Jaumann Derivative Research Articles

Efekty lepkosprężyste w warunkach nawadniania kolektora szczelinowo-porowego cieczą wodno-polimerową

The work is devoted to the numerical simulation of the flow of simple and viscoelastic (polymer solution) fluids through a fracture by using polymer solutions for enhanced oil recovery from a reservoir. Polymer solutions have viscoelastic properties. Therefore, when polymer solution flows through the slot, we use the well-known Maxwell’s fluid model with the Jaumann derivative to evaluate deformation characteristics of the flow (stream functions, distributions of the longitudinal velocity gradient and normal stress) resulting in the manifestation of abnormal (compared with the behaviour of the ordinary fluid) effects. The case of slow flow is considered. In this case, the inertial terms can be neglected, the velocities, stresses, and stream functions can be written as the decomposition by Weisenberg number, and we can assume that the Weissenberg number is less than one. The determined regularities of viscoelastic (polymer solution) liquid behaviour with longitudinal velocity gradient and the elastic deformations effects manifested in this case have a decisive meaning in understanding the mechanism of anomalously high oil recovery capacity of a reservoir by using water-polymer flooding of the fractured-porous reservoir. Understanding the nature of the effects of elastic deformations under the conditions of water-polymer flooding of the fractured-porous reservoir enables hydrodynamic calculations of the optimal flow of the polymer solution. One of the main issues that need to be solved when developing the technology for increasing oil recovery from formations using polymer solutions is to determine the optimal flow regime in the fractured-porous reservoir. The calculation results verify the ideas obtained from the experimental solution of this problem concerning the strain-stress state of polymer macromolecules (liquid elements) during polymer flow in the inlet area of the fracture in the oil reservoir and confirm the possibility of using numerical analysis of convergent polymer flow for calculating longitudinal velocity gradients in the inlet area of the fracture and can also serve as additional substantiation of the proposed earlier mechanism for increasing oil recovery from reservoirs by using polymer solutions.

Leray–Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba–Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.

Creep, recovery and vibration of an incompressible viscoelastic material of the rate type: Simple tension case

We consider three-dimensional nonlinear viscoelastic models that account for both stress relaxation and creep/recovery phenomena. These models are based on different frame indifferent time derivatives: the Oldroyd (or upper-convected) derivative, the Jaumann (or co-rotational) derivative and the Cotter–Rivlin (or lower-convected) derivative. Under a simple tension creep process, these constitutive equations predict the same stress relaxation but lead to different situations. The models based on the Oldroyd and the lower-convected derivative require restrictions on the values of the material parameters as well as on the traction/compression stress. The model based on the Jaumann derivatives does not require any restriction. All the constitutive models examined are used to study the finite amplitude, horizontal oscillatory motion of a mass attached to a rate-type viscoelastic string. In this way we generalize the classical results by Beatty and Zhou (1991).

A strongly objective expression for the average deformation rate with application to numerical integration algorithms

Although the Hughes-Winget rotation tensor is not strongly objective when stretching occurs during a time step it is very accurate when the incremental stretching during the time step is small, a condition that is almost always met in practice. Two types of evolution equations based on the Jaumann derivative are examined which show that the main source of error in the numerical algorithms is due to the approximation of the average deformation rate during the time step. A new strongly objective expression for the average deformation rate during a time step is developed and shown to correct the observed errors in tests of strong objectivity.

Positional finite element formulation for two-dimensional analysis of elasto-plastic solids with contact applied to cold forming processes simulation

In this work, we develop a position-based finite element formulation for elasto-plastic solids under contact situation. The proposed positional formulation employs a total Lagrangian description and naturally considers geometric nonlinearities. The employed elasto-plastic model is derived from the dissipation inequality, using the thermodynamic conjugacy between the plastic strain rate and the so-called Mandel stress. The formulation is based on the Kroner–Lee decomposition, in which the deformation gradient is multiplicatively split into its elastic and plastic parts. We apply the backward Euler method to solve the plastic evolutions and von Mises yield criterion to define the elastic limit. The adopted kinematic hardening model is a finite strain generalization of the Armstrong–Frederick law, which uses the objective Jaumann derivative for the evolution equation and the concept of back stress tensor as an internal variable. For the elastic parcel of strains, we adopt a neo-Hookean constitutive law. With respect to the 2D application, plane strain and plane stress approximations are considered, where the latter is solved numerically by a local Newton–Raphson numerical procedure. Regarding the contact problem, a classical node-to-segment algorithm is applied, considering both frictionless and frictional cases, with the introduction of Lagrange multipliers in order to enforce contact constraints. Representative numerical examples are used to validate and show the possibilities of the proposed formulation in macroscale simulation of metal cold forming manufacturing processes.

The Stress-Strain State of a Hypoelastic Cylindrical Layer under Finite Strain

The quasistatic problem of determining the stress-strain state in a cylindrical axisymmetric layer is solved. The layer is subjected to the action of tangential forces directed along the axis of the cylinder and uniformly distributed over the outer surface of the layer whose inner surface is fixed. A hypoelasticity model for an incompressible material under finite strain is adopted. This model is based on using the following objective derivatives of the corotation type: the Jaumann derivative and the Green-McInnis-Naghdi derivative. The analysis of this model shows the existence of the Kelvin and Poynting effects.

О единственности классического решения начально-краевой задачи течения вязкоупругой среды Максвелла с нулевыми скоростями на границе

Рассматривается система уравнений с вращательной производной Яуманна, описывающая движения вязкоупругой несжимаемой среды Максвелла в ограниченной области трехмерного пространства в приближении Стокса (без конвективных слагаемых как в уравнении движения, так и в реологическом соотношении). Выбор объективной производной Яуманна в реологическом соотношении обусловлен наличием энергетического тождества, справедливость которого не удается доказать для случаев использования верхней и нижней конвективных производных в качестве объективных производных. Доказывается единственность классического решения начально-краевой задачи с условием прилипания на границе области течения в классе достаточно гладких функций в предположении, что система обладает четырьмя различными вещественными звуковыми характеристиками. Доказательство основано на применении интегральных оценок и использовании условия гиперболичности системы. Для наглядности выкладки проводятся для двумерного случая, однако нигде не используются свойства двухмерности. Отличия от трехмерного случая носят чисто технический характер. Отмечается, что наличие конвективных членов в уравнениях движения не препятствует доказательству единственности. Единственность первоначально доказывается для малого промежутка времени, затем стандартным образом утверждение распространяется на промежуток произвольной длины. DOI 10.14258/izvasu(2018)4-17

Stress–strain measures used in the ANSYS package to solve elastoplastic problems at finite strains

It is shown that the Dienes (or the Green–McInnis–Naghdi) derivative is used in the ANSYS package rather than the Jaumann derivative as an objective derivative for the Cauchy stress tensor when solving elastoplastic problems, although the usage of the Jaumann derivative is stated in the ANSYS theory reference manual. In this manual it is also stated that, for these problems, the strain tensor is the Hencky logarithmic strain tensor; however, in reality, this strain tensor is the right nonholonomic strain tensor such that the left nonholonomic strain tensor associated with the right one is generated by the Dienes derivative.

Characteristic properties of the system of equations for an incompressible viscoelastic Maxwell medium

Characteristics of a system of equations that describe three-dimensional motion of an incompressible viscoelastic Maxwell medium with the upper and lower convective derivatives and the rotational Jaumann derivative being used in the rheological relation are calculated. An initial-boundary-value problem is formulated for the system linearized in the vicinity of the state at rest, and its unique solvability is established.

О ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ ОПРЕДЕЛЯЮЩИХ СООТНОШЕНИЯХ УПРУГОГО МАТЕРИАЛА

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process. The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form. In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation. As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor. The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration. Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways. An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), “not closed” stress trajectories and nonzero stress work for a closed deformation trajectory. In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor. When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations. The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case - elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures. Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various measures of strain state and their corotational derivatives are considered; a kinematic loading in a closed cycle (in the space of deformations) is applied. It is shown for isotropic elastic materials that stress trajectories are closed. When using work-conjugacy stress and strain measures the dissipation energy is absent. Thus it is possible to question the exclusive choice of logarithmic spin and the above-mentioned commonly used measures of stress-strain state. In formulating and solving the problems of solid mechanics there should be an opportunity to use different stress-strain measures and their objective derivatives, the choice of measures and constitutive relations should be justified from the point of physical analysis of the process.

A NURBS‐based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach

SummaryA numerical model to deal with nonlinear elastodynamics involving large rotations within the framework of the finite element based on NURBS (Non‐Uniform Rational B‐Spline) basis is presented. A comprehensive kinematical description using a corotational approach and an orthogonal tensor given by the exact polar decomposition is adopted. The state equation is written in terms of corotational variables according to the hypoelastic theory, relating the Jaumann derivative of the Cauchy stress to the Eulerian strain rate.The generalized‐α method (Gα) method and Generalized Energy‐Momentum Method with an additional parameter (GEMM+ξ) are employed in order to obtain a stable and controllable dissipative time‐stepping scheme with algorithmic conservative properties for nonlinear dynamic analyses.The main contribution is to show that the energy–momentum conservation properties and numerical stability may be improved once a NURBS‐based FEM in the spatial discretization is used. Also it is shown that high continuity can postpone the numerical instability when GEMM+ξ with consistent mass is employed; likewise, increasing the continuity class yields a decrease in the numerical dissipation. A parametric study is carried out in order to show the stability and energy budget in terms of several properties such as continuity class, spectral radius and lumped as well as consistent mass matrices. Copyright © 2015 John Wiley & Sons, Ltd.

On steady motion of viscoelastic fluid of Oldroyd type

We consider a mathematical model describing the steady motion of a viscoelastic medium of Oldroyd type under the Navier slip condition at the boundary. In the rheological relation, we use the objective regularized Jaumann derivative. We prove the solubility of the corresponding boundary-value problem in the weak setting. We obtain an estimate for the norm of a solution in terms of the data of the problem. We show that the solution set is sequentially weakly closed. Furthermore, we give an analytic solution of the boundary-value problem describing the flow of a viscoelastic fluid in a flat channel under a slip condition at the walls. Bibliography: 13 titles.

NURBS-based three-dimensional analysis of geometrically nonlinear elastic structures

A comprehensive kinematical description with a corotational approach, which is based on exact polar decomposition for solid elements using isogeometric analysis is the main objective of this work. A numerical model using NURBS-based FEM is developed to deal with nonlinear problems in solid mechanics involving large rotations. From the theoretical viewpoint, the rate type form of state equation is formulated within the framework of isotropic infinitesimal elasticity. It is shown in particular that the state equation relates the Jaumann derivative of the Cauchy stress to the Eulerian strain rate. Illustrative examples are analyzed to verify the efficiency in solving a wide variety of strong nonlinear problems. • NURBS-based analysis is employed to deal with nonlinear problems in solid mechanics. • Kinematical description with a corotational-based approach is proposed. • The rate form of state equation is formulated within framework of large rotations. • Several examples of shells structures are analyzed numerically.

A cell-centered Lagrangian finite volume approach for computing elasto-plastic response of solids in cylindrical axisymmetric geometries

A finite volume cell-centered Lagrangian formulation is presented for solving large deformation problems in cylindrical axisymmetric geometries. Since solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum and energy conservation laws. The total strain-rate realized in the material is split into an elastic and plastic response. The elastic and plastic components in turn are modeled using hypo-elastic theory. In accordance with the hypo-elastic model, a predictor–corrector algorithm is employed for evolving the deviatoric component of the stress tensor. A trial elastic deviatoric stress state is obtained by integrating a rate equation, cast in the form of an objective (Jaumann) derivative, based on Hooke’s law. The dilatational response of the material is modeled using an equation of state of the Mie-Grüneisen form. The plastic deformation is accounted for via an iterative radial return algorithm constructed from the J2 von Mises yield condition. Several benchmark example problems with non-linear strain hardening and thermal softening yield models are presented. Extensive comparisons with representative Eulerian and Lagrangian hydrocodes in addition to analytical and experimental results are made to validate the current approach.

Postbuckling elastoplastic state analysis of three-dimensional bodies taking into account finite strains

A technique for analyzing the stress strain state of elastoplastic bodies taking into account large displacements, rotations, finite strains, and buckling is presented. The step-by-step loading method with formulation of the resolvent variation equation in the current configuration is used. The governing equations for elastoplastic bodies that connect the Jaumann derivative of the Cauchy-Euler stress tensor with the velocity strain are applied. The spatial discretization is based on the finite element method (FEM) in the framework of the polylinear three-dimensional isoparameteric approximation. The numerical solutions of the problems are given.

Approximating-topological methods in some problems of hydrodynamics

We describe a new approach to the study of initial-boundary value problems for evolutionary equations of hydrodynamics, which is based on approximation of the problems and subsequent application of topological degree theory for investigation of weak solvability for approximating problems. Use of this method turned out to be especially effective in problems of non- Newtonian hydrodynamics. We demonstrate its application to the study of weak solvability and attractors of the initial-boundary value problem for the Oldroyd model with regularized objective Jaumann derivative.

Application of Anand's constitutive model on twin roll casting process of AZ31 magnesium alloy

Twin-roll thin strip casting process combines casting and hot rolling into a single process, in which thermal stress and thermal mechanical stress were involved. Considering the high temperature gradient, the existing of liquid and solid regions and rolling deformation, suitable constitutive model is the key to describe the process. Anand's model is a temperature-dependent, rate-dependent and unified of creep and plasticity model and the Jaumann derivative was employed in Anand's model which makes the constitutive model frame-indifferent or objective, therefore the highly nonlinearities behavior in the twin-roll casting process can be simulated. The parameters of the Anand's model were regressed based on the compression tests of AZ31 magnesium alloy. The simulation results reveal that the Anand's model can well describe the deformation characteristics of twin-roll casting process. Based on the simulation results, the form of evolution equations in Anand's model was discussed.

Numerical Investigation of Large Elastoplastic Strains of Three-Dimensional Bodies

A method of stress—strain analysis of elastoplastic bodies with large displacements, rotations, and finite strains is developed. The incremental loading technique is used within the framework of the arbitrary Lagrangian—Eulerian formulation. Constitutive equations are derived which relate the Jaumann derivative of the Cauchy—Euler stress tensor and the strain rate. The spatial discretization is based on the FEM and multilinear three-dimensional isoparametric approximation. An algorithm of stress—strain analysis of elastic, hyperelastic, and perfectly plastic bodies is given. Numerical examples demonstrate the capabilities of the method and its software implementation

Reynolds stress model involving the mean spin tensor

In this work, we develop a Reynolds stress model along the line of the approach presented by Commun. Nonlinear Sci. Numer. Simul. 9, 543 (2004)]], aiming to assess the role and contribution of the mean spin tensor in turbulence modeling. Here, the constitutive functional for the Reynolds stress depends on the mean spin tensor as well as the mean stretching tensor and its Jaumann derivative, the turbulent kinetic energy K , and the turbulent dissipation rate epsilon , which is at the complexity level of p=1,m=1 , and n=0 of a rate-type constitutive equation for the Reynolds stress proposed in the aforementioned paper. The explicit form for the Reynolds stress is obtained with recourse to the representation theorem and the theory of invariants developed in modern rational continuum mechanics, and, as an approximation, a nonlinear cubic K-epsilon model is worked out in which the model coefficients are analytically identified based on the experimental results of Tavoularis and Corrsin [J. Fluid Mech. 104, 311 (1981)]]. In addition, numerical results based on this model, in the forms of employing the Jaumann derivative and the Oldroyd derivative, respectively, for homogeneous turbulent shear flow and fully developed turbulent flow over a backward-facing step, are presented in comparison with those obtained based on a few previously proposed linear and nonlinear K-epsilon models, showing reasonably good agreement with the experimental results and the DNS data concerned and a better performance than the previously developed quadratic models.

Stability of rheometric viscoelastic fluid flow with respect to shear perturbations

Flow between two plates is considered for a fluid obeying the DeWitt rheological equation of state with the Jaumann derivative. It is found analytically that the steady-state Couette flow is stable or unstable with respect to plane shear perturbations when the Weissenberg numbers are less or greater than unity, respectively. The flow acceleration stage is studied analytically and numerically, a comparison with the case of an Oldroyd fluid is carried out, and the neutral stability curves are constructed. The fundamental role of perturbations of the type considered among the set of instability types which can act on the fluid in such a flow is noted.