Principle Of Material Objectivity Research Articles

On damage modeling of material interfaces: Numerical implementation and computational homogenization

A novel constitutive framework suitable for material interfaces undergoing large deformations in a geometrically exact setting was developed in Ottosen et al. (2016). In contrast to previous works, it permits the description of arbitrary material anisotropies by fulfilling all fundamental balance laws in physics as well as the principle of material objectivity. This paper deals with an efficient finite element implementation of the aforementioned framework in terms of the natural basis vectors. To be more precise, a different, more compact and more direct derivation of this framework is outlined first. It relies on the variational structure of the underlying problem. Subsequently, the aforementioned finite element approximation is elaborated which is finally embedded into a computational homogenization scheme. This scheme allows the analysis of the influence of the novel interface model on the resulting macroscopic (effective) material response. It is shown by numerical examples that the interaction of bulk energies and interface energies leads, in a very natural manner, to a complex size effect. It includes the frequently observed “the smaller the stiffer” relation, but also the less often observed “the smaller the softer” relation. However, since the overall response is usually a superposition of such relations, the effective properties cannot generally be characterized by one of the aforementioned limiting relations.

On non-equilibrium molecular dynamics with Euclidean objectivity

In continuum mechanics, the concept of objectivity requires that the qualitative and quantitative description of physical phenomena remains unchanged under a change in reference frame. It means that the observations in different reference frames are coherent and thus the observed motions can be regarded as objectively equivalent. On the other hand, concerning the material constitution, the principle of material objectivity (also often referred as principle of material frame-indifference, or simply the principle of objectivity) requires that constitutive equations of a material body must be independent of the observer (reference frame). Both objectivity and principle of objectivity have been widely used in modern continuum mechanics studies. However, in molecular dynamics (MD) simulation, which is a prevalent numerical method in nanoscience, consideration is rarely given to the concept of objectivity or principle of objectivity since most of the MD simulations have been performed in inertial systems, where Galilean-invariance is assumed. This research aims to investigate the conditions for Euclidean objectivity in non-equilibrium molecular dynamics (NEMD), where reference frames are non-inertial and multi-physics effects are present in the atomistic material system. By supplementing fictitious acceleration to the governing equation and requiring the principle of objectivity on the constitutive equation, this research arrives at a special solution for NEMD that is Euclidean-invariant. Several sets of simulations demonstrate that equivalent motions can be achieved in NEMD. This study may provide new directions for future MD applications.

Thermodynamically consistent phase field theory of phase transformations with anisotropic interface energies and stresses

The main focus of this paper is to introduce, in a thermodynamically consistent manner, an anisotropic interface energy into a phase field theory for phase transformations. Here we use a small strain formulation for simplicity, but we retain some geometric nonlinearities, which are necessary for introducing correct interface stresses. Previous theories have assumed the free energy density (i.e., gradient energy) is an anisotropic function of the gradient of the order parameters in the current (deformed) state, which yields a nonsymmetric Cauchy stress tensor. This violates two fundamental principles: the angular momentum equation and the principle of material objectivity. Here, it is justified that for a noncontradictory theory the gradient energy must be an isotropic function of the gradient of the order parameters in the current state, which also depends anisotropically on the direction of the gradient of the order parameters in the reference state. A complete system of thermodynamically consistent equations is presented. We find that the main contribution to the Ginzburg-Landau equation resulting from small strains arises from the anisotropy of the interface energy, which was neglected before. The explicit expression for the free energy is justified. An analytical solution for the nonequilibrium interface and critical nucleus has been found and a parametric study is performed for orientation dependence of the interface energy and width as well as the distribution of interface stresses.

Thermomechanical coupling and second sound in superfluid flows

The present paper is concerned with the modelling of the influence of the thermomechanical coupling on the propagation of temperature waves (second sound) in superfluids. For an adequate heat transfer analysis in superfluids, finite thermal wave speed must be considered. Besides hyperbolic heat transfer, turbulent flows are generally observed despite the lost of internal friction. Due to the thermo-mechanical coupling, density waves may induce moving heat sources or sinks while temperature waves propagate at a different and independent speed. In particular, a rotational flow in this kind of fluid can strongly affect heat propagation. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. Such a phenomenological continuum approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand its influence on the superfluid flow.

Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry

To say that a constitutive model has to verify “the principle of material objectivity” to ensure its frame-indifference has become a common wisdom. Objective transports are thus defined to serve as tensor rates. These operators are in particular applied to the Cauchy stress tensor. They are used as time derivatives to describe non-linear or dissipative phenomena observed during the finite transformations of a material continuum. Because an infinite number of such transports may be constructed and shown to be objective, the selection of the appropriate transport and its validity still constitutes an open and debatable question.Differential geometry, within its four-dimensional formalism, has proven its ability to describe physical fields and their variations in space and time while ensuring the covariance of any physical law. This description is here applied to the motion of a material continuum within the classical hypotheses of Newtonian physics. In this context, we show that the rate of a tensor as seen by a point of space–time is uniquely defined by the covariant rate; this quantity is not invariant with respect to superposed rigid body motions. The rate of a tensor as seen by a moving particle of matter is uniquely defined by the Lie derivative of the tensor. This operator is invariant with respect to superposed rigid body motions. Both, the covariant rate and the Lie derivative are independent of the observer and could thus be used in a constitutive model within a four-dimensional formalism. We show next that the projection of the Lie derivative of the Cauchy stress tensor within an inertial 3D Cartesian frame corresponds to Truesdell’s transport and that the other 3D objective stress transports, if they have the dimension of a rate, do not correspond to a time derivative of this tensor. The Truesdell transport is thus the only objective transport that represents a frame-indifferent time derivative of the Cauchy stress tensor.

Hyperbolic heat transfer in generalized Newtonian fluids

The present paper is concerned with the modeling of hyperbolic heat transfer in generalized Newtonian fluids. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. Such a thermodynamic approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand its influence on the fluid behavior.

Linear reduced cosserat medium with spherical tensor of inertia, where rotations are not observed in experiment

We consider a linear reduced Cosserat medium: a linear elastic continuum, whose point bodies possess kinematically independent translational and rotational degrees of freedom, but the strain energy does not depend on the gradient of rotation of particles. In such a medium the force stress tensor is asymmetric, but the couple stress tensor is zero. This model can be applied for description of soils and granular media. Since for the time being the experimental technique for measurement of rotational deformations is not well developed, we investigate how the presence of rotational degrees of freedom affects the dynamics of translational displacements. We consider the case of the spherical tensor of inertia and isotropy with respect to the rotational degrees of freedom. Integration of the equation of balance of torques lets us in several cases to put in correspondence a linear reduced Cosserat continuum with the spherical tensor of inertia with a classical (non-polar elastic linear) medium with memory with the same equation for the balance of forces, written in terms of translational displacements. This is possible for the isotropic case and also if the anisotropy is present only in the tensor of elastic constants corresponding to the classical strain tensor. If the material is isotropic with respect to rotational deformations but the (anisotropic) coupling between rotational and classical translational strains is present, then the corresponding classical medium does not exist. If we ignore the rotational degrees of freedom when this coupling is present, this will lead us to the conclusion that the principle of material objectivity is violated.

On thermodynamics, objectivity and the modeling of non-isothermal viscoelastic fluid behavior

The present paper is concerned with the modeling of non-isothermal viscoelastic fluid behavior. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. It is shown that the choice of the objective time derivative is important not only to assure a mathematically correct and physically realistic description of the viscoelastic behavior, but also to perform an adequate description of the thermomechanical coupling in the case of non-isothermal flows. Such a thermodynamic approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand their influence on the fluid behavior.

Comparison of two integration algorithms for finite plasticity combined with nonlinear kinematic and isotropic hardening at the example of springback

The constitutive modelling of combined nonlinear kinematic and isotropic hardening in the finite strain regime is still a very active area of research. Nowadays, hyperelasticity-based formulations using the multiplicative decomposition of the deformation gradient into elastic and plastic parts are widely accepted. The most characteristic phenomena of the hardening behaviour of metals are the Bauschinger effect and ratcheting. The first one is related to the fact that straining in one direction reduces the yield stress in the opposite direction, whilst the latter denotes the progressive increase of the mean strain as a result of unsymmetric stress cycles. A main focus of the present work is the comparison of two algorithms used for the numerical integration of the evolution equations. Due to the deviatoric character of the flow rule, an integration scheme which preserves the plastic volume is required. Dettmer & Reese [2004] proposed a new type of the exponential map algorithm which fulfills plastic incompressibility. Here, we develop this scheme further and use the spectral decomposition to evaluate the exponential tensor functions in closed form. This integration algorithm has the advantage that the symmetry of the internal variables is automatically preserved. In addition, we discuss a modified backward Euler schemes that preserve plastic incompressibility. The results show higher accuracy and improved performance when using large time steps in favour of the exponential algorithm. In the finite strain regime the use of the classical multiplicative decomposition F = Fe Fp of the deformation gradient into elastic and plastic parts is well established. In analogy to the additive split of the plastic strain into two parts modelling the deformation in the hardening spring and the dashpot, we carry out the multiplicative split Fp = Fpe Fpi . The Helmholtz free energy represents the elastic energy storage. Based on the principle of material objectivity, the free energy depends on the deformation only through the elastic right Cauchy-Green tensors Ce and Cpe . Thus, for the ArmstrongFrederick model of kinematic hardening the Helmholtz free energy per unit volume decomposes additively into the three parts

Continuum models of discrete heterogeneous structures and saturated porous media: constitutive relations and invariance of internal interactions

Approaches to mechanical (constructive) modeling of nonhomogeneous media with complicated structure are presented. The constitutive and motion equations of special type discrete systems are obtained as average equations demonstrating the Cosserat type properties of the systems. The equations for saturated porous media are proposed with special attention to the different types of inner interactions. The invariance properties of these interactions and their quantitative contributions are analyzed using the principle of material objectivity and the methods of measurement theory. The invariances of compound (force, moment) internal interactions are studied.

Theory of Porous Media: Highlights in Historical Development and Current State

3R29. Theory of Porous Media: Highlights in Historical Development and Current State. - R de Boer (Inst fur Mechanik, Univ Essen, Fachbereich 10 Bauwesen, Essen, D-45117, Germany). Springer-Verlag, Berlin. 2000. 618 pp. ISBN 3-540-65982-X. $95.00. Reviewed by N Katsube (Dept of Mech Eng, Ohio State Univ, 206 W 18th Ave, Columbus, OH 43210).This book consists of historical and recent development of porous material theory, can be used as a reference book in this area. In the early and classical era, general development of continuum mechanics such as Cauchy’s formulation of the stress concept, the development of linear elasticity theory, and the foundation of thermodynamics are summarized. At the end of each chapter, biographical notes of the important contributors are outlined. There the readers could learn about the lives of these prominent researchers, including their characters, and the nature of scholarly debates. The readers can also learn about how the lives of these researchers were influenced by the historical events, and how failure in one event had led to success in another. In the modern era, the emphasis is placed on the porous materials. The historical review includes the controversy between Fillunger and von Terzaghi and the subsequent development of the theory by Biot and others. Further development of elasticity and plasticity theories and a more modern framework of continuum mechanics and mixture theory are also discussed. The current state of porous material theory reflects the author’s work as well as the related work developed mostly in Europe. It is based on the assumption that the deformation gradient of the constituent can be separated into the part due to the real material and the part due to the change of the pores in size and shape through a multiplicative decomposition process. In developing constitutive theories, the principle of material objectivity as well as entropy inequality is employed. An extensive development of a porous material theory is presented based on these theoretical frameworks.Theory of Porous Media: Highlights in Historical Development and Current State is a great book to have if you would like to know about the historical development of porous material theory since many of the letters and articles written by the researchers in early days are not available in English. In addition, if you would like to follow the recent development of porous material theory, which is consistent with the above stated assumptions, this book will provide an extensive review. However, the reader should be aware that this is not the only approach available in the area of porous materials. There are a number of different approaches available for the development of constitutive behavior of porous materials. These other approaches are more phenomenologically based and often motivated by experimental observations. The developed models are verified by experimental data and used to solve important practical problems.

A new integration procedure for thermo-elasto-viscoplasticity

A numerical integration algorithm for thermo-elasto-viscoplastic constitutive equations is presented. This algorithm satisfies the principle of material objectivity with respect to the total motion (translation, rotation and strain) of a material element. For this purpose, the properties of convective description are used. The explicit-implicit integration scheme for the plastic flow rule plays the crucial role in the proposed algorithm. The method of determining the stress state for inelastic deformations is based on the iterative solution of the dynamic yield condition with respect to the norm of the viscoplastic deformation rate tensor. The constitutive model being the subject of numerical analyses is described. Results of numerical calculations, which show an excellent performance of the proposed procedure, are presented.

On the Influence of Material Objectivity on Electroelastic Dissipation in Polarized Ferroelectric Ceramics

An earlier system of rotationally invariant electroelastic equations for deformable insulators, which was derived from a well-defined macroscopic model, is extended to include the simplest mechanical and electrical viscous-type dissipation. The resulting nonlinear description satisfies the principle of material objectivity. When the resulting description is linearized, the viscous-type linear electrical dissipative variable contains the spin tensor because a biasing polarization is present in the ferroelectric ceramic.

Modified Stress Tensors for Anisotropic Behavior of Granular Materials

Effects of fabric anisotropy on the mechanical behavior of granular materials have become well known. Without the introduction of variables describing fabric anisotropy into the constitutive formulation, some important deformation features of granular materials can not be formulated in a reasonable way. Although the principle of material objectivity and the thermo-dynamical restrictions in the continuum mechanics give the framework to a class of reasonable constitutive equations associated with anisotropic features, the framework is still too wide to obtain a practical constitutive equation. This fact indicates the necessity of conventional methods within the fundamental restrictions. Modified stress tensors being a consecquence of the linear transformation depending on fabric anisotropy from the stress tensor are poposed and discussed, with a particular emphasis on the elasto-plastic behavior of anisotropic granular materials. It is shown that : (a) classical hardening models (isotropic, kinematic, and combined) are obtained as special cases of the present method ; (b) the shape change of yield surfaces during plastic deformation is a natural consequence if relevant terms are included in the transformation rule. Anisotropic deformation and strength behaviors of granular materials are formulated based on the modified Drucker-Prager yield condition and the non-associated flow model for two dimensional problems in order to demonstrate the effectiveness of the proposed method.

Configurational viscosity of dilute colloidal suspensions.

The colloid configurational contribution to the excess stress tensor of a dilute suspension interacting via a screened Coulomb potential in a fluid undergoing Couette flow is analyzed. The structure factor for the nonequilibrium system obtained by solving a fluctuating diffusion equation with a wave-vector-dependent diffusion constant is used to obtain shear-rate-dependent viscosity and normal stress functions. The viscometric functions are nonanalytic in the limit of vanishing shear rate, although the nonanalytic parts are very small. At high shear rates, the viscosity due to the colloid is inversely proportional to the shear rate. In general, the principle of material objectivity, often evoked in rheology, is not satisfied.

Kinetic theory and the principle of material frame-indifference in the mechanics of continuous media

The property of Boltzmann's kinetic equation in the presence of Euclidean transformations such as non-stationary rotations and translations of the frame of reference is discussed. It is shown that in the transition from an inertial frame to a rotating frame, in Boltzmann's equation additional intertial terms appear, and in the transition from non-inertial to non-inertial, the equation is invariant under the above transitions. The algorithms, and the results of an approximate method for solving the equation, of the Chapman-Enskog method in particular, are also invariant. The additional terms appear, specifically, in the expressions for stresses and the heat fluxes in Barnett's approximation, and in this sense these expressions are frame-dependent. Because of the condition that Knudsen's numbers should be small, this limits the domain of applicability of one of the basic postulates of the axiomatic theory of continuous media, namely the principle of material frame-indifference (or the principle of material objectivity), in accordance with which the constitutive (determining) relations should be invariant under continuous changes of the frame of reference. The existing papers on this subject are critically analysed.

Thermodynamics of an interfacial fluid membrane

We discuss a closed system of field equations for a semipermeable membrane which has particle and heat exchange with its surroundings. In this case we consider a surface with an arbitrary shape for specific quantities and mechanical properties. A representation of the constitutive equations follows from the principle of material objectivity in space as well as on surfaces. The constitutive equations can be restricted by an entropy principle. We present both the Gibbs equation and the entropy flux. Furthermore, we obtain the surface stress and the chemical potential in terms of the specific free energy of the membrane. Both the heat flux and the particle flux normal to the membrane depend on the mean curvature and the friction between the particle across the membrane. The interaction tangential to the interface is dependent up on gradients of the surface stress as well as the chemical potential of the interface.

On the frame dependence of electric current and heat flux in a metal

The ideas and methods of the kinetic theory of metal electrons are used to show that Ohm's law of electric conduction and Fourier's law of heat conduction in a metal both contain frame dependent terms. It follows that the principle of material objectivity does not hold rigorously.

Generalizations of the diffusion equation

AbstractThe mass flux entering the Fick's diffusion equation is considered as an arbitrary analytical function of concentration, concentration gradient, and of the gradient of concentration gradient. The restrictions imposed on the flux by the principle of material objectivity are stated and briefly discussed.