Plane-strain bulk modulus

The explicit formulae given by Rosen55 are also of value. They are derived from a model consisting of a random assemblage of composite cylinders (Hashin and Rosen56 ) and expressed in terms of the axial Young modulus E, the Poisson ratio for uniaxial stress in the fibre direction v, the transverse plane strain bulk modulus k, the axial shear modulus G and the transverse shear modulus G. ... 

For the other elastic properties, the self-consistent model is only able to give accurate predictions for the plane-strain bulk modulus. 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

The first case considered is solute desorption during unconfined compression. We consider a two dimensional plane strain problem, see Fig. 1. A sinusoidal strain between 0 and 15 % is applied at 0.001 Hz, 0.01 Hz, 0.1 Hz and 1 Hz. To account for microscopic solute spreading due to fluid flow a dispersion parameter is introduced. Against the background of the release of newly synthesized matrix molecules the diffusion parameter is set to the value for chondroitin sulfate in dilute solution Dcs = 4 x 10 7 cm2 s-1 [4] The dispersion parameter Dd is varied in the range from 0 mm to 1 x 10 1 mm. The fluid volume fraction is set to v = 0.9, the bulk modulus k = 8.1 kPa, the shear modulus G = 8.9 kPa and the permeability K = lx 10-13m4 N-1 s-1 [14], The initial concentration is normalized to 1 and the evolution of the concentration is followed for a total time period of 4000 s. for the displacement and linear discontinuous. For displacement and fluid velocity a 9 noded quadrilateral is used, the pressure is taken linear discontinuous. 

H6 and BCT4 both have a bulk modulus which, within the tight-binding model is comparable in magnitude to that of diamond. In accord with an empirical TB scheme [24] H6 seems to be even harder than diamond while SCF plane wave calculations [25,26] predict that the almost equal bulk moduli of BCT4 and H6 are 17% below the diamond value. The bulk moduli of the models have been determined by calculating the elastic compliances after applying suitable strains to the crystals and inversion of the volume compressibility [68]. 

Fig. 11.18 Map of cavitated zone in plane-strain region, showing dependence of zone boundary on the particle diameter when Kj = 1 MPa Critical mean stresses calculated with bulk modulus K = 3GPa and data of Dompas et al. (Dompas and Groeninckx 1994) (From Bucknall and Paul (2009) reproduced with permission of Elsevier)...

Fibrous fillers are often embedded in a laminar form. The fibres used have higher moduli than the resins in which they are embedded so that when the composite of resin plus fibre is strained in the plane of the fibrous layer the bulk of the stress is taken up by the fibre. This results in an enhancement of both strength and modulus when compared with the unfilled resin. 

Strained set of lattice parameters and calculating the stress from the peak shifts, taking into account the angle of the detected sets of planes relative to the surface (see discussion above). If the assumed unstrained lattice parameters are incorrect not all peaks will give the same values. It should be borne in mind that, because of stoichiometry or impurity effects, modified surface films often have unstrained lattice parameters that are different from the same materials in the bulk form. In addition, thin film mechanical properties (Young s modulus and Poisson ratio) can differ from those of bulk materials. Where pronounced texture and stress are present simultaneously analysis can be particularly difficult. 

If the film is acoustically thin 1), then displacements are constant across the film thickness, and only gradients in the plane of the film arise. The SAW-induced film deformation can be decomposed into three translations (in the x-, y-, and z-directions) and three strain modes, as shown in Figure 3.29 (page 94). An important parameter in determining the contribution of each strain mode in perturbing SAW propagation is the modulus — the ratio of stress to strain associated with each strain mode in Figure 3.29. These moduli are listed in Table 3.3 in terms of the intrinsic elastic properties of the film, represented in terms of the Lamd constants (A, p) and the bulk and shear moduli (AT, G). These sets of moduli are interrelated [51]. 

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