Continuum theory of plasticity

Khan, A.S. and Huang, S. (1995) Continuum Theory of Plasticity (Wiley, New York). 

Continuum Theory of Plasticity by A. S. Khan and S. Huang, John Wiley and Sons, New York New York, 1995. The subject of plasticity is lacking in texts that are directed at interpreting plastic deformation on the basis of the underlying mechanisms. This book provides an overview of much of the machinery that is invoked in current considerations of plasticity. 

Khan A.S., and S. Huang, 1995. Continuum theory of plasticity. New York Wiley. 

The demonstration of the validity of the continuum-based modelling approach to tablet compaction requires familiarity with fundamental concepts of applied mechanics. Under the theory of such a mechanism, powder compaction can be viewed as a forming event during which large irrecoverable deformation takes place as the state of the material changes from loose packing to near full density. Moreover, it is important to define the three components of the elastoplastic constitutive models which arose from the growing theory of plasticity, that is the deformation of materials such as powder within a die ... 

Non-linear yield surfaces for plastically anisotropic materials which are applicable in the continuum theory of inelastic composites are suggested by R. Hill [3], W. Prager [8], S. W. Tsai and E. Wu [15], F. Barlatetal [1], M. Gotoh [2],... 

The Mathematical Theory of Plasticity by R. Hill, Clarendon Press, Oxford England, 1967. The definitive work on the mathematical theory of plasticity. This book goes well beyond our treatment regarding continuum descriptions of plasticity without entering into the question of how plasticity emerges as the result of defect motion. 

Fox, N. (1968) On the continuum theories of dislocations and plasticity. Quarterly Journal of Mechanics and... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

As early as 1829, the observation of grain boundaries was reported. But it was more than one hundred years later that the structure of dislocations in crystals was understood. Early ideas on strain-figures that move in elastic bodies date back to the turn of this century. Although the mathematical theory of dislocations in an elastic continuum was summarized by [V. Volterra (1907)], it did not really influence the theory of crystal plasticity. X-ray intensity measurements [C.G. Darwin (1914)] with single crystals indicated their mosaic structure (j.e., subgrain boundaries) formed by dislocation arrays. Prandtl, Masing, and Polanyi, and in particular [U. Dehlinger (1929)] came close to the modern concept of line imperfections, which can move in a crystal lattice and induce plastic deformation. 

The task of this chapter is to introduce the key concepts (both continuum and discrete) used in thinking about dislocations with the aim of explaining a range of observations concerning plastic deformation in crystalline solids. In the opinion of the author, one of the primary conclusions to emerge from the discussions to be made here is that despite the fact that the theory of a single dislocation has reached a high level of sophistication, the promise of dislocation-based models of plasticity with predictive power remains elusive. One reason for this is the fact that true... 

Finite element modeling is a technique whereby a material continuum is divided into a number of patches, or finite elements, and the appropriate engineering theory is applied to solve a variety of problems. The initial (and probably still dominant) use of finite element modeling was for the solution of structural engineering problems. The technique is currently being applied by a number of companies and research institutions in the design of plastic products. CAD/CAM systems provide the means to create a mesh of finite elements directly from a product model database, by automatic and semiautomatic means. 

Green, A.E. and Naghdi, P.M., "A General Theory of an Elastic-Plastic Continuum", Arch. Rat. Mech. Anal., 18, pp.251-281, 1965. 

The constitutive theory is based on a decomposition of the foam in two parts (Figure 2(b)) a skeleton, i.e. the cell walls, and a nonlinear elastic continuum. The skeleton accoimts for the foam behavior in the linear elastic and collapse plateau regimes and is approached by a coupled plasticity and continuum damage theory. The diffuse nonlinear elastic continuum describes the densification of the foam caused by the compression of the enclosed gas and the interaction between the cell walls. 

Progress toward resolving such questions is summarized in this chapter. The discussion begins with the issue of fundamental dislocation interaction phenomena and nonequilibrium behavior of interacting dislocations. Attention is then shifted from consideration of films with low dislocation density to the modeling of inelastic deformation of thin films with a relatively high densities of dislocations. For this purpose, constitutive models for time-independent and time-dependent deformation of thin films are examined by appeal to continuum plasticity theory. Overall features of material behavior captured by such theories are then compared with available... 

The experimental results shown in Tables 7.1 and 7.2, as well as in Figures 7.29 and 7.30, indicate that the yield stress of metallic thin films varies inversely with film thickness /if. Such observations cannot be rationalized by recourse to conventional continuum plasticity theories which are devoid of intrinsic length scales. Trends in modeling the plastic response of thin films are discussed briefly in the next section. 

At the microscale, dislocation dynamics simulations implement the equations of continuum elasticity theory to track the motion and interaction of individual dislocations under an applied stress, leading to the development of a dislocation microstructure and single-crystal plastic deformation. In our multiscale modeling... 

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