Corotational Rate Research Articles

On The Objective Corotational Rates of Eulerian Strain Measures

In the present paper, some new basis-free expressions for an arbitrary objective corotational rate of the general Eulerian strain measures are provided which are in compact form. Moreover, a complete discussion on the requirements for the continuity of the objective corotational rates are presented.

Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains

The familiar small strain thermodynamic 3D theory of isotropic pseudoelasticity proposed by Raniecki and Lexcellent is generalized to account for geometrical effects. The Mandel concept of mobile isoclinic, natural reference configurations is used in order to accomplish multiplicative decomposition of total deformation gradient into elastic and phase transformation (p.t.) parts, and resulting from it the additive decomposition of Eulerian strain rate tensor. The hypoelastic rate relations of elasticity involving elastic strain rate \(\underline{{{\mathbf{d}}}}^{\rm e}\) are derived consistent with hyperelastic relations resulting from free energy potential. It is shown that use of Jaumann corotational rate of stress tensor in rate constitutive equations formulation proves to be convenient. The formal equation for p.t. strain rate \(\underline{{{\mathbf{d}}}}^{{\rm in}}\) , describing p.t. deformation effects is proposed, based on experimental evidence. Phase transformation kinetics relations are presented in objective form. The field, coupled problem of thermomechanics is specified in rate weak form (rate principle of virtual work, and rate principle of heat transport). It is shown how information on the material behavior and motion inseparably enters the rate virtual work principle through the familiar bridging equation involving Eulerian rate of nominal stress tensor.

General Corotational Rate Tensor and Replacement of Material-time Derivative to Corotational Derivative of Yield Function

General Corotational Rate Tensor and Replacement of Material-time Derivative to Corotational Derivative of Yield Function

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.

On stress rate and plasticity constitutive equations referred to a body-fixed coordinate system

Investigators have viewed the stress rate in two different ways: the material (body-fixed) point of view and the Eulerian point of view. We discuss the Zaremba–Jaumann rate and Oldroyd’s rate from the material viewpoint and apply them to the material formulation of a theory of plasticity for materials undergoing anisotropic plastic deformation. Significant advantages of the material formulation are that the derivation of equations is straight forward, the distortion of yield surface can be easily accounted for, and the issue of self-consistent elastic equation does not arise. A convected (body-fixed) coordinate system that is imbedded in the material element is used to define a convected material element. Constitutive equations in a theory of plasticity are written with respect to this element referred to the body-fixed coordinate system. A material type yield function is a function of stress components and it is invariant under either a superimposed rigid rotation of the body or an arbitrary change of observer. The kinematic hardening evolves by the co-rotational rate and it is disassociated from the general kinematic of deformation. The latter evolves by the convected rate.

Hardening materials modeling in finite elastic–plastic deformations based on the stretch tensor decomposition

In this paper, finite elastic–plastic deformations of hardening materials are analyzed based on the modified multiplicative decomposition of the left stretch tensor. This decomposition is the modified form of the Metzger and Dubey’s decomposition used in the frame work of the principal axes of the left stretch tensor. For this purpose, basis-free corotational constitutive equations are derived for elastic–plastic hardening materials with the Armstrong–Frederick kinematic hardening and isotropic hardening models. The proposed governing equations are solved with different corotational rates for the simple shear problem with the material properties of the stainless steel SUS 304. The results are compared with those obtained experimentally by Ishikawa. This comparison shows a good agreement between the proposed theoretical model and the experimental data. Also, there is no significant difference in the stress results, using different corotational rates in the proposed model. As another example, the Prager kinematic hardening equation is used. In this case, the stress results using log-rate are close to those obtained by Bruhns et al., in which they used the additive decomposition of the strain rate tensor in a constitutive modeling based on the log-rate. Based on this example, it is noted that for the same corotational rate, the choice of the kinematic decomposition affects the results slightly.

General Corotational Rate Tensor and Replacement to Corotational Derivative of Yield Function

Constitutive equations have to be formulated in an identical form independent of coordinate systems by which they are described so as to be independent of the superposition of rigid body rotation. This fact is attained by describing rate variables in terms of rate tensors with objectivity in constitutive equations in rate form. Besides, a plastic strain rate is derived by substituting the plastic flow rule into the consistency condition given as the material-time derivative of yield condition. In this note a general corotational rate tensor in arbitrary order is first shown and it is verified that rate variables in the material-time derivative of yield function involving arbitrary tensors can be replaced to the corotational rate tensors.

Spin tensors associated with corotational rates and corotational integrals in continua

In many cases of constitutive modeling of continua undergoing large deformations, use of corotational rates and integrals is inevitable to avoid the effects of rigid body rotations. Making corotational rates associated with specific spin tensors is a matter of interest, which can help for a better physical interpretation of the deformation. In this paper, for a given kinematic tensor function, say G , a tensor valued function F as well as a spin tensor Ω 0 is obtained in such a way that the corotational rate of F associated with the spin tensor Ω 0, becomes equal to G . In other words, F is the corotational integral of G associated with the spin tensor Ω 0. Here, G is decomposed additively into the principally diagonal and the principally off-diagonal parts. The symmetric tensors with the same principally diagonal parts have the same tensor valued corotational integrals, but associated with different spin tensors, which depends on the principally off-diagonal part. For the validity of the results, the Eulerian logarithmic strain tensor is shown to be the tensor valued corotational integral of the strain rate tensor, associated with the logarithmic spin. This result has separately been derived by Reinhardt & Dubey and Xiao et al. In addition, a specific corotational rate called Γ-rate and its associated spin tensor are introduced.

Modeling of finite strain viscoplasticity based on the logarithmic corotational description

It has been proven that the deformation rate is identical to the logarithmic corotational rate of the spatial logarithmic strain. Based on the logarithmic corotational description we present an extension of a Chaboche infinitesimal viscoplastic law for finite strain cases. An additive decomposition of the logarithmic corotational rate of the logarithmic strain, implicitly included in an internal dissipation inequality, is applied to the extension. Functionally, this additive decomposition corresponds to the additive decomposition of the material time derivative of the logarithmic strain used in infinitesimalinelastic problems. Using the additive decomposition and replacing the material time derivatives of all second-order tensors in the infinitesimal viscoplastic law with the corresponding logarithmic corotational rates, we arrive at a new finite viscoplastic law with nonlinear isotropic and kinematic hardening. The numerical algorithms suitable for the finite element implementation of this law are formulated and several numerical examples are presented. These numerical examples prove that the finite viscoplastic law and the algorithms are effective and reliable.

The effect of crystal rotation direction on the thermal and velocity fields of a Czochralski system with a low Prandtl number melt

The purpose of this study is to understand the effect of crystal co- and counter-rotation on the thermal and velocity fields of a Czochralski (Cz) melt in a large crucible. The physical modeling of the Cz system was performed using Wood's metal, which has a similar Prandtl ( Pr) number (=0.019) with Si melt and a low melting temperature (70 °C). The velocities and temperatures in a large crucible with 800 mm diameter were measured using an incorporated magnetic probe and thermocouples, respectively. The rate of crystal rotation affects the temperature and velocity fields. Without crystal rotation, the maximum thermal fluctuation region is located under the crystal due to buoyancy convection. With increasing crystal rotations rate, however, the maximum thermal fluctuation region migrates toward the crucible sidewall. When the rate of crystal counter-rotation is increased, the azimuthal velocities decrease and the thermal fluctuations are reduced at the edge of the crystal. With increasing crystal co-rotation rate, however, the azimuthal velocities increase considerably and the temperature fluctuations remain strongly at the edge of the crystal. The results of 3-D numerical simulations using the k– ε turbulent model are well in agreement with the experimental results.

Two Back Stress Hardening Models in Rate Independent Rigid Plastic Deformation

In the present work, the constitutive relations based on the combination of two back stresses are developed using the Armstrong-Frederick, Phillips and Ziegler’s type hardening rules. Various evolutions of the kinematic hardening parameter can be obtained by means of a simple combination of back stress rate using the rule of mixtures. Thus, a wide range of plastic deformation behavior can be depicted depending on the dominant back stress evolution. The ultimate back stress is also determined for the present combined kinematic hardening models. Since a kinematic hardening rule is assumed in the finite deformation regime, the stress rate is co-rotated with respect to the spin of substructure obtained by incorporating the plastic spin concept. A comparison of the various co-rotational rates is also included. Assuming rigid plasticity, the continuum body consists of the elastic deformation zone and the plastic deformation zone to form a hybrid finite element formulation. Then, the plastic deformation behavior is investigated under various loading conditions with an assumption of the J2 deformation theory. The plastic deformation localization turns out to be strongly dependent on the description of back stress evolution and its associated hardening parameters. The analysis for the shear deformation with fixed boundaries is carried out to examine the deformation localization behavior and the evolution of state variables.

Axial effects investigation in fixed-end circular bars under torsion with a finite deformation model based on logarithmic strain

In this paper the torsion problem of a circular bar with fixed ends is solved using a finite deformation constitutive model based on the corotational rates of the logarithmic strain. The logarithmic, Green–Naghdi and Eulerian corotational rates of the logarithmic strain are used in the model. The solution is based on a von Mises type yield function that incorporates isotropic and kinematic hardenings. For the kinematic hardening, a modified Armstrong–Fredrick hardening model with the corotational rate of the logarithmic strain is used. Assuming incompressible behavior, the fixed-end torsion problem is simplified to the simple shear problem. Solving the problem, the stress components are integrated to calculate the torque and axial force. It is qualitatively shown that the results based on the logarithmic corotational rate are in good agreement with the experimental results.

Origin and evolution of deep plasmaspheric notches

Deep plasmaspheric notches can extend over more than 2 RE in radial distance and 3 hours MLT in the magnetic equatorial plane, as observed by the extreme ultraviolet (EUV) imager on the IMAGE mission. They are among the largest evacuated features in the exterior plasmaspheric boundary. They can last for days and exhibit a variety of shapes. It appears that weak convection and limited erosion precedes notch formation at the westward, near‐Earth edge of the convection plume. Eighteen clear notch events were found and analyzed in 2000. Among these events, notches were found to drift as slowly as 44% of corotation. In only one case was a notch found to drift at the corotation rate within measurement error. On average, these notches drift at about 21.5 h d−1 or 90% of the corotational rate. Notches sometimes exhibit an interior structure that appears as an extended prominence of dense plasma, which forms a W‐ or M‐like feature in IMAGE/EUV images, depending on viewing perspective. Initial modeling suggests that notches and notch prominences may be caused in part by intense small‐scale potential structures that result from the localized injection of ring current plasma. Plasma filling rates during recovery are examined in three L shell ranges from L = 2 to L = 3.5 with rates ranging from 5 to 140 cm−3 d−1. Plasma loss during a minor substorm is found to extend to surprisingly low L shell with rates ranging from 100 to 130 cm−3 d−1 across the L shells examined.

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.

Non‐linear analysis of shells with arbitrary evolving cracks using XFEM

AbstractA new formulation and numerical procedures are developed for the analysis of arbitrary crack propagation in shells using the extended finite element method. The method is valid for completely non‐linear problems. Through‐the‐thickness cracks in sandwich shells are considered. An exact shell kinematics is presented, and a new enrichment of the rotation field is proposed which satisfies the director inextensibility condition. To avoid locking, an enhanced strain formulation is proposed for the 4‐node cracked shell element. A finite strain plane stress constitutive model based on the logarithmic corotational rate is employed. A cohesive zone model is introduced which embodies the special characteristics of the shell kinematics. Stress intensity factors are calculated for selected problems and crack propagation problems are solved. Copyright © 2004 John Wiley & Sons, Ltd.

Modeling of the monotonic and cyclic Swift effects using an isotropic, finite viscoplasticity theory based on overstress (FVBO)

The simulation of free-end torsion effects of a tubular specimen using isotropic, FVBO is presented. The limited capability of isotropic formulations to model deformation induced anisotropy is demonstrated. The logarithmic, the Jaumann and the modified Jaumann co-rotational rates are chosen as objective rates. Stress-like state variables should have the same non-zero components as the Cauchy stress to satisfy stress-boundary conditions once equilibrium is reached. This method will be called Case B. However, this is not always practiced. Some papers use different non-zero components. This method will be called Case A. In this paper, two methods are employed to investigate their influence on the free-end torsion behavior. It turns out that numerical experiments according to Method A can only reproduce the loading behavior. Method B can reproduce the cyclic Swift effect qualitatively. The newly introduced logarithmic spin [Acta Mech. 124 (1997) 124 89] had a positive influence on the shape of the curves.

Modeling of the monotonic and cyclic Swift effects using an isotropic, finite viscoplasticity theory based on overstress (FVBO)

The simulation of free-end torsion effects of a tubular specimen using isotropic, FVBO is presented. The limited capability of isotropic formulations to model deformation induced anisotropy is demonstrated. The logarithmic, the Jaumann and the modified Jaumann co-rotational rates are chosen as objective rates. Stress-like state variables should have the same non-zero components as the Cauchy stress to satisfy stress-boundary conditions once equilibrium is reached. This method will be called Case B. However, this is not always practiced. Some papers use different non-zero components. This method will be called Case A. In this paper, two methods are employed to investigate their influence on the free-end torsion behavior. It turns out that numerical experiments according to Method A can only reproduce the loading behavior. Method B can reproduce the cyclic Swift effect qualitatively. The newly introduced logarithmic spin [Acta Mech. 124 (1997) 124 89] had a positive influence on the shape of the curves.

On non–corotational rates of Oldroyd's type and relevant issues in rate constitutive formulations

The lower and upper Oldroyd rates are two wellknown objective noncorotational rates. A remarkable property of these rates is that they establish direct relationships between finite strain measures ...

IMAGE EUV observation of radially bifurcated plasmaspheric features: First observations of a possible standing ULF waveform in the inner magnetosphere

We present EUV observations of the plasmasphere from 1938 to 2211 UT on 28 June 2000 characterized by the presence of bifurcated radial enhancements of the He+ plasma distribution in the nightside sector. These features remain stable throughout the period of observation and are found to move about the Earth at 67% of the corotation rate. Two‐dimensional simulation of the plasmasphere assuming the presence of an azimuthal standing wave patterns at L = 1.8 and 2.5 suggests that the organization of the outer plasmasphere is the result of convective motion driven by a ULF standing wave. Preliminary analysis of ground‐based magnetometer data provided by the IMAGE magnetometer network during the period of EUV observation indicates the presence of several narrow‐frequency oscillations extending down to 0.68 mHz, consistent with simulation results. Similar narrow‐frequency features are found in preliminary analysis of ACE Solar Wind Electron Proton Alpha Monitor (SWEPAM) bulk solar wind velocity data. We speculate that the observed radial He+ enhancements of the nightside plasmasphere may be the result of a nonresonant oscillation driven by the solar wind.

On the Replacement of Material-Time Derivative to Corotational Rate of Yield Function : Mathematical Proof

ABSTRACT Constitutive equations have to be formulated in an indifferent form independent of the frame (i. e. coordinate systems) by which they are described or so as to be independent of the superposition of rigid body rotation. This fact is required by the principle of material-frame indifference (Oldroyd, 1950) and is attained conveniently by describing rate variables in terms of rate tensors with objectivity in constitutive equations in rate form. A plastic strain rate is derived by substituting the plastic flow rulr into the consistency condition given as the material-time derivative of yield condition. In this note the mathematical process demonstrates the fact that rate variables involved in the material-time derivative of yield function can be directly replaced with their objective rate tensors. Here, the yield function involves arbitrary tensors, whilst the special case that the yield function involves only a single tensor or second-order tensors has been discussed in the past.