Plasticity yield criterion

W. L. Johnson and K. Samwer, A Universal Criterion for Plastic Yielding pf Metallic Glasses with a (T/Tg)2S Temperature Dependence, Phys. Rev. Lett., 95, 195501 (2005). 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

Yield criterion, which defines the transition of elastic to plastic deformation... 

In the model derived by McClintock and Irwin the shape and size of the plastic zone were calculated by a combination of the stress field at the crack tip (e.g. Eq. (2)) with a yield criterion (e.g. von Mises, Tresca). This leads to the well known dog-bone type of plastic zone showing the influence of stress state. Its form is often approximated to by a circle of radius Tp, where... 

Since the stresses are singular at the crack tip, then clearly the yield oiterion is exceeded in some zone in the crack tip region. If this zone is assumed to be small, then it will not greatly disturb the elastic stress field so that the extent of the plastic zone will be defined by the elastic stresses. If it is assumed that the Von Mises yield criterion is applicable (a reasonable first approximation for polymers), then the shape and e of the plastic zone may be derived from the stresses given in Eq. (15). As-sumii a state of plane strain so that the transverse stress is given by 1/(0 + oee)> then for a yield stress of Oy, the dastic zone radius becomes ... 

For these calculations, three different material models were investigated a linear elastic, an elasto-brittle, and elasto-plastic ubiquitous joint model. The ubiquitous joint, elasto-plastic model, is defined by a two-dimensional yield criterion, composed of two Mohr-Coulomb criteria, along two predefined directions characterised by their normal vectors ni and n2. 

Suction changes may create irreversible strains. In the Barcelona model, this is modelled thanks a yield surface, the SI Suction Increase curve. When suction becomes higher than a suction level So, plastic strains are created. This yield criterion is introduced in our constitutive law ... 

Tresca and R. von Mises criteria are for isotropic materials. In 1948, Rodney Hill provided a quadratic yield criterion for anisotropic materials. A special case of this criterion is von Mises criterion. In 1979, Hill proposed a non-quadratic yield criterion. Later on several other criteria were proposed including Hill s 1993 criterion. Rodney HiU (1921-2011) was bom in Yorkshire, England and has tremendous contribution in the theory of plasticity. 

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

In accordance with the field sampling and rock mechanics test results, the rocks presented clear plastic deformation characteristics at different confining pressure conditions. In this paper, Mohr— Coulomb yield criterion was used to determine the... 

It should be emphasized that two fundamentally different types of craze tests were performed in this work. The test described initially, in which the craze stress below a notch was calculated from the slip line plasticity theory, without exposure to solvent, is a test in which the strain is changed as a function of time. The craze stress itself is calculated assuming that both slip line plasticity theory and the simple von Mises yield criterion are both applicable. The second test, used to determine the effect of solvent on crazing, is a surface crazing test under simple tension in which the strain... 

The idea of plastic yielding is also applied to granular materials, such as soils and powders. In this case, the shear stress required for deformation depends on the packing density, and the particle shape and surfaee characteristics. The shear resistance is commonly described by the Mohr-Coulomb yield criterion, i.e.. 

A yield criterion defines the limit of elasticity (or onset of plastic flow) under any combination of stresses. Figure 1 shows a stress-strain curve that might be derived from a simple uniaxial tensile test. 

The deviatoric stresses are thus related to the maximum shear stresses, a fact which is in accord with experimentally observed relatirMiships between shearing and plastic flow, thus lending support to this intuitive approach. On these grounds, the following type of yield criterion should be acceptable ... 

The strength-differential effect is also reflected prominently in the multi-axial yield criteria which translate the multi-axial stress driving forces for yield into an equivalent uniaxial state of extension (tension) or simple shear <7se that is most relevant to the mechanisms governing plastic flow. In a more mechanistieally relevant statement for polymers, the multi-axial yield criterion of von Mises defines a uniaxial equivalent stress Oe (or a o-se) as... 

Most amorphous solids and many crystalline ones, particularly non-metals and polymers, exhibit a Coulomb-Mohr-type (Coulomb 1773 Mohr 1900) yield criterion or plastic-shear resistance such that this resistance on the best shear plane is dependent on the normal stress acting across the plane of shear, resulting in a dependence of the type... 

A three-dimensional formulation of the Bingham plastic was developed by Hohenemser and Prager [H19] in 1932 using the von Mises yield criterion (see Reiner [R4] and Prager [P15]). This employs a deviatoric stress tensor T and has the form... 

Based on the mathematical theory of plasticity, the plastic deformation behavior of the material can be described by the three components of the rate-independent plasticity model, namely yield criterion, flow rule, and hardening rule. The yield criterion determines the stress level at which yielding is initiated. This is represented by the equivalent stress Ueq, which is a function of the individual stress vector components a. Plastic strain is developed in the metal parts when the equivalent stress is equal to a material yield parameter ay finally, the flow rule determines the direction of plastic straining ... 

We see that the Coulomb yield criterion therefore defines both the stress condition required for yielding to occur and the directions in which the material will deform. Where a deformation band forms, its direction is one that is neither rotated nor distorted by the plastic deformation, because its orientation marks the direction that establishes material continuity between the deformed material in the deformation band and the undistorted material in the rest of the specimen. If volume is conserved, the band direction denotes the direction of shear in a simple shear (by the definition of a shear strain). Thus for a Coulomb yield criterion the band direction is defined by Equation (11.6). 

A very simple explanation of the effect of notching has been given by Orowan [95], For a deep, symmetrical tensile notch, the distribution of stress is identical to that for a flat frictionless punch indenting a plate under conditions of plane strain [102] (Figure 12.31). The compressive stress on the punch required to produce plastic deformation can be shown to be (2 + 7t)K, where K is the shear yield stress. For the Tresca yield criterion the value is l.Sloy and for the von Mises yield criterion the value is 2.82oy, where 0 is the tensile yield stress. Hence for an ideally deep and sharp notch in an infinite solid the plastic constraint raises the yield stress to a value of approximately 2>Oy which leads to the following classification for brittle-ductile behaviour first proposed by Orowan [95] ... 

Since the Tresca yield criterion can be easily evaluated using Mohr s circle, it is often used in heuristic explanations. Using it in the calculation of plastic deformations, for example, with the method of finite elements, is problematic, though, as we will see on page 96. 

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