Flow theory, plasticity

It is assumed that the reader has an elementary knowledge of the mechanics of materials. However, appendices given at the end summarise the necessary principles and provide the heat and fluid flow theories relevant to plastics. 

An example of a material model based on the physics of material behavior is classical metals plasticity theory. This theory, often referred to as /2-flow theory, is based on a Mises yield surface with an associated flow rule, followed by rate-independent isotropic hardening (Khan and Huang 1995). Physically, plastic flow in metals is a result of dislocation motion, a mechanism known to be driven by shear stresses and to be insensitive to hydrostatic pressure. 

A variation of the return mapping algorithm (30)-(32) that also imposes exactly the isochoric plastic response in isochoric plastic models, like the classical /2-flow theory of metals, can be found in Armero Zambrana-Rojas, (2007). Similarly, we refer to Armero Romero (2001a, b) for variations of the stress formula (33) that incorporate a controllable high-frequency numerical energy dissipation to handle the usual high numerical stiffness in the systems of interest. 

We consider /2-flow theory of elastoplasticity, defined by the plastic evolution equations (10)-(13) and the classical von Mises yield snrface... 

The result of a simple tension experiment for a metal is schematically shown in Fig. 2.17 with axes of axial stress ai and axial sdain 1 or deviatoric stress s and deviatoric strain e. In metals the volumetric plastic strain can generally be ignored (sP = 0) therefore we can treat the behavior as a uniaxial response. On the other hand, the shearing behavior of geomaterials is inevitably accompanied by volume changes that are plastic, therefore we have to modify the original flow theory developed for metallic materials (Kachanov 2005 Lubliner 1990). Note that in small strain plasticity we assume that the plastic increment de can be decomposed into incremental elastic and plastic components ... 

Small Strain Plasticity Flow Theory The indicial form of D p is given by... 

Time dependence Viscoelastic deformation is a transition type behavior that is characterized by the occurrence of both elastic strain and time-dependent flow. It is the time dependence of the mechanical properties of plastics that makes the behavior of these materials difficult to analyze by mathematical theory. 

The model proposed by Bowden and Tabor has been regarded as the most successful one for presenting a simple and logical theory capable of explaining the Amontons friction law. However, suspicions concerning the two fundamental assumptions in the model were gradually aroused over past years. Friction has been attributed, in Bowden and Tabor s model, to the adhesion between asperities in contact and torn-off of the adhesive junctions when the shear stress exceeds a critical value. This implies that plastic flow and surface destruction may occur at the moment of slip, and that friction is dominated by the shear strength of the adhesive conjunctions, which is material dependent. 

The rheological characteristics of AB cements are complex. Mostly, the unset cement paste behaves as a plastic or plastoelastic body, rather than as a Newtonian or viscoelastic substance. In other words, it does not flow unless the applied stress exceeds a certain value known as the yield point. Below the yield point a plastoelastic body behaves as an elastic solid and above the yield point it behaves as a viscoelastic one (Andrade, 1947). This makes a mathematical treatment complicated, and although the theories of viscoelasticity are well developed, as are those of an ideal plastic (Bingham body), plastoelasticity has received much less attention. In many AB cements, yield stress appears to be more important than viscosity in determining the stiffness of a paste. 

In retrospect, it should not be surprising that a time independent theory modeled after elasticity theory does not apply to a plastic flow process. Elastic deformation is conservative with the work done on the material stored as elastic strain energy. Plastic deformation is non-conservative with the work done on the material dissipated as heat, or converted into internal defects... 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

This paper rerports an investigation of the yield behavior of several amine and anhydride cured DGEBA resin systems. The Argon theory is used to assess the controlling molecular parameters from the experimental results. Such parameters are then compared with the known chemical structures of the resins. The mechanisms of plastic flow in thermoset polymers such as epoxies is demonstrated. 

Solids 12, 59 - 65 (1964) "A Generalized Theory of Strain- Rate- Dependent Plastic Wave Propagation in Bars 86) M. Lutzky, "The Flow Field Behind a Spherical Detonation in TNT, Using the Landau- Stanyukovich Equation, USNOL-White Oak, NOLTR 64-40 Dec 1964)... 

The viscous dissipation term is normally not important. Its significance has been considered in connection with lubrication theory (VI), flow through tubes (B20), extrusion of plastics melts (BIO), and viscometry in rotating-cylinder systems (W6). There is also an additional contribution to the energy flux vector describing energy transport by radiation. See discussion in connection with Eq. (29). 

Ree,T., Eyring,H. Theory of non-newtonian flow. I. Solid plastic system. J. Appl. Phys. 26,793-800(1955). 

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