Corotational Rate Research Articles

Implementation of subloading surface model for hyperelastoplasticity with nonlinear kinematic/isotropic hardening based on reference and intermediate configurations

In this paper, the elastoplastic behavior of isotropic materials under finite strains is modelled and simulated under consideration of the subloading surface concept. Using the subloading surface model has many advantages, such as the ability of modeling more smoothly transition from elastic to plastic state and also more realistic modeling of the Bauschinger effect. For this purpose, three time integration algorithms are utilized in the reference and intermediate configurations in the present study. The first algorithm is based on a time integration scheme, available in the literature, and the second time integration is proposed by tensor exponential-based time integration and the third one is represented by a standard backward Euler time integration scheme. In the second proposed time integration, which is defined in the intermediate configuration, there is no need to use co-rotational objective time rates. However, in the third one, although it is based on the intermediate configuration, the Zaremba–Jaumann co-rotational rate is employed. Besides, in the second two time integrations, the constitutive equations are derived based on the unknown tensors defined in the intermediate configuration and as a consequence there is no restriction of using different hyperelastic models. For case studies, diverse deformation gradient tensors with and without volume changes are considered. These studied cases are accomplished for SUS 304 stainless steel and a glassy polymer (oriented PET). The obtained results of some cases are compared with available results in the literature. The agreement among the predicted results in this paper and the results from literature is good.

Rate-type models of dissipative compressible fluids

Two models of dissipative compressible fluids are established within the general setting of Rational Thermodynamics. It is a common feature underlying the analysis of the thermodynamic consistency that the entropy production and the free energy potential are viewed as constitutive functions of the same set of physical variables and no internal variable is involved. Also in light of late experimental observations, hysteretic models are established for the dependence of pressure on the mass density. Next relaxation properties are modelled via rate equations with an objective derivative which combines the corotational rate with the divergence of the velocity.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

The main goal of this work is to test the possibility of a newly introduced constitutive law to model the behaviour of the isotropic elastic-perfectly plastic material which is exposed to large elastoplastic deformations. The proposed constitutive relation is based on the hypo-elastic relation and the inelastic INTERATOM model. The verification of the model is done by its implementation into the commercial software ABAQUS/Standard via the user subroutine UMAT. For that purpose, the large simple shear problem is studied where selected objective corotational rates, i.e. the logarithmic rate, the Jaumann rate and the Green-Naghdi rate, are individually implemented in the aforementioned constitutive relations. The obtained results are compared mutually and with the relevant literature. The proposed constitutive model is also used to test the behaviour of the part of a real engineering structure, i.e. a seismic isolator, in order to obtain the correct input data for further analysis of superstructure behaviour due to seismic excitation.

Fractional derivative models for viscoelastic materials at finite deformations

Fractional derivative models, which are expressed by combining standard dashpots, fractional dashpots and elastic springs in series or parallel, are often utilized to account for the behaviors for viscoelastic materials. Even with the models extended to finite deformation, the precise definition of objective fractional derivative remains challenging. The proposed fractional derivative model is expressed by the combination of an elastic spring in series with two parallel fractional dashpots. We extend the fractional derivative model to finite deformation through a new approach without defining an objective fractional derivative and assuming the decomposition of the deformation rate into the elastic and inelastic parts. This proposed model can be reduced to the Maxwell model for finite deformation. Such reduction results in a model that stands in between the two existing Maxwell models in which the objective rate of the Cauchy stress is taken as the material corotational rate and the relative corotational rate respectively. The proposed model is applied to the simple shear deformation.

Numerical analysis of finite hypo-elastic cyclic deformation with large rotations

Constitutive relations which describe engineering materials behaviour during the finite elastoplastic deformations are usually presented in the form of rates of stresses and strains. One of the possible approaches in the constitutive relations formulation is the additive decomposition of the total deformation rate into its elastic part and its plastic part. The elastic deformation rate contributes to any elastoplastic deformation at any stage. Hence, its exact and well-considered formulation is of particular importance and it has to be properly predicted by the corresponding material law. This is of great importance in particular when deformation cyclic processes are considered, in which case small errors may accumulate, even if the total deformation is small. The implementation of the most frequently used corotational rates, i.e. the Jaumann rate and the Green-Naghdi rate, in the hypo-elastic constitutive relations regarding small and moderate rotations gives accurate results for low number of repeated deformation cycles. With increased number of cycles, however, the implementation of these rates results in different and physically non-admissible material responses. This instability in results is particularly observable during the cyclic deformations with large rotations, which is the main subject of this work. In contrast to the aforementioned objective rates, the results of the logarithmic rate implementation into the hypo-elastic constitutive relations for the case of pure elastic deformation describe a physically stable process.

Material Spin and Finite-Strain Hypo-Elasticity for Two-Dimensional Orthotropic Media

A constitutive material spin tensor in the case of purely elastic finite-strain deformation is introduced for a two-dimensional orthotropic media using the minimizing principle applied to obtain the reloaded configuration of the material volume. This material spin explains the rotation of the orthonormal vector frame which coincides with the material symmetry axes in the initial configuration of the material volume and uniquely corresponds to a set of these axes in the current configuration although it does not coincide with the latter. The given definition is followed by the exact expression which includes the deformation gradient tensor, unit vectors of the initial material anisotropy axes and their axial parameters. This definition allows obtaining a new variant of decomposing any elastic finite-strain motion onto rigid and deformational parts and introducing the material corotational rate. The latter is used for the formulation of the anisotropic rate-type elastic law in the current configuration based on the strain measure which does not belong to the Seth-Hill family. For isotropic as well as for tetragonal media, the introduced material rotation tensor coincides with the rotation tensor from the polar decomposition of a deformation gradient.

Kinematical Aspects of Levi-Civita Transport of Vectors and Tensors Along a Surface Curve

The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.

Directional distortional hardening at large plastic deformations

This paper extents the directional distortional hardening model of Feigenbaum and Dafalias (2007) into the range of large plastic deformations. This model allows the yield surface to deform such that a region of high curvature develops approximately in the direction of loading and a region of flattening develops on the opposite side. To extend this model into large deformations and in order to ensure positive dissipation and objectivity, hardening rules are derived from thermodynamic conditions in terms of corotational rates. Since this model includes a fourth order tensor-valued hardening internal variable, the corotational rates for fourth order tensors are examined in this work employing the concept of plastic spin. Several choices for plastic spins are presented and used for the simulation of the response under simple shear loading up to 1000% strain.

Hencky strain and logarithmic rates in Lagrangian analysis

An hypothesis conjectured independently by Lehmann et al. (1991), Reinhard and Dubey (1995, 1996) and Xiao et al. (1997) takes Hencky strain H=lnV as the strain measure of choice and supposes the existence of a co-rotational rate H• whose components are equivalent to those of the rate of deformation D for all values of stretch V. From this conjecture, they derived the governing spin tensor Z, known today as the logarithmic spin. Applying the same co-rotational rates to both stress and strain ensures consistency within a constitutive construction, even integrability for sufficiently simple models. Their derivations were done in the Eulerian frame of reference. Here their hypothesis is extended to the Lagrangian frame of reference. Adopting mixed tensor fields allows this construction to take on its most simple form.

Signatures of magnetospheric injections in Saturn's aurora

We report on the evolution of auroral structures based on two 3 h sequences of Saturn's northern hemisphere obtained with the UVIS instrument on board Cassini, and we discuss their possible association with injections in the magnetosphere of Saturn. Simultaneously with the UV auroral structures, we observe Energetic Neutral Atom (ENA) enhancements which are indicative of a rotating heated plasma region possibly related to magnetospheric injections. We examine the possibility that the UV auroral structures reported here are triggered by energetic particle injections by investigating (a) the evolution of the longitudinal extent of an injection and (b) its energy density, properties that change with time due to dispersion and ion/electron losses. We simulate the auroral counterpart of an injection considering that the precipitating energy flux could be provided to the ionosphere by pitch angle diffusion and electron scattering by whistler‐mode waves. We compare the brightness and size evolution of the simulated ionospheric signature with the observed values, and we demonstrate that the UV auroral structure behaves as an auroral signature of an injection. This comparative study defines characteristics of the injections such as the spectral index and the electron energy range as well as the magnetospheric corotation rate. Additionally, based on the simultaneous ENA‐UV emissions, we discuss the possibility that pitch angle diffusion and electron scattering may not be the only mechanism responsible for the observed auroral emissions. Field‐aligned currents driven by the pressure gradients along the boundaries of the injected hot plasma cloud could be also considered to play a role on how injections create auroral emissions at Saturn.

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part I: Rate-Form Hyperelasticity

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

Non-equilibrium thermodynamics for fully coupled thermal hydraulic mechanical chemical processes

In this paper, a pioneering approach of reactive transport in porous media is introduced, which model thermal–hydraulic–mechanical–chemical processes. The novelties of this approach are: (i) non-equilibrium thermodynamics which is used as a unifying framework relating generalized fluxes to forces and (ii) fully coupled integration of the multi-physics processes, introduced within the framework of large transformations including logarithmic finite strain and co-rotational rates. This formulation opens the horizons for complex simulations which were difficult to conduct previously because of the lacking bridges between non-linear computational mechanics and reactive transport processes. As an illustration of the model, a sample of simple geometry is subjected to a non-linear deformation beyond the reversible regime. This perturbation from equilibrium produces a permanent deformation, an overpressure and a temperature change. The subsequent thermodynamic conditions trigger chemical reactions among the aqueous species which are not necessarily in equilibrium with their environment. The deformation also induces a change of porosity which affects the permeability as well as the pore pressure distribution.

Poromechanics of saturated media based on the logarithmic finite strain

In this paper, we introduce the mathematical formulation and numerical implementation of a coupled thermo-hydro-mechanical model for saturated poromaterials undergoing logarithmic finite deformation and corotational rates. The model combines (i) the thermodynamics of standard materials, (ii) the frame indifferent hyperelastoplasticty, (iii) the orthogonality condition of maximum dissipation, as well as (iv) the principles of conservations of mass, energy and momenta. This formulation involves new developments based on the logarithmic strain measures and corotational rates which overcome the aberrant oscillations classically encountered in large simple shear. It also takes into accounts recent findings on the thermodynamics of dissipative materials which consist of deriving the yielding conditions and flow rules from suitable free energy and dissipation functions. This framework resulted in the implementation of a new finite element algorithm based on Galerkin’s method. The numerical procedures used in this paper involve the spectral decomposition of the logarithmic strain measures, the gradient split techniques as well as the return mapping method. The formulation is validated using the classical problems of Terzaghi and strip loading consolidation.

Constitutive modeling of temperature and strain rate dependent elastoplastic hardening materials using a corotational rate associated with the plastic deformation

In this paper, a constitutive model with a temperature and strain rate dependent flow stress (Bergstrom hardening rule) and modified Armstrong–Frederick kinematic evolution equation for elastoplastic hardening materials is introduced. Based on the multiplicative decomposition of the deformation gradient,new kinematic relations for the elastic and plastic left stretch tensors as well as the plastic deformation-dependent spin tensor are proposed. Also, a closed-form solution has been obtained for the elastic and plastic left stretch tensors for the simple shear problem.To evaluate model validity, results are compared with known experimental data for SUS 304 stainless steel, which shows a good agreement with the results of the proposed theoretical model.Finally, the stress-deformation curve, as predicted by the model, is plotted for the simple shear problem at room and elevated temperatures using the same material properties for AA5754-O aluminium alloy.

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green–Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius–Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.

Frame indifferent elastoplasticity of frictional materials at finite strain

This paper puts forward a finite strain formulation based on the (i) thermodynamics of non-associated materials, (ii) logarithmic strain measures and corotational rates, and (iii) numerical procedures of gradient split and return mapping. Unlike most of the existing finite strain formulations which use classical strain measures and objective corotational rates, the current approach emphasises the logarithmic objective description as an alternative. This formulation overcomes the aberrant oscillations or hyperbolic responses which appear in shear zones when classical spins are used. The model combines this kinematic description with recent developments which extended the classical thermodynamics of generalized standard materials to non-associated materials. This thermodynamic framework allows deducing the constitutive relationships, yielding limits and flow rules directly from the energy functions instead of introducing them on ad hoc basis as commonly accepted in the classic theories of soil and rock mechanics.

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.

An Eulerian multiplicative constitutive model of finite elastoplasticity

An Eulerian rate-independent constitutive model for isotropic materials undergoing finite elastoplastic deformation is formulated. Entirely fulfilling the multiplicative decomposition of the deformation gradient , a constitutive equation and the coupled elastoplastic spin of the objective corotational rate therein are explicitly derived. For the purely elastic deformation , the model degenerates into a hypoelastic-type equation with the Green–Naghdi rate. For the small elastic- and rigid-plastic deformations, the model converges to the widely-used additive model where the Jaumann rate is used. Finally, as an illustration, using a combined exponential isotropic-nonlinear kinematic hardening pattern, the finite simple shear deformation is analyzed and a comparison is made with the experimental findings in the literature.

Eulerian conjugate stress and strain

New results are presented for the stress conjugate to arbitrary Eulerian strain measures. The conjugate stress depends on two arbitrary quantities: the strain measure f(V) and the corotational rate defined by the spin \Omega. It is shown that for every choice of f there is a unique spin, called the f-spin, which makes the conjugate stress as close as possible to the Cauchy stress. The f-spin reduces to the logarithmic spin when the strain measure is the Hencky strain log(V). The formulation and the results emphasize the similarities in form of the Eulerian and Lagrangian stresses conjugate to the strains f(V) and f(U), respectively. Many of the results involve the solution to the equation AX-XA=Y, which is presented in a succinct format.

Plasma convection in Saturn's outer magnetosphere determined from ions detected by the Cassini INCA experiment

The Ion and Neutral Camera (INCA), one of three sensors comprising the Magnetospheric Imaging Instrument (MIMI) on the Cassini spacecraft, measures intensities of hydrogen and oxygen ions and neutral atoms in the Saturnian magnetosphere. The measured intensity spectrum and anisotropy of hot hydrogen and oxygen ions may be used to deduce the spectral parameters and the velocity of the ion population. The anisotropies are frequently convective in nature, allowing for the determination of a bulk velocity. Under the “frozen in” assumption, this is also the velocity of the cold plasma of magnetospheric ions. Initial analysis of selected measurements of nightside ion populations with strong anisotropies indicates nearly rigid corotation of magnetospheric plasma interior to Titan's orbit. Beyond this distance, these measurements infer that the plasma maintains at best a constant rotation velocity, falling farther behind the rigid corotation rate at increasing distance.