Elastic constants glasses

For the case where cr, = cTj = cTj which is known as hydrostatic stress (the situation when pressure is applied on a glass embedded in a material of low elastic constants - glass piece in steatite or in AgCl in a high pressure cell or simply embedded in a liquid) - then there are no shear strains. The hydrostatic stress is simply the pressure, P and the volumetric strain is Cm so that bulk modulus is also defined as... 

The elastic constants of bulk amorphous Pd-Ni-P and Pd-Cu-P alloys were determined using a resonant i rasound spectroscopy technique. The Pd-Ni-P glasses are slightly stiffer than the Pd-Cu-P glasses. Within each alloy system, the Young s modulus and the bulk modulus show little change with alloy composition. 

Specific results are calculated for SiC fiber-glass matrix composites with the elastic constants given in Table 4.1. A constant embedded fiber length L = 2.0 mm, and constant radii a = 0.2 mm and B = 2.0 mm are considered with varying matrix radius b. The stress distributions along the axial direction shown in Fig. 4.31 are predicted based on micromechanics analysis, which are essentially similar to those obtained by FE analysis for the two extremes of fiber volume fraction, V[, shown in Fig. 4.32. The corresponding FAS distribution calculated based on Eqs. (4.90) and (4.120), and IFSS at the fiber-matrix interface of Eqs. (4.93) and (4.132) are plotted along the axial direction in Fig. 4.32. 

Another class of thermodynamic barrier theories focuses on the large increases in the elastic constants that accompany glass formation. (These theoretical approaches seem especially appropriate to polymer fluids below the crossover temperature Fj.) In particular, the barrier height governing particle displacement in the shoving model [57] is taken to be on the order of the elastic energy GqoVo required to displace a particle on a scale comparable to the interparticle distance,... 

Figure 5.120 Elastic constants for unidirectional (a) glass fiber reinforced epoxy and (b) graphite fiber-reinforced epoxy laminae. Reprinled, by permission, from P. C. Powell and A. J. I. Housz, Engineering with Polymers, pp. 222, 223. Copyrighl 1998 by Stanley Thorned Publishers.

Fig. 12.5. A crack from an indent in glass (a) z = 0 (b) z = — 3.8 pm (c) z = —5.2 pm ELSAM, 1.5 GHz. The experimental line-scans superimposed on the images can be compared with the plots calculated using two-dimensional theory (eqns (12.2), (12.13), and (12.14)) with elastic constants from Table 6.3 and values of defocus (a) z = 0 (b) z = —4.2 pm (c) z = —6.8 pm. The values of z in the calculations were chosen for best fit the reason for the discrepancy is not known, though no doubt there are the usual uncertainties associated with thermal drift, the measurement of z, and the frequency and pupil function used (Briggs etal. 1990).

Entered into (17), the square-root dependence of the plateau value translates into the square-root anomaly of the elastic constant Go, and causes the increase of the yield stress close to the glass transition. 

Above the transition, the quiescent system forms an (idealized) glass [2, 38], whose density correlators arrest at the glass form factors fq from Fig. 10, and which exhibits a flnite elastic constant G , that describes the (zero-frequency) Hookian response of the amorphous solid to a small applied shear strain y cr = Geo/ for y 0 the plateau can be seen in Fig. 3 and for intermediate times in Fig. 12. If steady flow is imposed on the system, however, the glass melts for any arbitrarily small shear rate. Particles are freed from their cages and diffusion perpendicular to the shear plane also becomes possible. Any finite shear rate, however small, sets a finite longest relaxation time, beyond which ergodicity is restored see Figs. 1 l(b,c) and 12. 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

Elastic constants of solids can be related to the fundamental interaction energy terms. In fact it can be shown easily that the bulk modulus, K scales as the energy density, Ulr for an ionic material interacting through Bom-Mayer potential (see later in this section). Thus the molar volume which increases with the presence of larger ions in glasses can be expected to cause a decrease in Young s modulus. 

In this analysis, it is assumed that all the glass fibers are straight however, it is unlikely that this is true, particularly with fabrics. In practice, the load is increased with fibers not necessarily failing at the same time. Values of a number of elastic constants must be known in addition to strength properties of the resins, fibers, and combinations. In this analysis, arbitrary values are used that are low for elastic constants and strength values. Any values can be used here the theory is illxtstrated. 

In the uniaxial tension test (Fig. 2.8), there is usually a transverse strain, i.e., a strain perpendicular to the applied stress. This can be used to define a second elastic constant, Poisson s ratio (v), as the negative ratio of the transverse strain (e.j.) to the longitudinal strain (s ), i.e., v= -Sj/cl- For isotropic materials, it can be shown from thermodynamic arguments, that -1< u <0.5. For many ceramics and glasses, v is usually in the range 0.18-0.30. 

Use the Halpin-Tsai equations to determine the five elastic constants of a unidirectional fiber composite in which alumina fibers are dispersed in a glass matrix. The Young s modulus and Poisson s ratio of polycrystalline AljOj are 400 GPa and 0.23 and for the glass, 70 GPa and 0.20. 

The AFM tips, used in experiments, are conical and 2 pm long, they have a half-cone angle of 10° and the tip-end radius is approximately 10 nm. The cantilever elastic constant is 0.1 N/m. A hot stage is mounted between the AFM s piezoelectric scanner and the glass plate, allowing to control the sample temperature from room temperature up to 100°C with a precision of 0.1°C. 

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