Uniform strains

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

Other defects should be considered, such as uniform strain, which causes isotropic contraction/expan-sion of the cell giving rise to diffraction line shifts (but not broadening), chemical and/or phase segregation and... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Fundamental equations for piezoelectric effect due to uniform strain S are given in the form ... 

A conscious choice of such elements can be made but in general the equilibrium distribution of stress cannot be found except for particular geometries. The assumptions of uniform strain throughout the assembly or of uniform stress were respectively made by Voigt and by Reuss. Returning to the structures actually perceivable in polymers one may consider the spherulite in a semi crystalline polymer as being unsuitable as a RVE because the boundary is not included. However, an assembly of spherulites would be acceptable, since it would contain sufficient to make it entirely typical of the bulk and because such an assembly would have moduli independent of the surface tractions and displacements. The linear size of such a representative volume element of spherulites would be perhaps several hundred microns. 

An effective Hamiltonian for a static cooperative Jahn-Teller effect acting in the space of intra-site active vibronic modes is derived on a microscopic basis, including the interaction with phonon and uniform strains. The developed approach allows for simple treatment of cooperative Jahn-Teller distortions and orbital ordering in crystals, especially with strong vibronic interaction on sites. It also allows to describe quantitatively the induced distortions of non-Jahn-Teller type. 

In this chapter an effective Hamiltonian for a static cooperative Jahn-Teller effect is proposed. This Hamiltonian acts in the space of local active distortions only and possesses extrema points of the potential energy equivalent to those of the full microscopic Hamiltonian, defined in the space of all phonon and uniform strain coordinates. First we present the derivation of this effective Hamiltonian for a general case and then apply the theory to the investigation of the structure of Jahn-Teller hexagonal perovskites. 

For the sake of clarity we split the derivation in two parts. First the contribution from phonons is considered and then the effect of uniform strains is added. 

The above derivation of the effective Hamiltonian is only complete when, for some reasons, the uniform strains of the crystal are not relevant. This is clearly the case for crystals with low concentration of Jahn-Teller impurities. Contrary to that, bulk deformations often arise in its low-symmetry structural phases of Jahn-Teller crystals [2,11]. The uniform strains describing the bulk deformations of the crystal cannot be reduced to a combination of phonon modes, as it was first pointed out by... 

Bom and Huang [12]. In the following we generalize the approach by taking into account the effects of uniform strains. 

The uniform strains are independent degrees of freedom of the lattice, therefore they give an additional, elastic contribution to the potential energy (1) ... 

It is generally accepted in the theory of the cooperative Jahn-Teller effect to include the interaction with uniform strains in the way proposed by Kanamori [14], i.e., as additional terms of vibronic interaction at each Jahn-Teller ion. On the other hand, within one-centre-coordinate approach used here the vibronic interaction is fully described by means of one-centre active nuclear displacements qn. Therefore the interaction with uniform strains can be included implicitfy as additional terms in the Van Vleck expansion (3). Since phonons and uniform strains are independent degrees of freedom this new expansion is written as follows ... 

Substituting equation (19) into equation (9), the uniform strains contribution to in equation (14) becomes... 

Unlike Van Vleck coefficients, the expression within the brackets in the above equation, i.e., the expansion coefficient corresponding to uniform strains, is independent from n. This results from translational invariance of the coefficients c jr,(An/ta) and the obvious relations ... 

It follows from equation (5) that the matrix elements of are additive with respect to different nuclear modes if the zero-order deformation energy is diagonal after corresponding degrees of freedom. The same is expected when the contribution from uniform strains, equation (11), is taken into account. Using equations (5) and (11) we obtain... 

It is possible to distinguish the contributions from phonons and uniform strains in the effective force constants. After some transformations (see Appendix A) we obtain... 

Due to ferro-ordering of distortions in the ab planes and the lack of uniform strains [16-18], the problem of equilibrium structural phases of hexagonal perovskites is reduced to a single chain [25]. Passing to polar coordinates for... 

In uniformly strained materials, deformation structures can be readily observed using transmission electron microscopy. However, it is much more difficult to prepare a similar sample where the deformation is more localized, as is the case of nanoindentation. Recently this situation has been revolutionized by the development of focused ion beam techniques for semiconductor processing, so that it is possible to select the region to be thinned to within 100 nm (Overwijk et al., 1993 Saka, 1998). 

Although, by the converse effect, a bimorph will bend when a voltage is applied to it, this will not be according to the inverse of Eq. (6.88) because the deformation due to an applied field is governed by d = (dx/dE)x which is not the inverse of h = (dE/dx)D. Furthermore, the applied field will produce uniform strains along the length of the bimorph so that it will be bent in the form of a circular arc, in contrast with the more complex shape given by Eq. (6.79). 

Lumley (L3) has studied the distortion of homogeneous turbulence by uniform strain using a limited MRS closure. In homogeneous flow, the Ra equations become... 

The uniform shear stress through the layer will also cause a uniform strain rate given by... 

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