Inclusions elastic strain energy

Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure consists of all bad material. It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely, dilational, the particle was assumed incompressible, and the shape was generalized to that of an... 

Figure 19.7 Elastic strain energy function E c/a) for an incoherent ellipsoid inclusion of aspect ratio c/a.

In an elastically homogeneous system, the elastic strain energy per unit volume of the inclusion Age is independent of inclusion shape and is given by... 

The case of a pure dilational transformation strain in an inhomogeneous elastically isotropic system has been treated by Barnett et al. [10]. For this case, the elastic strain energy does depend on the shape of the inclusion. Results are shown in Fig. 19.9, which shows the ratio of A(inhomo) for the inhomogeneous problem to A<7 (homo) for the homogeneous case, vs. c/a. It is seen that when the inclusion is stiffer than the matrix, AgE (inhomo) is a minimum... 

Thus far, our results are primarily formal. As an exercise in exploiting the formalism, we will first examine the fields associated with spherical inclusions, and then make an assessment of the elastic strain energy tied to such inclusions. These results will leave us in a position to consider at a conceptual level the nucleation, subsequent evolution and equilibrium shapes associated with inclusion microstructures. 

The instability of the two lamellar structures may be understood in terms ofEshelby s inclusion theory [6,7]. According to the theory, a hard coherent precipitate with a dilata-tional misfit strain is elastically stable when it takes on a spherical shape in an infinite matrix. A soft coherent precipitate, on the other hand, takes on a plate-like shape as the minimum strain energy shape. Thus, the soft-hard-soft layered structure of Fig. 7 is simply a... 

Note that we have resorted to the use of /r and v for our description of the elastic properties for ease of comparison with many of the expressions that appear elsewhere. In order to compute the total elastic energy, we separate the volume integral of eqn (10.15) into two parts, one which is an integral over the inclusion, and the other of which is an integral over the matrix material. The reason for effecting this separation is that, external to the spherical inclusion, the displacements we have computed are the elastic displacements. On the other hand, within the inclusion, it is the total fields that have been computed and hence we must resort to several manipulations to deduce the elastic strains themselves. 

Unlike the case of toughening binary blends, where the primary variables controlling the fracture resistance are elastomer volume fraction and the size and distribution of elastomer inclusions (26-28), the toughness of composites with EILs is extremely sensitive to the thickness and elastic moduli of the interphase (6,7,31). A lower limit of strain energy release rate, Gc (or fracture toughness. 

We are also interested in the elastic energy external to the inclusion. To compute this, we demand the strains outside of the inclusion, which can be simply evaluated on the basis of eqn (10.13) with the result that... 

CNT-polymer composites at very low nanotube loadings exhibit substantial electrostrictive strains when exposed to an electric field that is dramatically lower than that required by neat polymers. Zhang et al. have shown that the crystallinity. Young s modulus, dielectric constant, electrostrictive strain, and elastic energy density of electrostrictive poly(vinylidene fluoride-trifluor-oethylene-chlorofluoroethylene) P(VDF-TrFE-CFE)] can be simultaneously improved by inclusion of only 0.5 wt% of MWNTs. At an applied electric field of 54 MV m the 0.5 wt% nanocomposite generates a strain of 2%, which nearly doubles that of pure P(VDF-TrFE-CFE) polymer. 

To date, results have been obtained for minimum-energy type simulations of elastic deformations of a nearest-neighbor face-centered cubic (fee) crystal of argon [20] with different inclusion shapes (cubic, orthorhombic, spherical, and biaxially ellipsoidal). On bisphenol-A-polycarbonate, elastic constant calculations were also performed [20] as finite deformation simulations to plastic unit events (see [21]). The first molecular dynamics results on a nearest-neighbor fee crystal of argon have also become available [42]. The consistency of the method with thermodynamics and statistical mechanics has been tested to a satisfactory extent [20] e.g., the calculations with different inclusion shapes all yield identical results the results are independent of the method employed to calculate the elastic properties of the system and its constituents (constant-strain and constant-stress simulations give practically identical values). 

The Mullins effect, which can be considered as a hysteretic mechanism related to energy dissipated by the material during deformation, corresponds to a decrease in the number of elastically effective network chains. It results from chains that reach their limit of extensibility by strain amplification effects caused by the inclusion of undeformable filler particles [24,25]. Stress-softening in filled rubbers has been associated with the rupture properties and a quantitative relationship between total hysteresis (area between the first extension and the first release curves in the first extension cycle) and the enei-gy required for rupture has been derived [26,27]. 

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