Onset of Plastic Deformation

In order to estimate the onset yield stress of the material, three common criteria are introduced. The Tresca criterion is based on maximum shear stress and is given as 

The third criterion is obtained from the maximum reduced stress and is expressed as 

No significant difference is found for the predictions of the yield stresses from these three criteria. However, Tresca s criterion is more widely used than the other two because of its simplicity. When two solid spherical particles are in contact, the principal stresses along the normal axis through the contact point can be obtained from the Hertzian elastic 

It can be proved from Eq. (2.156) that, for materials with Poisson s ratio of 0.3 (which is true for most solids), the maximum shear stress oz — or occurs at z/rc = 0.48. Consequently, according to Tresca s criterion, the yield stress Y in a simple compression is 0.62 p0. Therefore, when the hardness or the yield stress Y of the particle material is less than 0.62 times the maximum contact pressure, the sphere will, most likely, undergo plastic deformation. From the elastic collision of two solid spheres, the maximum contact pressure is given by Eq. (2.134). Thus, the relation between the critical normal collision velocity, Ui2Y. and the yield stress is given by 

Moreover, it can be shown from Eq. (2.156) that the maximum shear stress rz — or is insensitive to the variation of Poisson s ratio for solids, only deviating 5 percent when Poisson s ratio varies from 0.27 to 0.36 (see Problem 2.4). Thus, Eq. (2.157) can also be applied for solids with Poisson s ratio in the range of 0.27 to 0.36. 

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For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

The onset of plastic deformation in a material under load is called yielding [60]. In contrast to the experiments described in the previous sections, yielding causes a permanent deformation, i.e. a deformation that remains after the load is removed. The effects of crosslinks on the yield behaviour of polymers are demonstrated by three experiments ... 

Plastic deformation is the permanent change in shape of a specimen due to applied stress. The onset of plastic deformation is seen as curvature in the stress—strain curve. Plastic deformation is important because it allows pharmaceutical excipients and drugs to establish large true areas of contact during compaction that can remain on decompression. In this way, good, intact, tablets can be prepared. 

Several groups tried to unravel whether the cavitation occurs before or after the onset of plastic deformation. Real-time small angle X-ray scattering [162] and light scattering [163,164] techniques were used to study the deformation of... 

Constant-load SCC tests have been shown to be more severe than constant-deflection tests. Under a constant load, stress increases as the cross-section is reduced by cracking or corrosion. However, this condition produces decreasing stress when deflection is fixed. It has been suggested that SCC threshold stress is associated with the onset of plastic deformation, that is, the elastic limit of the alloy. The elastic limit is difficult to measure unambiguously, however, the stress at which 0.2% plastic deformation occurs is generally used. 

A high pr value indicates that there will be a high volume reduction of the product due to particle rearrangement. The constant A has been shown to be equal to the reciprocal of the mean yield pressure required to induce plastic deformation. A larger value for A (low yield pressure) indicates the onset of plastic deformation at relatively low pressure, a sign that the material is more compressible. 

Solidification and deformation processes are very seldom used to fabricate bulk articles from ceramics and other materials with low ductility and malleability. These substances are brittle and suffer fracmre before the onset of plastic deformation. Additionally, ceramics normally have exceedingly high melting points, decompose, or react with most cm-cible materials at their melting temperatures. Many ceramics are worked with in powder form since the products of most solid-state chemical syntheses are powders. Fabricating a bulk part from a powder requires a consolidation process, usually compaction followed... 

Figure 5.2. Stress vs. strain curves for various polymers around its glass-transition temperature. The maximum in the curve that occurs at Tg is referred to as the yield point (onset of plastic deformation).

Fig. 20a. Tensile strain at the onset of plastic deformation in PC as a function of temperature for samples having a molecxdar weight of 30,000 (O) and 150,000 ( ) . b Plot of the extension ratio in the crazes as a function of temperature for the same samples described in a...

Fig. 23. Tensile strain at onset of plastic deformation in PES as a function of temperature for samples having a molecular weight of47.000 Prior to deformation the samples were physically aged at 200 Cfor 1 hour ( ),4hours ( land 70hours (O)...

The onset of plastic deformation in ABS copolymer can be seen as a thermally and stress activated process. In that framework, the Eyring equation can be written in terms of strain rate and temperature as... 

From the standpoint of a stress-strain curve, the constitutive behavior discussed in the previous section corresponds to a limited range of loads and strains. In particular, as seen in fig. 2.10, for stresses beyond a certain level, the solid suffers permanent deformation and if the load is increased too far, the solid eventually fails via fracture. From a constitutive viewpoint, more challenges are posed by the existence of permanent deformation in the form of plasticity and fracture. Phenomenologically, the onset of plastic deformation is often treated in terms of a yield surface. The yield surface is the locus of all points in stress space at which the material begins to undergo plastic deformation. The fundamental idea is that until a certain critical load is reached, the deformation is entirely elastic. Upon reaching a critical state of stress (the yield stress), the material then undergoes plastic deformation. Because the state of stress at a point can be parameterized in terms of six numbers, the tensorial character of the stress state must be reflected in the determination of the yield surface. 

Fig. 8.3. Slip traces on crystal surface. Slip traces are a microstructural consequence of the onset of plastic deformation (courtesy of C. Coupeau).

As will be seen below, the presence of dislocations has the effect of producing local disturbances of the atomic-level geometry that are so severe as to produce scattering that is not consonant with that from the remainder of the crystal, giving rise to the contrast revealed in the figure. Evidently, the onset of plastic deformation reveals its presence at multiple scales simultaneously. Indeed, as was seen in fig. 1.9, the properties of individual dislocation cores can be observed using high-resolution transmission electron microscopy. 

Rubber particle size or ligament thickness has no effect on the onset of plastic deformation but may affect the extent of plastic deformation (i.e., the maximum draw ratio). A small change in draw ratio is of little consequence under isothermal conditions, but under adiabatic conditions it may result in enough heat generation for local melting and, hence, melt blunting. 

The behavior described by Equation 11.44 differs at a fundamental physical level from the behavior described by Equation 11.31 for the Gy of thermoplastics. As was discussed in Section ll.B.2.e, the chemical crosslinks do not typically have a strong effect on E, which is controlled mainly by interchain interactions on a more local length scale. On the other hand, the distance between the topological constraints created by the crosslinks in a typical thermoset is of a length scale that is relevant for yielding which defines the onset of plastic deformation. 

Criteria 2, 5, and 6 are generally used for yielding, or the onset of plastic deformation, whereas criteria 1,3, and 4 are used for fracture. The maximum shearing stress (or Tresca [3]) criterion is generally not true for multiaxial loading, but is widely used because of its simplicity. The distortion energy and octahedral shearing stress criteria (or von Mises criterion [4]) have been found to be more accurate. None of the failure criteria works very well. Their inadequacy is attributed, in part, to the presence of cracks, and of their dominance, in the failure process. 

Because of the constraint imposed under plane strain conditions, yielding (onset of plastic deformation) would occur at a higher stress level. A number of estimates were made with different assumed constraint and yielding criteria [9]. But, because of the approximate nature of these estimates, the plastic zone correction factor for plane strain is taken to be that given by Eqn. (3.50). 

Figure 6.12 shows a typical force-displacement curve for ice cream. The force initially increases linearly with the displacement (this is the elastic region). The Young s modulus is a measure of the resistance to deformation of the ice cream and is given by the initial slope of the force displacement curve, AFIAd. At a certain point, the slope of the force-displacement curve begins to decrease. This corresponds to the onset of plastic deformation. The maximum force (i ax) corresponds to the point at which a crack forms and the sample begins... 

Hydrogen incorporation in thin films can produce an extremely high out-of-plane expansion due to the clamping of the thin film to the substrate [20]. Within the sohd solution phase the expansion can be predicted by linear elastic theory [21]. However, deviations from linear elastic theory were reported for high H concentrations due to the onset of plastic deformation in the film [22, 23]. 

The proportional limit P corresponds to departure from linearity and is defined as the onset of plastic deformation. If the transition from elastic to plastic deformation is gradual it may be difficult to determine precisely where P is and sometimes it is better avoided. 

The yield stress (defining the onset of plastic deformation) = force at yield point/original cross-sectional area. 

The applied stress required to induce the onset of plastic deformation (yield point) is called yield stress. This stress is the most important value for structural design because it avoids the practical difficulties encountered in measuring the exact stress for the proportional limit or elastic limit at which a material starts to deform plastically. It is extremely sensitive to the structure and prior history of a material. 

During an accident, a channel tube rupture may be expected either on account of a steep pressure rise at near operational temperatures or as a result of thermomechanical deformation at rather high pressures in the circuit. The admissible pressure of hydraulic tests, equal to 13.4 MPa, may be taken as a conservative acceptance criterion for cases when the tube temperatures are close to their operating values (i.e. 50°C). The conservatism of this value can be easily proven by evaluating the pressure corresponding to the onset of plastic deformation of a tube. For the temperature of 300°C, the result is 20 MPa, and a real threat of pressure tube rupture will not appear until this pressure is exceeded. 

Equation (4.6) gives the resolved shear stress . The product in the equation is known as the Schmid factor and determines whether the orientation is favorable for shp. The conditions for slip are given by Schmid s Law and the value of Eq. (4.6), often represented in the literature by Tj, indicating the onset of plastic deformation and called critical resolved shear stress . CRSS is a structure-sensitive property, since it is very dependent on impurities and the way the crystal was grown and handled. 

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