Objective Stress Rate Research Articles

Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

Review of energy conservation errors in finite element softwares caused by using energy-inconsistent objective stress rates

The paper briefly summarizes the theoretical derivation of the objective stress rates that are work-conjugate to various finite strain tensors, and then briefly reviews several practical examples demonstrating large errors that can be used by energy inconsistent stress rates. It is concluded that the software makers should switch to the Truesdell objective stress rate, which is work-conjugate to Green’s Lagrangian finite strain tensor. The Jaumann rate of Cauchy stress and the Green-Naghdi rate, currently used in most software, should be abandoned since they are not work-conjugate to any finite strain tensor. The Jaumann rate of Kirchhoff stress is work-conjugate to the Hencky logarithmic strain tensor but, because of an energy inconsistency in the work of initial stresses, can lead to severe errors in the cases of high natural orthotropy or strain-induced incremental orthotropy due to material damage. If the commercial softwares are not revised, the user still can make in the user’s implicit or explicit material subroutines (such as UMAT and VUMAT in ABAQUS) a simple transformation of the incremental constitutive relation to the Truesdell rate, and the commercial software then delivers energy consistent results.

Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since it is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the Truesdell objective stress rate, which is work-conjugate to the Green–Lagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green–Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated.

Comparison of Various Object Stress Rates under Simple Shear

Object Stress rates to predict the behavior of material have been researched based on numerically and theoretically for researchers who study continuum mechanics due to its complexity. This study focused on the various objective stress rates which assumed the finite deformation theory. Eight object stress rates (Oldroyd, Truesdell, Cotter–Rivlin, Jaumann, Green–Naghdi, Eulerian, grangian, and logarithmic object stress rates) were introduced using continuum mechanics and analyzed to derive the numerical solution to the simple shear problem. Numerical results from each object stress rate were analyzed and compared with the results of the other stress rates. Finally, the appropriate object stress rate for the simple shear problem was determined based on the numerical results from eight objects stress rates.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

In this paper, we present a consistent spatial formulation for discontinuous Galerkin (DG) methods applied to solid mechanics problems with finite deformation. This spatial formulation provides a general, accurate, and efficient DG finite element computational framework for modeling nonlinear solid mechanics problems. To obtain a consistent formulation, we employ the Incomplete Interior Penalty Galerkin (IIPG) method. Another requirement for achieving a fast convergence rate for Newton’s iterations is the consistent formulation of material integrators. We show that material integrators that are well developed and tested in continuous Galerkin (CG) methods can be fully exploited for DG methods by additionally performing stress returning on element interfaces. Finally, for problems with pressure or follower loading, stiffness contributed from loaded surfaces must also be consistently incorporated. In this work, we propose the Truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems. Two formulations based on the co-rotational and multiplicative decomposition-based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity, respectively. Two new terminologies, the so-called standard surface geometry stiffness and the penalty surface geometry stiffness, are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals. The performance of our DG formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations, as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes. Fast convergence rates for Newton’s iterations have been achieved in our IIPG implementation.

Hypo- and hyper-inelasticity applied to modeling of compacted graphite iron machining simulations

In the present paper we are concerned with constitutive modeling and validation of the thermomechanically coupled Compacted Graphite Iron (CGI) machining problem. Particular emphasis is placed on the significance of the choice of different hypo-inelastic models induced by different objective stress rate formulations. We also relate to a thermodynamically consistent hyper-elastic–inelastic formulation based on multiplicative decomposition of the deformation gradient . The consequently induced tangent material behavior is then derived in the spatial setting in terms of the Oldroyd stress rate, and it is compared to the hypo–formulations. The Johnson–Cook (JC) model is taken as the main prototype for the modeling of isotropic hardening , strain rate and temperature dependencies , which is considered reframed within the Perzyna visco-plasticity framework, thereby highlighting the quasistatic and rate dependent properties of the model. The different models are compared both on the material point level (simple shear and uniaxial tensile–compressive tests) and on the structural level (FE-analysis of a 2D shear test and in representative CGI-machining simulations) and the resulting mechanical isothermal behavior obtained from the different ways of establishing the objective stress rate are surprisingly similar. Based on the results obtained a hypo-inelastic formulation based on a modified Oldroyd stress rate is proposed due to its link to thermo-mechanical consistency and relative computational efficiency. ► Validation of the thermomechanically coupled CGI-machining problem. ► Comparison is made between different hypo- and hyper-inelastic formulations. ► The Johnson–Cook model is used to represent major inelastic mechanisms. ► The models are compared in simple shear, and in a CGI-machining simulation. ► The resulting mechanical isothermal behaviors are surprisingly similar.

Length scale effects in finite strain micromorphic linear isotropic elasticity: finite element analysis of three-dimensional cubical microindentation

A three-dimensional finite strain micromorphic materially linear isotropic elastic model is formulated in two ways for finite element implementation: (i) direct finite strain elasticity; and (ii) rate form with semi-implicit time integration. The model is based upon the finite strain isotropic micromorphic elasticity model proposed by Eringen and Suhubi in 1964. For (i), the direct formulation, the constitutive equations are calculated in the reference configuration, and the resulting stresses are mapped to the current configuration. For (ii), the rate formulation, the constitutive equations are integrated in time in the current configuration using the Truesdell objective stress rates. After formulating the coupled weak form, the balance of linear momentum and the balance of first moment of momentum are linearized to construct the consistent tangent for solution by the Newton–Raphson method. Three-dimensional numerical examples are analysed to compare the two formulations (i) and (ii) for standard finite strain isotropic elasticity, and the formulation (i) is used to demonstrate the elastic length scale effects that come through the higher-order couple stress in the micromorphic theory.

슬관절 전방 십자 인대의 반복 변형하에서의 역학적 거동에 관한 수치적 연구

Anterior cruciate ligament(ACL) of human body experiences a large deformation. May during everyday when large deformation is repeated by various activities such as outdoor activity, ACL easily get damaged. In order to acknowledge the effect of the cyclic large deformation to ACL, the constitutive equations for ACL are derived from experiment data. The concept of the objective stress rate plays a important role wherever large deformation occurs. In order to obtain the objective stress rates the eigenprojection technique is used. A comparison is made for four different cases: Jaumann rate, Green-Naghdi rate, logarithmic rate and twirl tensor of Eulerian triad rate for an isotropic material subject to cyclic deformation, such as simple shear motion. Four different materials are studied to compare the behavior of the materials for ACL using different objective rates. Finally, more complicated model with fibers for soft tissues is used to calculate the behavior subjected to cyclic large deformation.

Investigation on the Large Strain Elastoplastic Constitutive Model with Logarithmic Stress Rate Based on Torsion Experiments

A complete analysis was made for an isotropic hardening rate-type elastoplastic constitutive model with the logarithmic stress rate utilizing solid shafts torsion test in the large strain range. The deformation rate, the logarithmic spin, the Kirchhoff stress and the logarithmic stress rate of the Kirchhoff stress were obtained for the free-end solid shaft torsion test when considering Swift effect. Utilizing the results obtained from the solid shaft torsion test, the plastic rigidity function in the isotropic hardening elastoplastic constitutive model was determined at finite strain range. It was shown that the influence of Swift effect on finite strain constitutive model was related to the varying rate of axial deformation and the varying rate of radius deformation to shear strain, and the plastic rigidity function corresponding to the logarithmic stress rate was the same as that corresponding to the Jaumann stress rate when neglecting Swift effect. Solid shaft can achieve very large strain without buckling in torsion test. They can be used to accurately determine the large strain elastoplastic behavior [1-3]. Many researchers are interesting in the theoretical study and experimental study on large strain torsion deformation [1-7]. Jaumann objective stress rate is often adopted for study on constitutive models [1], but Nagtegaa J C [8] discovered that Jaumann stress rate may render oscillatory stress response for anisotropic kinematics hardening simple shear problem. After that, great interests focus on the issues of choosing an appropriate objective stress rate in rate type constitutive models. The expression of logarithmic stress rate with logarithmic spin can be found in Xiao H [9]. It had been proved that among all rate type elastoplastic constitutive models, only those with the newly discovered logarithmic stress rate fulfill the self-consistency criterion. In this paper, the parameter in large strain isotropic hardening elastoplastic constitutive model is determined with logarithmic rate of Kirchhoff stress utilizing solid shaft torsion test.

Alternative stress-integration schemes for large-deformation problems of solid mechanics

In solving nonlinear problems of solid mechanics by the finite-element method, stresses at integration points are usually obtained by integrating nonlinear constitutive equations, given known incremental strains. In a large-deformation analysis, stress–strain relationships must be frame independent such that any rigid-body motion does not induce strain within the material. This principle is generally satisfied by introducing an objective stress rate, such as the Jaumann or Truesdell stress rates, into the constitutive equations. This paper investigates three alternative algorithms for integrating stress–strain relationships in a large-deformation analysis. It is shown that the effect of rigid-body motion is equivalent to a stress transformation and this transformation can be introduced before, after or during integration of the stress–strain constitutive equations. Although there is no theoretical advantage, in terms of accuracy, for selecting one of these strategies over the others, in terms of efficiency of algorithms one is more advantageous than the others. Performance of the proposed algorithms is studied and compared by means of numerical examples. The results of this study can be used in the development of fast and robust algorithms for stress integration of constitutive equations in nonlinear finite-element analysis.

Objective Stress Rates and Residual Strains in Stress Cycles

widely‐used hypoelastic model for four well‐known objective stress rates under a four‐phase stress cycle associated with axial tension and/or torsion of thin‐walled cylindrical tubes. Here, two kinds of models based upon the Cauchy stress and the Kirchhoff stress will be treated. The reduced systems of differential equations of these rate constitutive equations are derived and studied for Jaumann, Green‐ Naghdi, logarithmic and Truesdell stress rates, separately. Analytical solutions in some cases and numerical solutions in all cases are obtained using these reduced systems. Comparisons between the residual deformations are made for different cases. It may be seen that only the logarithmic stress rate results in no residual deformation. In particular, results indicate that Green‐Naghdi rate would generate unexpected residual deformation effect that is essentially different from that resulting from Jaumann rate. On the other hand, it is realized that this study accomplishes an alternative, direct proof for the nonintegrability problem of Truesdell’s hypoelastic rate equation with classical stress rates. This problem has been first treated successfully by Simo and Pister in 1984 using Bernstein’s integrability conditions. However, such treatment needs to cope with a coupled system of nonlinear partial differential equations in Cauchy stress. Here, a different idea is used. It is noted that every integrable hypoelastic equation is just an equivalent rate form of an elastic equation and hence should produce no residual deformations under every possible stress cycle. Accordingly, a hypoelastic model with a stress rate has to be non‐integrable, whenever a stress cycle can be found under which this model generates residual deformation. According to this idea of reductio ad absurdum, a well‐designed stress cycle is introduced and the corresponding residual deformations are calculated. Unlike the treatment of Bernstein’s integrability conditions, it may be a simple and straightforward matter to calculate the final deformations for a given stress cycle. This has been done in this study for several well‐known stress rates.

Objective Stress Rates and Residual Strains in Stress Cycles

widely‐used hypoelastic model for four well‐known objective stress rates under a four‐phase stress cycle associated with axial tension and/or torsion of thin‐walled cylindrical tubes. Here, two kinds of models based upon the Cauchy stress and the Kirchhoff stress will be treated. The reduced systems of differential equations of these rate constitutive equations are derived and studied for Jaumann, Green‐ Naghdi, logarithmic and Truesdell stress rates, separately. Analytical solutions in some cases and numerical solutions in all cases are obtained using these reduced systems. Comparisons between the residual deformations are made for different cases. It may be seen that only the logarithmic stress rate results in no residual deformation. In particular, results indicate that Green‐Naghdi rate would generate unexpected residual deformation effect that is essentially different from that resulting from Jaumann rate. On the other hand, it is realized that this study accomplishes an alternative, direct proof for the nonintegrability problem of Truesdell’s hypoelastic rate equation with classical stress rates. This problem has been first treated successfully by Simo and Pister in 1984 using Bernstein’s integrability conditions. However, such treatment needs to cope with a coupled system of nonlinear partial differential equations in Cauchy stress. Here, a different idea is used. It is noted that every integrable hypoelastic equation is just an equivalent rate form of an elastic equation and hence should produce no residual deformations under every possible stress cycle. Accordingly, a hypoelastic model with a stress rate has to be non‐integrable, whenever a stress cycle can be found under which this model generates residual deformation. According to this idea of reductio ad absurdum, a well‐designed stress cycle is introduced and the corresponding residual deformations are calculated. Unlike the treatment of Bernstein’s integrability conditions, it may be a simple and straightforward matter to calculate the final deformations for a given stress cycle. This has been done in this study for several well‐known stress rates.

On the definitions of failure and critical states in hypoplasticity

A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.

Obstacle Test for Large Deformation Plasticity Problems

An obstacle test for large deformation plasticity problems is proposed for evaluation of mathematical models and numerical algorithms. The obstacle test consists of a suite of verification and validation problems. We evaluate performance of several well-known hypoelastic and hyperelastic models in the obstacle test. Among the hypoelastic formulations we consider those based on Jaumann and Green Naghdi objective stress rates. The two hyperelastic formulations evaluated are those of Simo et al. [8, 9, 10] and Eterovic and Bathe [11]. We report several interesting anomalies.

Computational determination of in-plane shear mechanical behaviour of textile composite reinforcements

The knowledge of the mechanical behaviour of woven fabrics is necessary in many applications in particular for the simulation of textile composite forming. This mechanical behaviour is very specific due to the possible motions between the fibres and the yarns. In this paper, the in-plane shear behaviour is analysed from virtual tests on the Representative Unit Cell. The in-plane shear strains can be very large (up to 50°) in case of draping on a double curved surface. These virtual tests avoid performing tricky experimental tests. The presented 3D finite element analyses involve two main specific aspects. Firstly the boundary conditions have to render the periodicity at large deformations and, in some cases, the evolution of contacts between neighbouring yarns during the motion. Secondly the yarn that is made of thousand of fibres is modelled as a continuous medium but its constitutive law has to take its fibrous nature into account. For that reason a rate constitutive equation using a specific objective stress rate is used. It is based on the rotation of the fibre. The analysis is performed for two unit cells. Both results are in good agreement with the experiments, but the use of one of the cells turns out to be much easier.

A unified stress update algorithm for explicit transient shell dynamics with combined isotropic-kinematic hardening in Eulerian rate-type phenomenological finite elasto-plasticity models

This paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration. We propose a unified stress update algorithm via establishing the Eulerian rate-type constitutive equations of hypoelastic–plastic materials. The constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero, i.e., the fourth-rank constant isotropic tensor. Several objective stress rates are derived through the Lie derivative of the Cauchy or Kirchhoff stress tensor. Those stress rates cover, for instance, the Jaumann, Green–Naghdi, Truesdell, Oldroyd, and Cotter–Rivlin stress rates. Further, the unified stress update algorithm is implemented along with the so-called radial return method and the phenomenological plasticity theory with combined isotropic/kinematic hardenings. Among the objective stress rates employed here, we emphasize especially the stress update procedure of the Green–Naghdi stress rate for transient shell dynamics, because this algorithm developed in this work is not shown even in the representative commercial codes. In the implementation of the unified stress update algorithm, we applied our algorithms to the Belytschko–Lin–Tsay shell theory to validate its accuracy and effectiveness. Moreover, a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed. The ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models. Several numerical examples are demonstrated, including non-contact transient shell dynamics examples and contact–impact examples such as a ball-plate impact problem, by which the accuracy and effectiveness of the unified stress update algorithm, developed in this work, are addressed.

Objective stress rates, cyclic deformation paths, and residual stress accumulation

As one of the basic constitutive ingredients in finite elastoplasticity, Truesdell's hypoelastic equation of grade zero has been widely used to present an objective Eulerian rate formulation of the behaviour. Although it has been known that a stress rate employed in this basic rate equation would actually result in path-dependence behaviour inconsistent with the notion of hyperelasticity, sometimes it has been believed that the definition of the objective stress rate in this basic rate equation might be insubstantial, on account of the fact that the elastic may be small for metals and alloys, etc. Towards clarifying this issue, here we present a unified, direct study on the path-dependence properties of Truesdell equation for all possible deformation paths. Towards this goal, explicit closed-form solutions for the stress response for any given deformation path are derived for classical stress rates including Zaremba-Jaumann rate, Green-Naghdi rate, Oldroyd rate, Cotter-Rivlin rate, and Truesdell rate, etc. From these solutions in terms of explicit path integrations, it may become straightforward to understand how the stress response relies on deformation paths. In particular, cyclic deformation paths composed of any number of deformation cycles are taken into consideration. Explicit exact expressions in simple form are established between the residual stress and the cycle number. These results clearly suggest that the monotone accumulation phenomena of residual stresses with increasing cycle number, found in previous numerical examples, may actually be general facts for all possible cyclic deformation paths. These phenomena imply that, even for a cyclic deformation path undergoing small deformation, the final residual stress would reach such a noticeable level that in any event it could not be regarded to be negligible. Examples are given in the cases of the simple shearing and smooth cyclic plane deformation paths.

Choice of objective rate in single parameter hypoelastic deformation cycles

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 that•the 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.

Finite element study of the energy dissipation and residual stresses in the closed elastic deformation path

In this paper, energy dissipation and residual stress developments are numerically studied in three-dimensional closed deformation paths. Different objective stress rates coded in a finite element program are compared. In order to update the stresses, implicit integration algorithm based on mid-point rule for corotational and non-corotational objective rates is used. Several corotational objective rates such as Jaumann, Green–Naghdi, Eulerian and Lagrangian triad-based rates and non-corotational rates such as Truesdell and Cotter–Rivlin rates are considered. It is shown in this work that in some cases also a non-integrable model may exhibit no dissipation energy at the end of a closed deformation path. This study underlines some results previously obtained by other researchers, i.e. among all considered stress rates the logarithmic rate manifests the best result in respect of elasticity requirements. Copyright © 2006 John Wiley & Sons, Ltd.

Objective stress rates in repeated elastic deformation cycles

AbstractThe Eulerian elastoplastic description of initially isotropic materials essentially depends on the accurate formulation of the elastic deformation. This is often represented by Truesdell's hypoelastic equation, which involves an objective stress rate. Many objective stress rates have been presented in the past. With Bernstein's integrability conditions it has been shown that only a particular rate, namely the logarithmic stress rate, makes the hypoelastic relation exactly integrable.Here, we propose a simple, one‐parameter closed deformation cycle to demonstrate the error which will considerably accumulate over several cycles, whenever a non‐appropriate stress rate is in use. For this, several objective stress rates are compared in this cycle and the residual stresses at the end of the cycles are examined. By such a simple procedure the mathematical results of non‐integrability are evidenced by example. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)