Plastic deformation parameters

It is usual in the classical theory to assume that the stress rate is independent of the hardening parameters, since the elastic behavior is expected to be unaffected by plastic deformation. Consequently, the stress rate relation (5.23) reduces to... 

A typical shock-compression wave-profile measurement consists of particle velocity as a function of time at some material point within or on the surface of the sample. These measurements are commonly made by means of laser interferometry as discussed in Chapter 3 of this book. A typical wave profile as a function of position in the sample is shown in Fig. 7.2. Each portion of the wave profile contains information about the microstructure in the form of the product of and v. The decaying elastic wave has been an important source of indirect information on micromechanics of shock-induced plastic deformation. Taylor [9] used measurements of the decaying elastic precursor to determine parameters for polycrystalline Armco iron. He showed that the rate of decay of the elastic precursor in Fig. 7.2 is given by (Appendix)... 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

The representation of the results from slow strain-rate tests may be through the usual ductility parameters such as reduction in area, the maximum load achieved, the crack velocity or even the time to failure, although as with all tests, metallographic or fractographic examination, whilst not readily quantifiable, should also be involved. Since stress-corrosion failures are usually associated with relatively little plastic deformation, the ductility... 

In relatively recent years, it has been found that that indentations made in covalent crystals at temperatures below their Debye temperatures often result from crystal structure changes, as well as from plastic deformation via dislocation activity. Thus, indentation hardness numbers may provide better critical parameters for structural stability than pressure cell studies because indentation involves a combination of shear and hydrostatic compression and a phase transformation involves both of these quantities. 

Measured and calculated parameters related to the plastic deformation in epoxy resins... 

The plastic deformation in several amine and anhydride cured epoxy resins has been studied. The experimental results have been reasonably interpreted by the Argon theory. The molecular parameters determined from the data based on the theory reflect the different molecular structures of the resins studied. However, these parameters are in similar enough range to also show the structural similarity in these DGEBA based systems. In general, the mechanisms of plastic deformation in epoxy resins below T are essentially identical to those in amorphouE thermoplastics. The yield stress level being related to the modulus that controls the intermolecular energy due to molecular deformation will, however, be affected by the crosslinks in the thermosets. 

Strain rate sensitivity of (or the effect of press speed on) the formulation is of primary concern in scale-up. Whether the product development work was performed on a single-stroke press or a smaller rotary press, the objective in operations will be to increase efficiency, in this case the tablet output rate and, therefore, the speed of the press. For a material that deforms exclusively by brittle fracture, there will be no concern. Materials that exhibit plastic deformation, which is a kinetic phenomenon, do exhibit strain rate sensitivity, and the effect of press speed will be significant. One must be aware that although specific ingredients (such as calcium phosphate and lactose) may exhibit predominately brittle fracture behavior, almost everything has some plastic deformation component, and for some materials (such as microcrystalline cellulose) plastic deformation is the predominant behavior. The usual parameter indication is that target tablet hardness cannot be achieved at the faster press speed. Slowing the press may be the only option to correct the problem. 

According to Eq. (20) the compression of isotropic solid polymers having positive thermal expansivity must be accompanied by the internal energy inversion. AU inversion at compression has been estimated71 to occur at strains 5-15%. At compression, irreversible plastic deformations occur which prevents a correct experimental determination of AU. With inversion parameters, AU for isotropic poly-... 

Various models 1-2,42 43) have been proposed to describe the extent and shape of the localised plastic deformation zone at the crack tip. From these models one may define a parameter known as the crack opening displacement, 5, (see Fig. 16) and the value of 5,c for the onset of crack growth is given by... 

By necessity, the treatment of solid state kinetics has to be selective in view of the myriad processes which can occur in the solid state. This multitude is mainly due to three facts 1) correlation lengths in crystals are often much larger than in fluids and may comprise the whole crystal, 2) a structure element is characterized by three parameters instead of only by two in a liquid (chemical species, electrical charge, type of crystallographic site), and 3) a crystal can be elastically stressed. The stress state is normally inhomogeneous. If the yield strength is exceeded, then plastic deformation and the formation of dislocations will change the structural state of a crystal. What we aim at in this book is a strict treatment of concepts and basic situations in a quantitative way, so far as it is possible. In contrast, the often extremely complex kinetic situations in solid state chemistry and materials science will be analyzed in a rather qualitative manner, but with clearcut thermodynamic and kinetic concepts. 

Dislocations are line defects. They bound slipped areas in a crystal and their motion produces plastic deformation. They are characterized by two geometrical parameters 1) the elementary slip displacement vector b (Burgers vector) and 2) the unit vector that defines the direction of the dislocation line at some point in the crystal, s. Figures 3-1 and 3-2 show the two limiting cases of a dislocation. If b is perpendicular to s, the dislocation is named an edge dislocation. The screw dislocation has b parallel to v. Often one Finds mixed dislocations. Dislocation lines close upon themselves or they end at inner or outer surfaces of a solid. 

This relation is shown in Fig. 6.3.4 for the example of Si ceramics. The threshold parameters of plastic deformation in hardness tests can be determined to a good approximation according to Evans and Law (1977) from... 

Morris, D. G. Munoz-Morris, M. A. Relationships between mechanical properties, grain size, and grain boundary parameters in nanomaterials prepared by severe plastic deformation, by electrodeposition and by powder metallurgy methods. J. Metastable Nanocrystal. Mater. 15-16, 585-590 (2003). 

Since the force-displacement curve contains information about the whole indentation process, the elastic deformation of the sample can be measured and used to calculate the stiffness S=dFldh at h=hmax, where F is the force and h is the indentation. As already explained in Sect. 3.1.1., in order to relate the stiffness to the Young s modulus, it is necessary to make assumptions about the contact area. The depth of the permanent indentation (plastic deformation), i.e. the depth DFdi shown in Fig. 26b, and the maximum indentation (sum of the plastic and of the elastic deformation) can be used to calculate a parameter that describes the relative weight of the elastic and of the plastic response. 

In a subsequent investigation, with Roos and Kampschreur (1989), Northolt extended the modified series model to include viscoelasticity. For that an additional assumption was made, viz. that the relaxation process is confined solely to shear deformation of adjacent chains. The modified series model maybe applied to well-oriented fibres having a small plastic deformation (or set). In particular it explains the part of the tensile curve beyond the yield stress in which the orientation process of the fibrils takes place. The main factor governing this process is the modulus for shear, gd, between adjacent chains. At high deformation frequencies yd attains its maximum value, ydo at lower frequencies or longer times the viscoelasticity lowers the value of gd, and it becomes a function of time or frequency. Northolt s relations, that directly follow from his theoretical model for well-oriented fibres, are in perfect agreement with the experimental data if acceptable values for the elastic parameters are substituted. 

The equation of Heckel has been discussed again and again. One main issue of critique is that pharmaceutical powders are not purely plastically deforming materials and thus particle size and deformation mechanisms influence the derived parameters [129, 130]. Already very small errors in displacement determination or the measurement of true density can induce huge errors in the derived parameters [75-77, 129, 131, 132], Spnnergaard [126] referred the equation of Walker and Bal shin for his characterization of materials. He criticized further that the yield strength derived from the Heckel equation is directly dependent on the true density of the powders [127]. 

Fig. 12 Instantaneous plastic deformation for the set of craze parameters B at loading rate iCj° 3 x 10-2 MPaVrn/s. a Prior to craze fibril breakdown b, c during crack propagation, with K / (so /r ) 1.32 (from [22])...

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