A weakened form of Ilyushin's postulate and the structure of self-consistent Eulerian finite elastoplasticity

From a general standpoint in terms of internal variables, we formulate a general theory of self-consistent Eulerian finite elastoplasticity based on the additive decomposition of the Eulerian strain rate, i.e., D=D^e+D^p, as well as two consistency criteria. In this theory, the elastic behaviour is characterized by an exactly integrable elastic rate equation for D^e with a general form of complementary elastic potential. It is assumed that the yield function depends in a general manner on the Kirchhoff stress and the internal variables. Moreover, the plastic rate equation for D^p and the evolution equation for each internal variable are allowed to assume general forms relying on the just-mentioned variables and the stress rate. It is indicated that two consistency criteria, i.e., the self-consistency for the elastic rate equation and Prager's yielding stationarity, lead to the unique choice of objective rates, i.e., the logarithmic rate.The structure of the above theory is further studied and examined by virtue of a weakened form of Ilyushin's postulate. In a spinning frame defining the logarithmic rate, we introduce the notion of standard elastoplastic strain cycle, which starts at a point not on but inside a yield surface and incorporates only one infinitesimal plastic subpath. We show that this type of strain cycle is always possible. Then, by ruling out strain cycles starting at points on yield surfaces we propose a weakened form of Ilyushin's postulate, which says that the changing rate of the stress work done along every standard strain cycle should be non-negative, whenever the incorporated plastic subpath tends to vanish. By virtue of simple, rigorous procedures, we demonstrate that this weakened form of Ilyushin's postulate is adequate to ensure direct results concerning the normality rule and the convexity of the yield surface in the context of the foregoing Eulerian finite elastoplasticity theory. Specifically, with an exactly integrable elastic rate equation defining D^e, we prove that, in the space of the Kirchhoff stresses, the difference (D−D^e) is just the gradient of the yield function multiplied by a plastic multiplier, and thus bears the very kinematical and physical feature of plastic strain rate. Furthermore, we prove that, in the space of the Kirchhoff stresses, the elastic domain bounded by each yield surface should be convex. The main results are derived in a self-contained manner within the context of an Eulerian theory of finite elastoplasticity, without involving issues concerning how to define intermediate stress-free states and plastic strains, etc.     

Micromorphic continuum. Part II: Finite deformation plasticity coupled with damage

It is demonstrated how a micromorphic plasticity theory may be formulated on the basis of multiplicative decompositions of the macro- and microdeformation gradient tensor, respectively. The theory exhibits non-linear isotropic and non-linear kinematic hardening. The yield function is expressed in terms of Mandel stress and double stress tensors, appropriately defined for micromorphic continua. Flow rules are derived from the postulate of Il’iushin and represent generalized normality conditions. Evolution equations for isotropic and kinematic hardening are introduced as sufficient conditions for the validity of the second law of thermodynamics in every admissible process. Finally, it is sketched how isotropic damage effects may be incorporated in the theory. This is done for the concept of effective stress combined with the hypothesis of strain equivalence.     

Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaireDetermination of the macroscopic strength criterion of a porous medium by nonlinear homogenization

A porous medium, which matrix is a perfectly plastic solid, is considered. This paper proposes a method to determine the macroscopic admissible stress states. The method is based on a homogenization technique which takes advantage of the equivalence, under certain conditions, between a problem of limit analysis and a ficticious nonlinear elastic problem. The particular case of a Drucker–Prager solid matrix is considered. The method provides an analytical expression for the complete macroscopic strength criterion. To cite this article: J.-F. Barthélémy, L. Dormieux, C. R. Mecanique 331 (2003).     

损伤力学中的能量等效假设及其实验验证

大多数损伤模型在构造受损材料本构方程时所采用的应变或应力等效假设在一般情况下并不能满足热力学理论的基本规则.本文提出的能量等效假设,在弹性、塑性和蠕变损伤等普遍情形均证明了其有效性.同时,能量等效假设配合以应变能密度准则,可以很好地描述穿孔铜薄板试件的破断行为.     

Three-dimensional Maxwell stress-strain relations in viscoelasticity

Conclusion In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically.In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function.Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function.From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain.If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot.It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.     

Degradation of elastic modulus in elastoplastic coupling with finite strains

An elastic potential W is postulated for the case of the finite-strain theory of elastoplastic coupling with damage effects. The potential is defined in terms of the invariants of two internal variables p and q. The internal variables are used to express the degradation of the elastic stiffness tensor due to the accumulation of plastic strains. The material damage is independently introduced to both Lame's coefficients H and G. The physical significance of this softening of the elastic stiffness is demonstrated experimentally in uniaxial loading, reverse loading of metals at finite strains.

For elastoplastic coupling, the Il'Iushin postulate does not yield normality of the plastic strain increment. An associative flow rule is postulated in this work for the combined components of the plastic strain increment and the elastic coupling strain increment.

The formulation is implemented in the Langrangian coordinate system. Through the use of the Oldroyd or Truesdell stress rate, the equivalent consistent spatial coordinate formulation is presented.     

Geotechnical yield criteria and constitutive relations in strain space

Based on Hyushins postulate this paper deals with the necessity and features ofresearching the geotechnical elasto-plastie theory in strain space.In the paper,weestablished the relations between stress in variants and elastic strain invariants.broughtabout the transformation from the stress yield surfaces into the strain yield surfaces,derived and discussed the strain expressions from 12 yield criteria expressed by stress.Bynormality rule.we also derived 12 constitutive relations for ideal plastic materialsassociated with the above expressions.The results presented here can be applied to practiceand are helpful to the study of the plastic theory in strain space.     

On the non-conservativeness of a class of anisotropic damage models with unilateral effectsSur le caractère non-conservatif de quelques modèles d'endommagement anisotrope avec conditions unilatérales

The aim of this Note is to show that a class of anisotropic elastic-damage models including unilateral effects can be considered, for constant damage values, as non-linear and non-conservative elastic. The conservative character of corresponding constitutive models is related to the symmetry of the Hessian tensor. For the models under consideration, it is shown that the condition of conservativeness (existence of the elastic potential energy function) is obtained only when there is coaxiality of the strain and damage tensors. To cite this article: N. Challamel et al., C. R. Mecanique 334 (2006).     

Relationship between plastic and intrinsic dissipations

In this paper, the relationship between the plastic and intrinsic dissipations is addressed within the normality structure of [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455; Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, pp. 23–79.] It is shown that the plastic dissipation is generally not equal to the intrinsic dissipation. Within the normality structure, the microscale and macroscale thermodynamic fluxes and forces are related by the conditions of energy and dissipation equivalence. If the plastic dissipation is required to be equal to the intrinsic dissipation, J₂ potential and the Levy–Mises equation are recovered from the condition of dissipation equivalence for incompressible plastic flows.     

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.     

Limit-analysis of a circular cylinder obeying the Green plasticity criterion and loaded in combined tension and torsion

This short paper presents the limit-analysis of a cylinder with circular basis, made of an ideal-plastic material obeying Green’s yield criterion and subjected to combined tension and torsion. The exact solution of the problem is provided in the form of a statically and plastically admissible stress field and a kinematically admissible velocity field, associated via the normality rule. The overall yield locus, that is the set of pairs [tension force, torsion torque] for which unrestrained plastic flow occurs, is expressed first in parametric form, then explicitly upon elimination of the parameter involved. The explicit expression of this yield locus also entails that of the overall flow rule via the overall normality property. The impact of these results is two-fold. First, they provide a fresh example of a solution to a limit-analysis problem exceptionally combining three generally mutually exclusive features: be non-trivial, exact and explicit. Second, they provide a way of using simple experiments of combined tension and torsion of cylinders to determine the parameter characterizing the influence of the mean stress in Green’s criterion.     

Energy elastic effects and the concept of temperature in flowing polymeric liquids

The incorporation of energy elastic effects in the modeling of flowing polymeric liquids is discussed. Since conformational energetic effects are determined by structural features much smaller than the end-to-end vector of the polymer chains, commonly employed single conformation tensor models are insufficient to describe energy elastic effects. The need for a local structural variable is substantiated by studying a microscopic toy model with energetic effects in the setting of a generalized canonical ensemble. In order to examine the dynamics of flowing polymeric liquids with energy elastic effects, a thermodynamically admissible set of evolution equations is presented that accounts for the evolution of the microstructure in terms of a slow tensor, as well as a fast, local scalar variable. It is demonstrated that the temperature used in the definition of the heat flux is directly related to the Lagrange multiplier of the microscopic energy in the generalized canonical partition function. The temperature equation is discussed with respect to, first, the dependence of the heat capacity on the polymer conformation and, second, the possibility to measure experimentally the effects of the conformational energy.

Flow of a non-Newtonian fluid between heated parallel plates

The flow of a fluid of grade three between heated parallel plates is examined for two cases. In the first instance we postulate constant heat flux at the walls and via a similarity transformation calculate the Nusselt number as a function of both Γ, the parameter controlling viscous dissipation, and Λ, the non-Newtonian parameter. In the second case we restrict the temperature to change only normal to the plates; solutions in this case are obtained for two temperature-viscosity models, μ = μ(θ).     

Modelling of quasi-brittle behaviour: a discrete approach coupling anisotropic damage growth and frictional sliding

The paper provides development of the model of anisotropic damage by microcracking proposed by Bargellini et al. 2006. This model is based on a discrete approach, which introduces a finite set of microcrack densities associated with fixed directions. This approach avoids inconveniences encountered when using a single second order tensor damage variable D (non uniqueness of the free energy) and strain decomposition into positive and negative parts (spurious dissipation at crack closure). Frictional sliding on closed microcracks is introduced as an additional dissipative mechanism; it is represented by a second order sliding variable in each damage direction. Corresponding sliding criteria and non-associated sliding evolution laws, formulated in the strain space for the model coherence, permit to account for hysteretic phenomena. Unilateral effect is taken into account; Young's and shear moduli are correctly restored at microcrack closure. The crucial requirements of continuity of the energy and of stress–strain response are ensured through relevant conditions on parameters and sliding variables values at opening-closure. The discrete approach, associated with some hypotheses concerning damage evolution, permits to couple damage and dissipative sliding. The pertinence of the proposed theory is illustrated by simulating first elastic properties at constant damage, then by considering a specific loading path involving both damage and friction evolutions.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

Multi-scale analysis of subgrid stress and energy dissipation in turbulent channel flow

In present study, the subgrid scale （SGS） stress and dissipation for multiscale formulation of large eddy simulation are analyzed using the data of turbulent channel flow at Ret = 180 obtained by direct numerical simulation. It is found that the small scale SGS stress is much smaller than the large scale SGS stress for all the stress components. The dominant contributor to large scale SGS stress is the cross stress between small scale and subgrid scale motions, while the cross stress between large scale and subgrid scale motions make major contributions to small scale SGS stress. The energy transfer from resolved large scales to subgrid scales is mainly caused by SGS Reynolds stress, while that between resolved small scales and subgrid scales are mainly due to the cross stress. The multiscale formulation of SGS models are evaluated a priori, and it is found that the small- small model is superior to other variants in terms of SGS dissipation.     

An admissible form of an inelastic constitutive equation—I. General theory

From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.