Modulus misfit

For nickel, cobalt, and hon-base alloys the amount of solute, particularly tungsten or molybdenum, intentionally added for strengthening by lattice or modulus misfit is generally limited by the instability of the alloy to unwanted CJ-phase formation. However, the Group 5(VB) bcc metals rely on additions of the Group 6(VIB) metals Mo and W for sohd-solution strengthening. 

The allowable dimensional variation (the tolerance) of a polymer part can be larger than one made of metal - and specifying moulds with needlessly high tolerance raises costs greatly. This latitude is possible because of the low modulus the resilience of the components allows elastic deflections to accommodate misfitting parts. And the thermal expansion of polymers is almost ten times greater than metals there is no point in specifying dimensions to a tolerance which exceeds the thermal strains. 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

Here, G denotes the shear modulus, and f(c/r) is a function of the ratio c/r in which c and r are the spheroidal semiaxes of the precipitate. For spheres, f(c/r= 1) = 1 = /max. For discs as well as for rods, /< 1. In principle, shear stress energies and energies arising from misfit dislocation networks also have to be added. They influence AG by additional energy terms. 

Analyses of the plastic strains caused by matrix cracks, combined with calculations of the compliance change, provide a constitutive law for the material. The important parameters are the permanent strain, e0 and the unloading modulus, E. These quantities, in turn, depend on several constituent properties the sliding stress, r, the debond energy, T, and the misfit strain, il. The most important results are summarized below. 

Fig. 14.8. The critical thickness for introduction of misfit dislocations in a c-axis-oriented film according to [14.38]. The Burgers vector [100] is 0.389 nm at 700 °C, the biaxial modulus is 248 GPa, the Poisson s ratio is 0.281 and the shear modulus is 70 GPa [14.39].

It can be readily established from (6.49)-(6.52) that the elastic modulus mismatch between the film and the substrate can strongly influence the propensity for formation of misfit dislocations for a given mismatch strain as well as the preferred location for dislocation formation. These effects of modulus mismatch are illustrated quantitatively in the following numerical example. 

Although Conrad et al. have frequently justified their results in terms of conventional lattice-defect theory (including the size-misfit and modulus-defect formalisms), they have gone on to consider the effects of chemical interaction between the solute and solvent atoms. In doing so, the interaction mechanism was deduced, with the aid of atomic-orbital theory, to take the form of covalent bonding between the interstitial atom and the... 

Experiments bear this out. For example, the rheometric measurements of Mason and Weitz [34] of the frequency-dependent viscoelastic moduli of colloidal glasses give values on the order of 0.1-1 Pa for the storage moduli at low frequencies (static shear moduli). The methods discussed in this article also allow determination of the elastic moduli, albeit more indirectly. In Section 6 it is shown how a value of the shear modulus can be determined from the propagation of misfit dislocations. The calibration of such a measurement is based on the viscosity of the suspension fluid. It is also possible to determine the moduli from... 

The coupling between shrinkage (i.e. misfit value in Eq. 3,19) and the modulus of elasticity, as predicted from Eq. 3.20, has been the subject of several experimental studies to determine the frictional bond, reported by Pinchin and Tabor [21,22], Beaumont and Aleszka [23] and Stang [24],... 

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