non-equilibrium phenomena related to light water reactor safety, in relation to problems to be solved for the improvement of safety analysis codes. As for the problems related to flow conditions, fundamental researches on basic conservation equations and constitutive equations for transient two-phase flow were reviewed. Regarding to the problems related to thermal non- equilibrium phenomena, fundamental researches on film boiling in pool and forced convection, transient boiling heat transfer and flow behavior caused by pressure transients were reviewed.

Rational approach to thermodynamic rrocesses and constitutive equations in isothermal and non-isothermal kinetics

Isothermal and non-isothermal kinetics are classified according to the viewpoint of rational approach. The appropriate selection of basic quantities and constitutive equations is stressed. The extensive discussion recently focused to the meaning of the partial derivatives is reinvestigated and clarified considering the origin of following equation $$\alpha = f(T,t)$$ whereα is the extent of reaction,T andt are the temperature and time respectively, andf represents a function. The meaning of partial derivatives is demonstrated in details. The disagreement sometimes claimed between the data evaluated by means of isothermal and non-isothermal kinetics is also reviewed, but no fundamental differences are established.

Constitutive equations for interacting continua

Using a general thermo-dynamical theory of interacting continua in the form developed recently by Green and Naghdi basic constitutive equations are given for a mixture of a non-linear elastic solid and a linear viscous fluid, and also for a mixture of two non-linear elastic solids. The equilibrium of a fluid in an elastic solid is also discussed.

Simple microfluids

The basic field equations, jump conditions and constitutive equations of, what we call, ‘simple microfluent’ media are derived and discussed. These fluids are shown to be a generalization of the Stokesian fluids in which local micro-motions are taken into account. Special cases in which gyrations are small and micro-deformation rates are linear are discussed. The partial differential equations of the constitutively linear theory are obtained.

Nonlinear theory of simple micro-elastic solids—I

The present work is concerned with the formulation of the basic field equations, boundary conditions and constitutive equations of what we call ‘simple micro-elastic’ solids. Such solids are affected by the ‘micro’ deformations and rotations not encountered in the theory of finite elasticity. The theory, in a natural fashion, gives rise to the concept of stress moments, inertial spin and other types of second order effects and their laws of motion. The mechanism of the surface tension is contained in the theory. In a forthcoming paper (Part II) explicit expressions of constitutive equations of several simple micro-elastic solids will be given and applied to some special problems.

On the static nonlinear theory of electromagnetic thermoelastic solids—I

The basic field equations, boundary conditions, and constitutive equations, necessary for the treatment of problems involving the nonlinear, steady state behaviour of isotropic elastic solids subject to large deformations, electromagnetic fields and thermal gradients, are derived and employed to solve three specific problems. In Part I the basic equations are formulated to include Maxwell's equations, laws of conservation of mass and of energy, and principles of balance of momentum and balance of moment of momentum. The principle of equipresence and the invariance requirements of the theory of continuous media are employed to develop a constitutive theory in which the stress, dielectric displacement, electric current, magnetic flux density, and heat flux are assumed to be analytic tensor point functions of strain, electric field, magnetic field and thermal gradient. The elastic solids so characterized are further restricted by the assumption of hemihedral or holohedral isotropy with respect to mechanical, electrical, magnetic and thermal properties. In Part II (pp. 97–114) are given a number of special theories deducible from the general consitutive equation developed in Part I, e.g. rigid motion, electroelasticity, and magnetoelasticity. Linear, second degree, and other approximate constitutive equations are also discussed in detail. The problems solved are: Simple shear of an infinite, dielectric slab the faces of which are maintained at different temperatures and potentials; Torsion of an incompressible, circular, conducting cylinder with axial electric field; and Response of a rigid, annular, dielectric cylinder to an axial magnetic field and radial electric field. Part I is made self-contained through the appendices which give the expansion for any symmetric or anti-symmetric 3 × 3 matrix which can be expressed as a polynomial in one symmetric and three anti-symmetric matrices.