Large Strain Response of Kinematic Hardening Elastoplasticity with the Logarithmic Rate: Swift Effect in Torsion

Recently, a new Eulerian rate type kinematic hardening elastoplasticity model has been established by utilizing the newly discovered logarithmic rate. It has been proved that this model is unique among all the objective kinematic hardening elastoplastic models with all possible objective corotational stress rates and other known objective stress rates by virtue of the self-consistency criterion: the hypoelastic formulation intended for elastic behaviour must be exactly integrable to deliver a hyperelastic relation. The finite simple shear response of this model has been studied and shown to be reasonable for both shear and normal stress components. The objective of this work is to further study the large deformation response of this model, in particular the well-known Swift effect, in torsion of thin-walled cylindrical tubes with free ends. Analytical perturbation solution and numerical solution are presented for the case of linear kinematic hardening at large compressible deformation. It is shown that the prediction from the foregoing model is in good accord with experimental data reported in literature.     

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.     

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.     

Large-strain response of isotropic-hardening elastoplasticity with logarithmic rate: Swift effect in torsion

Summary  Recently, a new Eulerian rate-type isotropic-hardening elastoplasticity model has been established by utilizing the newly discovered logarithmic rate. It has been proved that this model is unique among all isotropic hardening elastoplastic models with all possible objective corotational stress rates and other known objective stress rates by virtue of the self-consistency criterion: the hypoelastic formulation intended for elastic behaviour must be exactly integrable to deliver a hyperelastic relation. The simple shear response of this model has been studied and shown to be reasonable for both the shear and normal stress components. The objective of this work is to further study the large deformation response of this model, in particular, the second-order effects, including the well-known Swift effect, in torsion of thin-walled cylindrical tubes with free ends. An analytical perturbation solution is derived, and numerical results are presented by means of the Runge–Kutta method. It is shown that the prediction of this model for the shear stress is in good accord with experimental data, but the predicted axial length change is negligibly small and much less than experimental data. This suggests that the strain-induced anisotropy may be the main cause of the Swift effect. Received 10 December 1999; accepted for publication 20 July 2000     

Kinematic hardening in large strain plasticity

A finite strain hyper elasto-plastic constitutive model capable to describe non-linear kinematic hardening as well as non-linear isotropic hardening is presented. In addition to the intermediate configuration and in order to model kinematic hardening, an additional configuration is introduced – the center configuration; both configurations are chosen to be isoclinic. The yield condition is formulated in terms of the Mandel stress and a back-stress with a structure similar to the Mandel stress.It is shown that the non-dissipative part of the plastic velocity gradient not governed by the thermodynamical framework and the corresponding quantity associated with the kinematic hardening influence the material behaviour to a large extent when kinematic hardening is present. However, for isotropic elasticity and isotropic hardening plasticity it is shown that the non-dissipative quantities have no influence upon the stress–strain relation.As an example, kinematic hardening von Mises plasticity is considered, which fulfils the plastic incompressibility condition and is independent of the hydrostatic pressure. To evaluate the response and to examine the influence of the non-dissipative quantities, simple shear is considered; no stress oscillations occur.     

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.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate

The objective of this article is to suggest new Eulerian rate type constitutive models for isotropic finite deformation elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening etc. The main novelty of the suggested models is the use of the newly discovered logarithmic stress rate and the incorporation of a simple, natural explicit integrable-exactly rate type formulation of general hyperelasticity. Each new model is thus subjected to no incompatibility of rate type formulation for elastic behaviour with the notion of elasticity, as encountered by any other existing Eulerian rate type model for elastoplasticity or hypoelasticity. As particular cases, new Prandtl-Reuss equations for elastic-perfect plasticity and elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening, respectively, are presented for computational and practical purposes. Of them, the equations for kinematic hardening and combined isotropic–kinematic hardening are, respectively, reduced to three uncoupled equations with respect to the spherical stress component, the shifted stress and the back-stress. The effects of finite rotation on the current strain and stress and hardening behaviour are indicated in a clear and direct manner. As illustrations, finite simple shear responses for the proposed models are studied by means of numerical integration. Further, it is proved that, among all possible (infinitely many) objective Eulerian rate type models, the proposed models are not only the first, but unique, self-consistent models of their kinds, in the sense that the rate type equation used to represent elastic behaviour is exactly integrable to really deliver an elastic relation. ©     

A nonlocal strain gradient plasticity theory for finite deformations

Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes.     

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity.Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains.To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening.For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived.The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

A strain space nonlinear kinematic hardening/softening plasticity model

A strain space plasticity theory based on the nonlinear kinematic hardening and softening rule is developed in order to accommodate work-hardening, work-softening, and elastic-perfectly plastic materials with one set of constitutive equations, and to facilitate strain controlled calculations. A generalized hardening/softening parameter is proposed, and the potential of linking the parameter to micro-mechanical material changes is discussed. The theory is used to investigate work-softening materials numerically and highlights a need for additional experimental results in this area.     

Simple shear of a strain-hardening elastoplastic hollow circular cylinder

Stress and deformation analysis of the simple shear at finite strain of a strain-hardening elastoplastic hollow circular cylinder is given. Both isotropic and anistotropic hardening models are considered. In the case of isotropic hardening, there is a closed from analytical solution. No normal stresses exist in this case. Purely kinematic hardening with a Mises-type yield condition is utilized as a model of anisotropic hardening. Conventional (average) spin is taken to construct the objective Jaumann derivative needed in the structure of the corresponding constitutive laws. Governing partial differential equations are derived and solved numerically to give stress and deformation distribution following the advance of plastic flow. The extent or range of the appropriateness of the considered constitutive model is also established.     

Numerical integration method for kinematic hardening rules with partial activation of dynamic recovery term

In order to express steady state ratcheting, Abdel-Karim and Ohno formulated kinematic hardening rules on the assumption that dynamic recovery term of back stress is partially activated before it reaches its critical value. These rules have a feature that only the projection of plastic strain rate into the direction of back stress contributes to the dynamic recovery. They are in contrast with the other rules in which either/both accumulated plastic strain rate or/and plastic strain rate enters into the dynamic recovery term. In this paper, another feature, which has not been investigated by Abdel-Karim and Ohno, is introduced. Discussing this feature, characterized by decomposition of back stress into radial and tangential components, a numerical integration method based on radial-return mapping using radial components only is developed. This method consists of a trial predictor and final corrector depending on the current state of back stress and plastic strain increment. The applicability of the method is examined carefully and critically. It is shown that this method is effective and its accuracy is indistinguishable to the radial-return method for elastic-perfectly plastic materials.     

Anisotropic plasticity for sheet metals using the concept of combined isotropic-kinematic hardening

Hill's 1948 anisotropic theory of plasticity (Hill, R., 1948. A theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A193, 281–297) is extended to include the concept of combined isotropic-kinematic hardening, and the objective of this paper is to validate the model so that it may be useful for analyses of sheet metal forming. Isotropic hardening and kinematic hardening may be experimentally observed in sheet metals, if yielding is defined by the proportional limit or by a small proof strain. In this paper, a single exponential term is used to describe isotropic hardening and Prager's linear kinematic hardening rule is applied for simplicity. It is shown that this model can satisfactorily describe both the yield stress and the plastic strain ratio, the R-ratio, observed in tension test of specimens cut at various angles measured from the rolling direction of the sheet. Kinematic hardening leads to a gradual change in the direction of the plastic strain increment, as the axial strain increases in the tension test; while in the traditional approach for sheet metal, this direction does not change due to the use of isotropic hardening.     

Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.     

Strain gradient effects on cyclic plasticity

Size effects on the cyclic shear response are studied numerically using a recent higher order strain gradient visco-plasticity theory accounting for both dissipative and energetic gradient hardening. Numerical investigations of the response under cyclic pure shear and shear of a finite slab between rigid platens have been carried out, using the finite element method. It is shown for elastic-perfectly plastic solids how dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. For linearly hardening materials it is quantified how dissipative and energetic gradient effects promote hardening above that of conventional predictions. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. Furthermore, it is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. For dissipative gradient effects it is found that dissipation has a maximum value for some none zero value of the material length parameter, which depends on the magnitude of the deformation cycles.     

Non-local crystal plasticity model with intrinsic SSD and GND effects

A strain gradient-dependent crystal plasticity approach is presented to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. In order to be capable of predicting scale dependence, the heterogeneous deformation-induced evolution and distribution of geometrically necessary dislocations (GNDs) are incorporated into the phenomenological continuum theory of crystal plasticity. Consequently, the resulting boundary value problem accommodates, in addition to the ordinary stress equilibrium condition, a condition which sets the additional nodal degrees of freedom, the edge and screw GND densities, proportional (in a weak sense) to the gradients of crystalline slip. Next to this direct coupling between microstructural dislocation evolutions and macroscopic gradients of plastic slip, another characteristic of the presented crystal plasticity model is the incorporation of the GND-effect, which leads to an essentially different constitutive behaviour than the statistically stored dislocation (SSD) densities. The GNDs, by their geometrical nature of locally similar signs, are expected to influence the plastic flow through a non-local back-stress measure, counteracting the resolved shear stress on the slip systems in the undeformed situation and providing a kinematic hardening contribution. Furthermore, the interactions between both SSD and GND densities are subject to the formation of slip system obstacle densities and accompanying hardening, accountable for slip resistance. As an example problem and without loss of generality, the model is applied to predict the formation of boundary layers and the accompanying size effect of a constrained strip under simple shear deformation, for symmetric double-slip conditions.     

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).