Objective Rate

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A. Meyers - One of the best experts on this subject based on the ideXlab platform.

  • Choice of Objective Rate in single parameter hypoelastic deformation cycles
    2006
    Co-Authors: A. Meyers, Heng Xiao, Otto T. Bruhns
    Abstract:

    Hill Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950] demonstRated that "infinitesimal-displacement theory, used in classical elastoplasticity, may no longer be valid in elastic-plastic analysis because the convected terms in the Rate of change of the stress acting on material particle may then not be negligible" Lee EH. Some anomalies in the structure of elastic-plastic theory at finite strain. In: Carroll MM, Hayes M. editors. Nonlinear effects in fluids and solids, New York: Plenum Press; 1996. p. 227-49]. From this we may deduce thatthe elastic deformation part may have considerable influence on the total deformation, even when it is relatively small, i.e. we are confronted with the vital requirement of properly computing elastic deformations.Further, finite deformation kinematics should be applied, i.e. they should take account of possibly large rotations, e.g. through a formulation in an Eulerian frame.Xiao et al. Xiao H, Bruhns OT, Meyers ATM. Self-consistent Eulerian Rate type elasto-plasticity models based upon the logarithmic stress Rate. Int J Plast 1999;15:479-520] gave the mathematical proof, that Bernstein's consistency criterion Bernstein B. Relation between hypo-elasticity and elasticity. Trans Soc Rheol 1960;4:23-8; Bernstein B. Hypoelasticity and elasticity. Arch Ration Mech Anal 1960;6:90-104] is fulfilled in a hypoelastic law of grade zero if, and only if, the Objective logarithmic stress Rate Xiao H, Bruhns OT, Meyers A. Hypo-elasticity model based upon the logarithmic stress Rate. J Elasticity 1997;47:51-68] has been applied. This proof is of complicated mathematical nature. Here, we compare several Objective Eulerian stress Rates of corotational and non-corotational type for the hypoelastic law cited above in closed single parameter deformation cycles. It is found that the logarithmic stress Rate returns the element to its stress-free original state after the closed cycle, thus confirming the findings in Xiao et al. (1999). We show that for some other Objective Rates the errors are accumulating to considerable amounts after several cycles, even when the deformation in investigation is relatively small. Interestingly, for Jaumann stress Rate, the error may vanish for specified deformation measures.

  • A self-consistent Eulerian Rate type model for finite deformation elastoplasticity with isotropic damage
    International Journal of Solids and Structures, 2001
    Co-Authors: Otto T. Bruhns, Heng Xiao, A. Meyers
    Abstract:

    Abstract Continuum models for coupled behaviour of elastoplasticity and isotropic damage at finite deformation are usually formulated by first postulating the additive decomposition of the stretching tensor D into the elastic and the plastic part and then relating each part to an Objective Rate of the effective stress, etc. It is pointed out that, according to the existing models with several widely used Objective stress Rates, none of the Rate equations intended for characterizing the damaged elastic response is exactly integrable to really deliver a damaged elastic relation between the effective stress and an elastic strain measure. The existing models are thus self-inconsistent in the sense of formulating the damaged elastic response. By consistently combining additive and multiplicative decomposition of the stretching D and the deformation gradient F and adopting the logarithmic stress Rate, in this article, we propose a general Eulerian Rate type model for finite deformation elastoplasticity coupled with isotropic damage. The new model is shown to be self-consistent in the sense that the incorpoRated Rate equation for the damaged elastic response is exactly integrable to yield a damaged elastic relation between the effective Kirchhoff stress and the elastic logarithmic strain. The Rate form of the new model in a rotating frame in which the foregoing logarithmic Rate is defined, is derived and from it an integral form is obtained. The former is found to have the same structure as the counterpart of the small deformation theory and may be appropriate for numerical integration. The latter indicates, in a clear and direct manner, the effect of finite rotation and deformation history on the current stress and the hardening and damage behaviours. Further, it is pointed out that in the foregoing self-consistency sense of formulating the damaged elastic response, the suggested model is unique among all Objective Eulerian Rate type models of its kind with infinitely many Objective stress Rates to be chosen. In particular, it is indicated that, within the context of the proposed theory, a natural combination of the two widely used decompositions concerning D and F can consistently and uniquely determine the elastic and the plastic parts in the two decompositions as well as all their related kinematical quantities, without recourse to any ad hoc assumption concerning a special form of the elastic part F e in the decomposition F = F e F p or a related relaxed intermediate configuration. As an application, the proposed general model is applied to derive a self-consistent Eulerian Rate type model for void growth and nucleation in metals experiencing finite elastic–plastic deformation by incorporating a modified Gurson's yield function and an associated flow rule, etc. Two issues involved in previous relevant literature are detected and raised for consideration. As a test problem, the finite simple shear response of the just-mentioned model is studied by means of numerical integration.

  • Some comments on Objective Rates of symmetric Eulerian tensors with application to Eulerian strain Rates
    Acta Mechanica, 2000
    Co-Authors: A. Meyers, P. Schieße, O. T. Bruhns
    Abstract:

    In recent years the role of a convenient Objective Rate of Objective quantities has been passionately discussed. There is a large number of well-justified formulations, e.g., [8], [13], [16]. For an overview of some selected derivatives see, e.g., [21]. However, unreliable results obtained in specific computations [11] complicate the right choice. Moreover, from a physical point of view there exist some additional requirements on time derivatives besides the principle of objectivity [5]. In this paper we try to show that there is a need for using corotational Rates. For that purpose we give different approaches. In an application to the aforementioned facts we prove that only the Hencky strain [6] can have an Objective corotational Rate. We do that by identifying the Objective strain Rate and the deformation Rate. Moreover, the spin involved in this Rate is the logarithmic spin as defined in [23].

  • On the Consistency of some Eulerian Strain Rates
    ZAMM, 1999
    Co-Authors: A. Meyers
    Abstract:

    In the past a variety of Objective Eulerian strain Rates has been considered. By use of the eigenproject concept it is shown that some of those Rates, though being Objective, are not mathematically admissible. Moreover, when the deformation Rate and the Objective strain Rate are equalized, only a single strain, namely the Hencky strain remains valid. It is related to a single Objective Rate, the logarithmic Rate.

Jing Wang - One of the best experts on this subject based on the ideXlab platform.

  • Power Allocation and Relay Selection for Two-Way Relaying Systems by Exploiting Physical-Layer Network Coding
    IEEE Transactions on Vehicular Technology, 2014
    Co-Authors: Lihua Pang, Yang Zhang, Jiandong Li, Jing Wang
    Abstract:

    In this paper, we study the power allocation and relay selection stRategy for two-way relaying systems using physical-layer network coding (PNC), in which two information symbols can be exchanged in two time slots. Our approach is based on maximizing the Objective Rate under a total power budget consumed by the transmission of the two information symbols. Two optimization Objectives, namely, the minimum of the achievable Rates of the two directions and the sum Rate of the system, are considered here. Since the Objectives are not continuously differentiable functions, we propose a method based on a suboptimal solution to solve the original problems. It is shown that the main problems have closed-form solutions, and the stRategies can be implemented in a distributed manner. The numerical results verify the effectiveness of our proposal.

Armin Abedini - One of the best experts on this subject based on the ideXlab platform.

  • shear confusion identification of the appropriate equivalent strain in simple shear using the logarithmic strain measure
    International Journal of Mechanical Sciences, 2017
    Co-Authors: Clifford Butcher, Armin Abedini
    Abstract:

    Abstract There is significant confusion surrounding the appropriateness of the logarithmic (Hencky) strain measure to describe simple shear deformation for finite strain. In simple shear loading of plastically deforming materials, the principal stress and strain directions do not remain coaxial which has led to conflicting derivations of the equivalent strain throughout the literature. The source of this confusion is attributed to a misapplication of the formulas for the von Mises equivalent strain and its increment that are only valid for proportional coaxial loading. In this work, a closed-form solution for the work-conjugate equivalent strain for an arbitrary yield function was derived for simple shear loading that is readily amenable to experimental characterization and is entirely consistent with the logarithmic strain measure. An analytical stress and strain integration of an elastic-plastic material was performed using the logarithmic Objective Rate to demonstRate that the stress, logarithmic strain, and principal directions are correctly determined within a hypo-elastic-plastic framework to finite strains. It was demonstRated that the integRated equivalent plastic strain is work-conjugate and the logarithmic strain measure is appropriate for finite simple shear. A review of the recent experimental literature for shear characterization has found that the misapplication of the von Mises equivalent strain formula for coaxial loading in simple shear loading is pervasive. The coaxial effective strain formula is the default measure in commercial digital image correlation (DIC) software and may significantly underestimate the equivalent strain in the simple shear loading condition. If the major principal strain in a simple shear test is lower than 50%, the error between the coaxial and work-conjugate equivalent strains is negligible. Otherwise, the error grows in a hyperbolic manner. Within the context of a finite element simulation of a simple shear test, if a hypo-elastic-plastic formulation is employed as in most commercial codes, the equivalent strain will be correctly computed from work conjugacy but the cumulative and principal logarithmic strain tensors will be incorrect at finite strains unless the logarithmic Rate is employed.

Angelo Morro - One of the best experts on this subject based on the ideXlab platform.

  • Objective Rate equations and memory properties in continuum physics
    Mathematics and Computers in Simulation, 2020
    Co-Authors: Angelo Morro, Claudio Giorgi
    Abstract:

    Abstract The paper deals with the modelling of material behaviours in continuum physics by means of Rate equations. The research has a twofold purpose. First, to review the structure of Objective time derivatives, namely invariant derivatives within the set of Euclidean transformations; known derivatives occurring in the literature are shown to be particular cases of the whole family of Objective time derivatives. Second, to investigate the thermodynamic consistency of some models involving Objective time derivatives. In particular, two topics are developed. One is the improvement of the constitutive equation of viscous fluids. The other topic is a possible Rate equation for the stress. The thermodynamic consistency is shown in connection with the co-rotational derivative.

  • Nonlinear waves in thermoelastic dielectrics
    Evolution Equations & Control Theory, 2019
    Co-Authors: Angelo Morro
    Abstract:

    This paper is addressed to the analysis of wave propagation in electroelastic materials. First the balance equations are reviewed and the entropy inequality is established. Next the constitutive equations are considered for a deformable and heat-conducting dielectric. To allow for discontinuity wave propagation, an appropriate Objective Rate equation of the heat flux is considered. The thermodynamic consistency of the whole set of constitutive equations is established. Next the nonlinear evolution equations so determined are tested in relation to wave propagation properties. Waves are investigated in the form of weak discontinuities and the whole system of equations for the jumps is obtained. As a particular simple case the propagation into an unperturbed region is examined. Both the classical electromagnetic waves and the thermal waves are found to occur. In both cases the mechanical term is found to be induced by the electrical or the thermal wave discontinuity.

  • Modelling of elastic heat conductors via Objective Rate equations
    Continuum Mechanics and Thermodynamics, 2018
    Co-Authors: Angelo Morro
    Abstract:

    A thermoelastic solid is modelled by letting the heat flux be given by a Rate equation. As any constitutive property, the Rate equation has to be Objective and consistent with thermodynamics. Accordingly, firstly a theorem is given that characterizes Objective time derivatives. This allows the known Objective time derivatives to be viewed as particular elements of the set so specified. Next the thermodynamic consistency is established for the constitutive models involving Objective time derivatives within appropriate sets. It emerges that the thermodynamic consistency holds provided the stress contains additively terms quadratic in the heat flux vector in a form that is related to the derivative adopted for the Rate of the heat flux.

  • Thermodynamic consistency of Objective Rate equations
    Mechanics Research Communications, 2017
    Co-Authors: Angelo Morro
    Abstract:

    Abstract The paper addresses the modelling of solids via Rate equations for the heat flux and the stress. As with any constitutive property, the Rate equations are required to be both Objective and consistent with the second law of thermodynamics. Upon a review of a connection between Objective time derivatives, attention is restricted to Rate equations where the time derivatives are those named after Jaumann, Green and Naghdi, Oldroyd, and Truesdell. Equations involving the Truesdell Rate are investigated within the material description thanks to the identity between the time derivative of the material fluxes and the Truesdell Rate in the current configuration. The remaining equations are examined within the spatial description. The occurrence of a skew tensor in the Objective derivative results in a further restriction on the constitutive properties. The thermodynamic requirements are found to be satisfied and the corresponding free energy is determined by direct integrations.

Lihua Pang - One of the best experts on this subject based on the ideXlab platform.

  • Power Allocation and Relay Selection for Two-Way Relaying Systems by Exploiting Physical-Layer Network Coding
    IEEE Transactions on Vehicular Technology, 2014
    Co-Authors: Lihua Pang, Yang Zhang, Jiandong Li, Jing Wang
    Abstract:

    In this paper, we study the power allocation and relay selection stRategy for two-way relaying systems using physical-layer network coding (PNC), in which two information symbols can be exchanged in two time slots. Our approach is based on maximizing the Objective Rate under a total power budget consumed by the transmission of the two information symbols. Two optimization Objectives, namely, the minimum of the achievable Rates of the two directions and the sum Rate of the system, are considered here. Since the Objectives are not continuously differentiable functions, we propose a method based on a suboptimal solution to solve the original problems. It is shown that the main problems have closed-form solutions, and the stRategies can be implemented in a distributed manner. The numerical results verify the effectiveness of our proposal.