Mathematical Theory Of Plasticity Research Articles

Revisiting the Influence of Contact Length and Surrounding Medium on Metal Cutting Tribology

Metal cutting processes involve severe frictional conditions at the tool-chip contact interface, physico-chemical phenomena that are not fully understood and theoretical models with a strong empirical basis. In the literature, it is common to find values for the friction coefficient which are higher than the unit, despite the inconsistency of these findings with the mathematical theory of plasticity used for accurate modelling of the chip formation mechanics. Thus, the present work seeks to contribute to a better understanding of metal cutting tribology based on well-controlled and specially designed experimental conditions. To enable the stress state at the contact interface to be determined and to provide a known real contact area, polished cutting inserts with prepared rake faces were used to restrict the contact length in orthogonal cutting tests under controlled atmospheres. These conditions also allowed the contribution of the sticking mechanisms to be minimized. Cutting tools of restricted contact lengths have been found to reduce process forces and the friction coefficient, while industrial standard inserts make cutting operation sensitive to the chemical composition of the surrounding atmosphere.

Galileo, Coulomb, the origins of Yield Design Analyses

The Memoir presented to the French Academy of Sciences by Coulomb in 1773 is often considered as the starting point for “yield design” stability analyses in geotechnics and, more generally, in various fields of civil and construction engineering. In this Essai (1773), Coulomb implemented a reasoning based on the necessary condition that the equilibrium of the structure under study and the resistance of its constituent materials should be mathematically compatible with each other. A previous out sketch of this rationale can be identified in Galileo’s solution to the problem of the bearing capacity of a cantilever beam in the Discorsi (1638). In the present contribution, after outlining the philosophy of Galileo’s argument, we will highlight Coulomb’s clarifying insight and essential input before the fundamental concepts of the mechanics of continuous media, such as the notion of stress and the principle of virtual velocities, were established. Various aspects of yield design or “limit equilibrium” stability analyses in geotechnics will be evoked. The theory of Yield design, in line with Coulomb’s reasoning, will be briefly presented, making it possible, in particular, to overcome the difficulties encountered when implementing “classical” methods and clarifying a possible confusion with the mathematical theory of plasticity.

Non-Quadratic Pseudo Dual Potentials for Plastic Flow Modeling

The existence of dual flow potentials is well established in mathematical theory of plasticity since the seminal work by Hill in 1987. For a metal undergoing plastic flow, a flow stress potential is used to compute its plastic strain increments when the applied yield stress is known. On the other hand, the corresponding dual flow strain-rate potential is used to compute the stress on the flow surface when the plastic strain increments are given. This work examines some issues associated with plasticity modeling using non-quadratic dual flow potentials. Unlike the quadratic case where flow stress and strain-rate potentials are the exact dual to each other, it is often difficult if not impossible to obtain analytically the dual of a non-quadratic flow stress or strain-rate potential. The study instead focuses on formulating and assessing various non-quadratic pseudo dual flow potentials that approximate the actual flow surfaces in either stress or strain-rate space. The difference and connection between the yield surface and flow surface in non-associated plasticity are also investigated. Although only one of the dual flow potentials is actually needed for their applications in associated and non-associated plasticity modeling, the unique advantage of having both dual flow potentials on hand even in their pseudo forms is pointed out for new computational analyses.

К 60-летию со дня рождения проф. Юрия Николаевича Радаева

10 февраля 2022 г. исполнилось 60 лет известному ученому в области механики деформируемого твердого тела и прикладной математики, педагогу, организатору науки и высшего образования в России Юрию Николаевичу Радаеву. В статье приведены основные библиографические данные Ю. Н. Радаева, в краткой форме представлены главные научные направления и результаты научной деятельности по фундаментальным проблемам математической теории пластичности, механике растущих тел и рассеянного накопления поврежденности, теории трещин, микрополярной упругости, связанной гиперболической термоупругости и термомеханике, общим математическим вопросам механики сплошных сред, механике сыпучих и гранулированных сред.

Thermo-mechanical coupled incremental variational formulation for thermosetting resins subjected to curing process

This study presents an incremental variational formulation (IVF) for the thermo-mechanical problem of resin products subjected to curing, for which a dual dissipation potential (DDP) for the cure state is originally derived in conjunction with that for viscoelasticity. Following the thermodynamically consistent formulation for the time evolution equation of the degree of cure (DOC) of a thermosetting resin and the linear viscoelastic law, we derive these DDPs by way of Legendre-Fenchel transformations with respect to characteristic dissipations. The evolution of DOC is then casted into the flow rule in mathematical theory of plasticity with the introduction of the curing multiplier as a new internal state variable in an analogous fashion to Perzyna’s viscoplastic regularization theory. Then, the pseudo stress is introduced as a new internal state variable to parameterize the derived flow rule of DOC. By virtue of this pseudo stress, we naturally define an incremental variational problem constrained by the parameterized flow rule in order to construct a semi-implicit scheme for the variational constitutive update. The finite element discretization is applied to the governing equations that satisfy the stationary conditions so that we could perform thermo-mechanical analyses of a thermosetting resin subjected to curing with the strong coupling procedure. The verification analysis is conducted to confirm the validity of the numerical result obtained by the proposed method in comparison with that of the conventional framework and followed by a demonstrative numerical example to simulate the stress development and relaxation in a virtual device subjected to different thermal histories.

К вопросу об определяющих законах связи общей математической теории пластичности

The paper discusses the question of the reliability and applicability of the general laws of the mathematical theory of plasticity. In a new direction of the theory of plasticity (the theory of elastic-plastic deformation processes) the isotropy postulate is given, which establishes the invariance of the connection between stresses and strains. However, this invariance during orthogonal transformations of the image of the process and its vectors in the linear coordinate space can be violated due to a change in the invariants of the form of the stress-strain state. However, numerous experiments show that the influence of these invariants is weak and can be neglected. In the theory of flow, the main hypothesis is the assumption of the decomposition of total deformations into elastic and plastic parts. Such decomposition under complex loading is impossible and contradicts the concept of the complete and incomplete plastic states of the material. This article shows that the flow theory is a special case of the theory of processes. An extended version of the theory of flow is obtained, which can be used for medium-curvature deformation trajectories, and which makes it possible to use the hypothesis of decomposition of total deformations in the theory of flow.

Some general remarks on hyperplasticity modelling and its extension to partially saturated soils

The essential ideas and equations of classic plasticity and hyperplasticity are successively recalled and compared, in order to highlight their differences and complementarities. The former is based on the mathematical framework proposed by Hill (The mathematical theory of plasticity. Oxford University Press, Oxford, 1950), whereas the latter is founded on the orthogonality hypothesis of Ziegler (An introduction to thermomechanics. Elsevier, North-Holland, 1983). The main drawback of classic plasticity is the possibility of violating the second principle of thermodynamics, while the relative ease to conjecture the yield function in order to approach experimental results is its main advantage. By opposition, the a priori satisfaction of thermodynamic principles constitutes the chief advantage of hyperplasticity theory. Noteworthy is also the fact that this latter approach allows a finer energy partition; in particular, the existence of frozen energy emerges as a natural consequence from its theoretical formulation. On the other hand, the relative difficulty to conjecture an efficient dissipation function to produce accurate predictions is its main drawback. The two theories are thus better viewed as two complementary approaches. Following this comparative study, a methodology to extend the hyperplasticity approach initially developed for dry or saturated materials to the case of partially saturated materials, accounting for interface energies and suction effects, is developed. A particular example based on the yield function of modified Cam–Clay model is then presented. It is shown that the approach developed leads to a model consistent with other existing works.

数理塑性理論における異方性降伏条件式

数理塑性理論における異方性降伏条件式

К 85-летию со дня рождения профессора Д. Д. Ивлева

6 сентября 2015 года Дюису Даниловичу Ивлеву исполнилось бы 85 лет. В настоящей статье приводятся некоторые сведения, не известные широкой научной публике, и список всех его основных работ.

Rabotnov damageparameter and description of delayed fracture: Results, current status, application to fracture mechanics, and prospects

This paper presents a review of studies of delayed fracture and fracture mechanics problems in which the hypotheses and ideas of Yu. N. Rabotnov and L. M. Kachanov on the mechanisms of delayed fracture under creep conditions are extended to describe fracture processes using scalar and tensor measures of damage. The results of current research in the theory of elasticity, the mathematical theory of plasticity and creep, the mechanics of composites, and linear and nonlinear fracture mechanics, with material damage taken into account.

数理塑性理論によるコピー用紙の引張負荷における応力-ひずみ関係の解析

数理塑性理論によるコピー用紙の引張負荷における応力-ひずみ関係の解析

Rodney Hill. 11 June 1921 — 2 February 2011

Rodney Hill was born on 11 June 1921 in Leeds, and educated at Leeds Grammar School. He went up to Cambridge University in October 1939, with a Major Scholarship at Pembroke College. He graduated BA with first-class honours in 1942 in the Mathematical Tripos. Volunteering for war work immediately, he worked in full-time government service on ballistics in the Cambridge Mathematical Laboratory, and on the plasticity of metals in the Cavendish Laboratory. In 1943 he moved to the Armament Research Department at Fort Halstead in Kent, for three years. Here he was involved in, for example, the modelling of armour penetration by projectiles. This established his expertise in the Mathematical Theory of Plasticity, in which he became a world-recognized leader via the writing of a renowned book with that title (still in print after 60 years) and 170 research articles with eventually 26 collaborators. He had more than 10 research students. In 1963 he wrote a textbook, Principles of dynamics , based on his lectures to undergraduates. Subsequent appointments were at the British Iron and Steel Research Association in Sheffield, at Bristol University, and then as Professor of Applied Mathematics at Nottingham University (1953–62), and at Cambridge University. He retired in 1979 but continued with active research for more than another 20 years. Hill was elected a Fellow of the Royal Society in 1961, whose Royal Medal he received in 1993. He received the Honorary Degree of DSc from the Universities of Manchester (1976) and Bath (1978). He was awarded the von Karman Medal of the American Society of Civil Engineers in 1978, and the Panetti Medal of the Turin Academy in 1988. In 1982 The Rodney Hill 60th Anniversary Volume called Mechanics of solids was published, edited by H. G. Hopkins and M. J. Sewell. It contains 19 articles by 23 contributors in 693 pages. The Rodney Hill Prize in Solid Mechanics (US $25 000, at four-yearly intervals) has been established by Elsevier Ltd. It was awarded first in 2008 (Ortiz) and then in 2012 (Gao). A principal relaxation of Hill for 50 years was in extended botanical expeditions with his wife, Jeanne, in many parts of the English countryside, searching for, and identifying and recording, many species of wild flowers and fungi; and in the cultivation of a garden at home. Rodney Hill married Jeanne Wickens in 1945. She died in 2003. They had one daughter, Caroline, who survives them. Rodney died in Cambridge on 2 February 2011.

Physical-mathematical theory of irreversible strains in metals

We give a systematic presentation of the physical-mathematical theory of irreversible strains in metallic materials. The fundamental postulates of the classical mathematical theory of plasticity are stated as corollaries of the fundamental statements of the physical-mathematical theory of plasticity.

Residual stresses in cold-deformed thin items and sheet tempered metal

Using solutions of the mathematical theory of plasticity for a thin elastoplastic layer, four theoretically possible types of distributions of residual stresses over the thickness of cold-deformed items and tempered sheet metal are determined. The results are obtained by studying different types of diagrams presenting the dependence of the stress intensity on the deformation intensity. The obtained theoretical dependences are confirmed by experimental studies. The effect of the residual stresses on the mechanical properties and formability of sheet cold-rolled steel is considered.

Thermodynamically Consistent Anisotropic Plasticity Model

AbstractThe objective of this paper is to consider the thermodynamically consistent anisotropic plasticity model based on full decomposition of stress tensor into generalized deviatoric part and generalized spherical part of stress tensor. Two fundamental tensors αij and βij, which represent anisotropic material properties, are defined and can be considered as generalizations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors αij and βij, the concept of total generalized “pressure” and pressure corresponding to the volumetric deformation is redefined. It is shown that the formulation of anisotropic plasticity in the case of incompressible plastic flow must be considered independently from the generalized hydrostatic pressure. Accordingly, a modification to the anisotropic Hill criterion is introduced. Based on experimental research, which has been published, the modified Hill (1948, “A Theory of the Yielding and Plastic Flow of Anisotropic Metals,” Proc. R. Soc. London, Ser. A, 193(1033), pp. 281–297;1950, Mathematical Theory of Plasticity, Clarendon, Oxford) criterion for anisotropic elastoplasticity is validated. The results are presented and discussed, and future studies are outlined.

A numerical examination of shear banding and simple shear non-coaxial flow rules

Strain localisation and shear band formation is frequently observed during the handling and flow of dense phase particulate materials. However, a complete understanding of how shear bands form and what happens inside shear bands is still lacking. In order to address this problem, discrete particle simulations have been carried out to examine the detailed processes that occur at the grain scale associated with the initiation and development of shear bands. To reliably identify the continuum model applicable within a shear band is difficult due to the small number of particles/contacts involved. However, it is normally accepted that the mode of deformation within a shear band is one of simple shear. Consequently, simple shear simulations have been performed in order to determine the evolution of the stress tensor, dilation rate, and the principal directions of stress and strain-rate. It is demonstrated that the corresponding non-coaxial flow rule is equivalent to that suggested by Tatsuoka et al. (Géotechnique 38 148 (1988)). Furthermore, at fully developed flow when there is no further change in volume, the stress and strain-rate directions are coaxial and the flow rule is that proposed by Hill (The Mathematical Theory of Plasticity (Oxford University Press, 1950) p. 294).

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.

On Microscopic Thermodynamic Mechanisms of Damage Evolution Laws

Most existing phenomenological damage evolution laws can be covered by phenomenological equations or linear irreversible thermodynamics. In this paper, general microscopic thermodynamic mechanisms leading to nonlinear phenomenological equations are explored within the framework of ‘normality structures’ by Rice (Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity, Journal of the Mechanics and Physics of 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.) at the level of microstructural rearrangements. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, are cornerstones of the normality structure. It is revealed in this paper that nonlinear phenomenological equations and Onsager reciprocal relations emerge naturally from the normality structures if each rate is a homogeneous function of degree q in its conjugate force. Furthermore, the nonlinear phenomenological coefficient matrix is identical to the Hessian matrix of the flow potential function in conjugate forces scaled by q. Finally, as an application and demonstration, some fundamental issues on damage evolution laws for microcracked solids have been addressed based on the revealed remarkable properties. It is shown that the deduced flow potential functions of microcracked solids can be expressed in the forms of well-established Hill (Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford.) anisotropic yield function and Karafillis and Boyce (Karafillis, A.P. and Boyce, D.B. (1993). A General Anisotropic Yield Criterion Using Bounds and a Transformation Weighting Tensor, Journal of the Mechanics and Physics of Solids, 46: 85-113.) isotropic yield surface.

On the structure of anisotropic damage yield criteria

This paper addresses explicit yield criteria coupled with anisotropic damage in Cauchy stress space for materials initially obeying J 2 plasticity. It is shown in this paper that the original Kachanov–Rabotnov damage model is inherently microplane-based and generally cannot ensure the existence of the effective stress tensors. It is further assumed that the effective stress approach still holds if the effective stress tensor is replaced by the effective stress vector. The effective counterpart of J 2 is calculated based on the generalized effective stress approach and the hypothesis of mean square shear stress averaged over all microplanes for J 2. It is revealed that the effective counterpart of J 2 takes the form of Hill [Hill, R., 1950. The Mathematical Theory of Plasticity, Clarendon Press, Oxford] anisotropic yield function in Cauchy stress space. The six parameters of Hill [Hill, R., 1950. The Mathematical Theory of Plasticity, Clarendon Press, Oxford] anisotropic yield function can be determined uniquely by a fourth-order fabric tensor.

Patterns, fabric, anisotropy, and soil elasto-plasticity

This paper proposes a new method in the theory of soil plasticity – an advance on Hill [The Mathematical Theory of Plasticity, Clarendon Press, Oxford]. The method assumes that soil fabric consists of inter-locking, inter-twining, inter-laced, juxtaposed, and superposed elementary units called “patterns”. A mechanics of patterns is developed. As well as elastic and plastic components, a third strain-increment component is deduced which helps explain non-associated flow. The proposed method leads to explanations of critical states, anisotropy, sensitivity, the Bauschinger effect, and swept-out memory. All these appear in the method as near-inescapable features of plastic solids. Results are illustrated in detail for plane strain biaxial processes.