Slip-line theory

This is similar to the analysis obtained by Ainsley and Smith (see Chhabra, 1992) using the slip line theory from soil mechanics, which results in a dimensionless group called the plasticity number ... 

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. 

Several researchers tried to replace the single-shear plane model by a shear zone model. Lee and Shaffer (1951) provided a slip-line solution by applying the theory of plasticity. In the slip-line model, the metal is assumed to flow along the line of maximum shear lines. The slip-line field solution cannot be applied easily to three-dimensional as well as strain-hardening cases. Sidjanin and Kovac (1997) applied the concept of fracture mechanics in chip formation process. Atkins (2003) demonstrated that the work for creation of new surfaces in metal cutting is significant. He also points out that Shaw (1954) has shown this work to be insignificant. However, when this work is included based on the modem ductile fracture mechanics, even the Merchant analysis provides reasonable results. 

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 actual behavior of the blunting crack requires for its analysis numerical approaches that we consider below. However, here we try first to capture the essential features of the flow pattern from the ideally plastic, non-hardening material solutions using slip-line-field approaches of plasticity theory. 

In the BOC fan the slip lines undergo a smooth rotation from BO to CO by an angle Af = ujl, which, according to slip-line-field theory, results in a monotonic change with of the mean normal stress by... 

Based on the slip-line field theory [e.g., see Hill (1950)], Adachi and Yoshioka (1973) also extended the analysis of Ansley and Smith (1967) for spheres to include the creeping cross-flow over cylinders and obtained the following approximation expression for X ... 

A large class of solutions was obtained for plane strain, plane stress and axisymmetric problems [3,4,5], using the method of slip-lines or providing lower and upper bounds to limit load factors. However, experimental studies indicate that in many technical problems such as coaxial plate cutting (Fig.la),punch indentation into a quarter-space (Fig.lb), etc., the failure modes are three-dimensional and their form is much effected by both geometrical and material parameters. Numerous papers were devoted to the problem of plate cutting.for ductile or brittle materials [6,7,8]. Some experimental verification was provided in [9,10] and the agreement between theory and experiment was obtained for cases where plane strain condition was predominant. 

Liquid viscosities have been observed to increase, decrease, and remain constant in microfluidic devices as compared to viscosities in larger systems. ° Deviations from the no-slip boundary condition have been observed to occur at high shear rates. One important deviation from no-slip conditions occurs at moving contact lines, such as when capillary forces pull a liquid into a hydrophilic channel. The point at which the gas, liquid, and solid phases move along the channel wall is in violation of the no-slip boundary condition. Ho and Tai review discrepancies between classical Stokes flow theory and observations of flow in microchannels. No adequate theory is yet available to explain these deviations from classical behavior. ... 

The non-slip boundary condition is discussed in an excellent paper by Huh and Scriven They take note of the fact that, previous workers seem not to have been well informed by fluid mechanics , in aUuding to the essentially surface chemical analyses of spreading dynamics. Another point they address is that except for very smooth surfaces and non-adsorbing hquids the advancing or receding of the contact line proceeds in a slip-stick and discontinuous fashion a fact which is the focus of attention in current analyses of contact angle hysteresis using the theory of random fluctuations... 

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