Elastic-Plastic Ratios

Another important ratio is H IE that has units of pressure and is representative of the film resistance to plastic deformation (Johnson, 1985). This ratio is widely employed in the development of hard coatings for tribological applications (Musil, 2000). Various elastic-plastic ratios and units are summarized in Table H.l. 

Let us analyse the above data on the basis of Lawn and Howes analysis 29). Based on the mechanics of hardness identation - assuming the loading cycle to be elastic-plastic and unloading to be elastic — these authors have recently derived an interesting expression of the residual impression parameter (relative depth recovery) as function of the ratio MH/E. Accordingly ... 

The plasticity characteristic /, was calculated as the ratio of the work spent for plastic deformation of the material to the total work spent for elastic - plastic deformation of a material under indenter. It is shown (Table 1), that X for 400 nm thickness molybdenum coatings is higher, than its value for chromium coatings of the same thickness. 

The value of the elastic Poisson ratio used is shown in Table I. However, the analysis was carried out using the assumption that plastic deformation takes place essentially at constant volume that is, with a Poisson ratio of 0.5. The Poisson ratio of any element is the weighted sum of the elastic Poisson ratio and 0.5, weighted via the proportion of elastic and plastic strain. However, it should be noted that this complex behavior leads to considerable difficulty regarding directional loading of the spherical model for elastic-plastic behavior. 

One of the major areas of RubCon application is in structures that operate in aggressive environments where crack resistance of the material is important. Our experiments have shown that steel fibers show the greatest opportunity to increase crack resistance and allow production of a material with high elastic-plastic properties. For this reason, we have undertaken research on the influence of fiber reinforcement ratio and aspect ratio on RubCon strength at compression, tension, and bend. 

After nucleation, the median crack propagates away from the elastic-plastic zone boundary. Stable propagation occurs on increasing the external load, and the ratio of the diameter of the median crack, D, to the indent size is given by ... 

For the common textile uses, fibres are characterised by flexibility, fineness and a high ratio of length to width (McIntyre and Daniels, 1995), but, they must also have an intermediate extensibility. Most have at least partially recoverable extensions up to typical break extensions of 7 to 50%, much higher than for brittle solids or the yield extension of elastic-plastic materials and much lower than for elastomers. Such properties are achieved by partially oriented, partially crystalline polymers, and are almost completely satisfied by six chemical types cellulose, protein, polyamide, polyester, polyacrylonitrile and polypropylene. 

The indentation confidence depth was determined to be a function of the coating to substrate yield strength ratio for three different substrate configurations. Tilbrook et al. (2007) incorporated microstructural failure mechanisms into finite element simulations of nanoindentation of TiN thin films on elastic—plastic substrates. Intergranular sliding that occurs as a result of the columnar grain structure is incorporated into the model via anisotropic property definitions and nodal coupling. 

Both Eqs. (11.1) and (11.2) account for the effect of transverse strain on plastic strain intensity factor characterized by the modified Poisson s ratio, V. In Eq. (11.1), this is accounted for by the ratio Sy/Sa, whereas in Eq. (11.2) the ratio Eg/E serves the same purpose as will be shown later. The modified Poisson s ratio in each case is intended to account for the different transverse contraction in the elastic-plastic condition as compared to the assumed elastic condition. Therefore this effect is primarily associated with the differences in variation in volume without any consideration given to the nonlinear stress-strain relationship in plasticity. Instead the approaches are based on an equation analogous to Hooke s law as obtained by Nadai. Gonyea uses expression (rule) due to Neuber to estimate the strain concentration effects through a correction factor, K, for various notches (characterized by the elastic stress concentration factor, Kj). Moulin and Roche obtain the same factor for a biaxial situation involving thermal shock problem and present a design curve for K, for alloy steels as a function of equivalent strain range. Similar results were obtained by Houtman for thermal shock in plates and cylinders and for cylinders fixed to a wall, which were discussed by Nickell. The problem of Poisson s effect in plasticity has been discussed in detail by Severud. Hubei... 

For a vessel of iniiiite oEtemal raldius the maximum value of the integral for the example given (see page 511 of Reference 239) is 185,800 psi. In the example quoted the elastic-plastic interface was placed at a radius ratio of 3.0. 

MPa the elastic modulus of the SiC whisker E = 400 GPa, and Poisson s ratio Vf = 0.16 the interfacial layer of the material is isotropic, Poisson s ratio is 0.2, and interfacial elastic modulus is E. The elastic-plastic modulus is calculated by using a different interfacial modulus E value. The exterior stress Oq is set at 0.85 MPa, and the influence of different interfacial bonding states on stress concentration coefficient, interfacial shear stress, and axial stress of whisker end face is simulated. Figure 4.12 is the stress concentration coefficient... 

Fig. 4.14. A hierarchical point of view of interface fracture advance, whereby the complexities of material separation are lumped into a representative phenomenological cohesive rule that is representative of the system its essential features are the work per unit area Fo required for separation of the surfaces and the maximum cohesive traction <7 that arises in the process. The cohesive traction must be imposed by the surrounding film and substrate materials, viewed as elastic-plastic continua. The tendency for significant plastic deformation in either material is determined by the ratio of a to the yield stress of that material. The driving force necessary to effect separation is characterized by an energy release rate Q. To sustain crack growth, its value must be large enough to overcome Fq plus plastic dissipation per unit area Fp. Adapted from Hutchinson and Evans (2000).

The temperature Tr,ei is also indicated in Figure 7.15. If the bilayer is subjected to thermal cycles between the temperature limits T = 0 and T = Tr,el) there is no net accumulation of plastic strain with increasing number of thermal cycles, and the system is said to have undergone elastic shakedown. The relative magnitudes of Ty, Tr ei and Tpi are strongly influenced by the elastic modulus ratio m and the thickness ratio rj for the film-substrate system. 

The attempt to relate the nonlinear and steady flow behavior with the plasticity (the yield stress) of the solutions of styrene (S) and butadiene (B) block copolymers were carried out by using a concentric cylinder rheometer [8]. Nonsinusoidal responses of the outer cylinder found in the SB sample were analyzed to obtain G - and G J by Eq. (78). Figure 54 shows the frequency dependence of the nonlinear dynamic modulus G - and GJ (/ = 1 and 3) for the system. Typical nonlinear behavior, that is, a large contribution of the higher odd harmonics and a plateau region in the fundamental harmonics, can be seen in the data with 60 = 2 and 4, whereas in those with 1.1° the response is nearly linear and elastic. The ratio of the amplitude of the 7th harmonics to that of the fundamental harmonics, Gj/Gj, can be used as a measure of the nonlinearity of the system, where Gj = [(Gj)+(GJ)]. Figure 55 shows the frequency dependence of Gj/G for the SB system with various... 

The elastic-plastic contact area coeffieient ka represents the ratio of plastic deformation Api to... 

The minimum required thickness can be determined by using closed-form equations for cylindrical and spherical shells, blind ends, threaded closures, and clamp connections. The elastic-plastic analysis can be used in most cases and for all D/d ratios. The linear elastic analysis is permitted only if D/d< 1.25 [17]. 

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