Plastic deformation of aluminium alloy under abruptly-changing loading or strain paths

The effect of deformation history on the plastic behaviour of thin-walled tubular specimens of aluminium alloy 5056 was examined for three types of abruptly-changing loading or strain paths: namely, reverse loading after pre-loading, orthogonal straining after pre-strain, and orthogonal reloading after pre-loading and then perfect unloading, by applying combined loadings of axial force and torque. The experimental results revealed the following trends. The relation between the magnitudes of stress and strain after the comer is expressed for every pre-strain by a single curve parallel to the extension of the pre-loading curve, when the strain after the corner exceeds 1.2 per cent. Moreover, the relations between the stress reduced by pre-stress and strain after the corner for various values of the same type of pre-strain agree with each other; however, the relation for the tensile pre-strain differs clearly from that for the torsional pre-strain. Thus, A.A. Il'yushin's postulate of isotropy does not hold accurately for the above loading histories for the aluminium alloy even if the effect of the third invariant of stress deviator is eliminated.     

Dissipative flow model based on dissipative surface and irreversible thermodynamics

Summary A dissipative flow model is presented to describe dissipative deformation processes in a macroscopic solid continuum. Dissipative process may consist of material plasticity, material damage and other intrinsic mechanical phenomena represented by internal variables. The concept of a dissipative surface is introduced in the paper to distinguish between conservative and dissipative processes. Conventional plastic yielding and damage initiation are expressed by a unique condition which may else include other possible intrinsic mechanical dissipations. The proposed model is based on the principles of irreversible thermodynamics and the minimum free energy theorem. A modified material stability postulate, modified Drucker's postulate, in thermodynamic stress space is also used to obtain the same results. Received 1 July 1998; accepted for publication 13 January 1999     

A robotic model for Codman's paradox simulation and interpretation

A robotic approach based on Denavit–Hartenberg parametrization is proposed for simulating and interpreting Codman's paradox. A 3-degree-of-freedom robot model of the glenohumeral joint, driving the arm reduced to its long humerus, is considered for simulating the two-step rotational sequence of Codman's paradox. We propose to use the classical distinction made in robotics between the joint space, i.e. the inner space of joint angles, and the operational space, i.e. the outer physical space, for interpreting this historical version of the paradox, as there is some kind of confusion between these two spaces to be considered for arm movement definition. In its extended form, developed by MacConnail, the three-step rotational sequence of Codman's paradox would highlight the motor redundancy of the shoulder joint, necessitating for its simulation, according to our robotic approach, a 4-axis model of the shoulder spheroid joint. Our model provides a general prediction of the conjunct rotation angle in full accordance with clinical observation for a two-step or three-step version of Codman's paradox. The relation of the paradox with a possible general law of motion is finally discussed.     

Analysis of the problem on a plastic layer flow between rigid plates approaching each other

The present paper deals with the solution of the boundary value problem on a plastic layer flow between rigid plates approaching each other in A. A. Il’yushin’s setting. After averaging over the layer thickness, the problem is reduced to a two-dimensional spreading problem for a domain with free boundary occupied by a plastic medium.     

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.     

A. A. Il’yushin’s scientific heritage and development of his ideas in mechanics

We present a survey of the main results and scientific ideas due to the 20th century prominent scientist Aleksey Antonovich Il’yushin, which can be regarded today, at his hundredth birthday anniversary, as his scientific heritage. The survey material is arranged (mainly, chronologically) in the key directions of Il’yushin’s activities such as the theory of viscoplastic flow, hydrodynamic stability, dynamics of deformable media, supersonic aerodynamics and related flatter problems, theory of elastoplastic processes, theory of plastic flow, thermoviscoelasticity and thermodynamics, strength of polymer bodies and structures, general theory of constitutive relations in classical continuum mechanics, and nonclassical models of continuum.     

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.     

Review of Drucker’s postulate and the issue of plastic stability in metal forming

Drucker’s postulate defines a class of stable work hardening materials that are classified as non-energetic and is equivalent to the associated flow rule (AFR). The postulate has been shown to be a sufficient condition for plastic stability. However, experiments indicate that plastic deformation of aluminum and steel alloys does not adhere to the constraints of the AFR. Therefore, the requirement for accuracy suggests that the metal forming industry should also consider material models that are based on non-associated flow. But Drucker’s work raises the issue of stability when considering the use of non-associated flow in material models. While this concern is merited and many types of instability arises from certain types of non-associated flow, this has led to a widely accepted view that Drucker’s postulate is a necessary condition for stability. This perception is inhibiting the acceptance or consideration of more accurate material models that are suggested from the experimental observations about violations of the AFR. This paper proposes a specific class of material models based on non-associated flow and derives the constraints on this class of models to ensure stability. The existence of this class of non-AFR models proves that Drucker’s postulate is a sufficient but not necessary condition for stability. Furthermore, the class of models described in this paper is quite general and provides a framework for consideration of potentially more accurate material models while guaranteeing the same level of stability as typically associated with materials that satisfy Drucker’s postulate.     

Lebesgue measure for creep

The Lebesgue strain measure for creep has been developed by taking the Lebesgue integral of the function of a weighted function instead of the power of a weighted function as was taken in Seth's strain measure. The problem of a spherical shell under internal pressure is considered and it is shown that the results obtained by using Seth's concept of measure can be derived from the more general analysis presented herein.     

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned. The project supported by the National Natural Science Foundation of China (10272055).     

A. A. Il’yushin’s postulate for anisotropic materials and a version of constitutive relations

The particular isotropy postulate proposed by A. A. Il’yushin permits stating and implementing programs for construction and experimental justification of relations determining the thermomechanical properties of initially isotropic materials for trajectories of various degree of complexity.     

Un algorithme efficace d'intégration plastique pour un matériau obéissant au critère anisotrope de HillAn efficient algorithm for plastic integration of material satisfying the Hill's anisotropic yield criterion.

This Note deals with an efficient algorithm to carry out the plastic integration and compute the stresses due to large strains for materials satisfying the Hill's anisotropic yield criterion. The classical algorithm of plastic integration such as ‘Return Mapping Method’ is largely used for nonlinear analyses of structures and numerical simulations of forming processes, but it requires an iterative schema and may have convergence problems. A new direct algorithm based on a scalar method is developed which allows us to directly obtain the plastic multiplier without an iteration procedure; thus the computation time is largely reduced and the numerical problems are avoided. To cite this article: I. Titeux et al., C. R. Mecanique 332 (2004).     

Elasto-plastic coupling within a thermodynamic strain space formulation of plasticity

Elasto-plastic coupling is studied within a general thermodynamic train space formulation of rate-independent plasticity by means of plastic internal variables. The strain space formulation offers a unified approach in treating both stable and unstable material behavior simultaneously. The conditions on elasto-plastic coupling are imposed by the second law of thermodynamics and the consistency equation in the strain space formulation, and are expressed by two inequalities. This is further illustrated by specific examples where explicit necessary and sufficient conditions on the elastic moduli and their change with plastic deformation are derived for the two inequalities to be satisfied.     

Elastoplastic analysis of nonlinearly hardening variable thickness annular disks under external pressure

Based on von Mises’ yield criterion, deformation theory of plasticity and Swift’s hardening law, elasto-plastic deformation of variable thickness annular disks subjected to external pressure is studied. A nonlinear shooting method using Newton’s iterations with numerically approximated tangent is designed for the solution of the problem. Considering a thickness profile in the form of a general parabolic function, the condition of occurrence of plastic deformation at the inner and outer edges of the annular disk is investigated. A critical disk profile is determined and the corresponding elastic–plastic stresses as well as the residual stress distribution upon removal of the applied pressure are computed and discussed.     

Buckminster Fuller's “Tensegrity” structures and Clerk Maxwell's rules for the construction of stiff frames

Maxwell has shown that b bars assembled into a frame having j joints would, in general, be simply stiff if b = 3j?6. Some of Buckminster Fuller's “Tensegrity” structures have fewer bars than are necessary to satisfy Maxwell's rule, and yet are not “mechanisms” as one might expect, but are actually stiff structures. Maxwell anticipates special cases of this sort, and states that their stiffness will “be of a low order”. In fact, the conditions under which Maxwell's exceptional cases occur also permit at least one state of “self-stress” in the frame.Linear algebra enables us to find the number of “incipient” modes of low-order stiffness of the frame in terms of the numbers of bars, joints and independent states of self-stress. Self-stress in the frame has the effect of imparting first-order stiffness to the frame, and it seems from experiments that a single state of self-stress can stiffen a large number of modes. It is this factor which Fuller exploits to make satisfactory structures.     

Stability charts for Hill's equation

A numerical method for computing stability boundaries for Hill's equation is presented.     

A constitutive equation derived from Lodge's network theory

A constitutive equation is presented, derived from Lodge's molecular network theory. The equation is of a form similar to the one presented previously by Carreau. The rate of creation of junctions and the probability of loss of junctions depend here also on the second invariant of the rate-of-strain tensor. The dependence, however, is through simple exponential functions, resulting in easy-to-use equations.Material functions are presented for the viscosity, the primary normal stress coefficient, the complex viscosity, the stress relaxation after cessation of steady simple shear, the stress growth after onset of steady simple shear and for elongational flow. The usefulness of the simplified (series truncated) equations is discussed and the model is evaluated with typical viscoelastic data.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.     

Modèles standards généralisés à convexe fixe en endommagementDamage standard models with a fixed convex domain

A thermodynamic framework is introduced for damage models. This framework, which consists in adding internal variables called complementary variables, lies within the framework of standard materials whose convex domain of admissible forces does not depend on the present state of internal variables. Thus its advantages are kept. Marigo's model is put back in this framework and another example combining isotropic and kinematic hardening is given. To cite this article: A. Cimetière et al., C. R. Mecanique 331 (2003).