Compressibility data

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

It should also be noted that in this case the material was loaded in compre-sion whereas the tensile creep curves were used. The vast majority of creep data which is available is for tensile loading mainly because this is the simplest and most convenient test method. However, it should not be forgotten that the material will behave differently under other modes of deformation. In compression the material deforms less than in tension although the efrect is small for strains up to 0.5%. If no compression data is available then the use of tensile data is permissible because the lower modulus in the latter case will provide a conservative design. 

For the two explosive loading systems used, the initial pressure wave into the powder is relatively low, varying from perhaps 1.5-4 GPa. In such cases the most relevant compression characteristic of the powder compact is its crush strength , i.e., the pressure required to compress the porous compact to solid density. In the simulations, this strength can be varied over a wide range with the P-a model. The wavespeed of the initial waves was modeled on the basis of shock-compression data on rutile at densities from 44% to 61% of solid density [74T02]. 

Most comparisons of a failure criterion with failure data will be for the glass-epoxy data shown in Figure 2-36 as a function of off-axis angle 0 for both tension and compression loading [2-21]. The tension data are denoted by solid circles, and the compression data by solid squares. The tension data were obtained by use of dog-bone-shaped specimens, whereas the compression data were obtained by use of specimens with uniform rectangular cross sections. The shear strength for this glass-epoxy is 8 ksi (55 MPa) instead of the 6 ksi (41 MPa) in Table 2-3. 

The compressive data are of limited design value. They can be used for comparative material evaluation and design purposes if the conditions of the test approximate those of the application. The data are of definite value for materials that fail in the compressive test by a shattering fracture. On the other hand, for those that do not fail in this manner, the compressive information is arbitrary and is determined by selecting a point of compressive deformation at which it is considered that a complete failure of the material has taken place. About 10% of deformation are viewed in most cases as maximum. 

Zubarev Telegin (Ref 40a) developed an equation of state for detonation products based on the dynamic compressibility data for these substances and the requirement that product... 

The integration of the left side of equation (7.118) requires that we know how Vf changes with p. Compressibility data would provide this information. However, p is not a large pressure and liquid solvents are not usually very compressible, so that it is a reasonable approximation to assume Vf is independent of pressure, in which case, integration of equation (7.118)... 

The observed deviations from Gaussian stress-strain behaviour in compression were in the same sense as those predicted by the Mooney-Rivlin equation, with modulus increasing as deformation ratio(A) decreases. The Mooney-Rivlin equation is usually applied to tensile data but can also be applied compression data(33). According to the Mooney-Rivlin equation... 

What this comparison indicates is that as a first approximation, the terms A and B in equation (45) are not strongly dependent upon temperature and concentration. The pressure dependence of K for electrolyte solutions can be thus estimated from the properties of pure water. Since K = 1/B , the reciprocal of the 1 atm compressibility, it thus becomes possible to make reasonable estimates of vp from 1 atm specific volume data (v°) and compressibility data (B ). 

Combining thermal expansion and compressibility data for compounds in the (Mg,Fe)2Si04 mixture, Hazen (1977) proposed the following equation of state for olivines of the series Mg2Si04-Fe2Si04 ... 

Table 5.48 summarizes thermal expansion and compressibility data for amphibole end-members according to the databases of Holland and Powell (1990) and Saxena et al. (1993). Isobaric thermal expansion (a, K ) and isothermal compressibility (jS, bar ) may be retrieved from the listed coefhcients by applying the polynomial expansions... 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

W.S. Cramer, "Bulk Compressibility Data on Several Explosives , NavOrdRept... 

In the in situ consolidation model of Liu [26], the Lee-Springer intimate contact model was modified to account for the effects of shear rate-dependent viscosity of the non-Newtonian matrix resin and included a contact model to estimate the size of the contact area between the roller and the composite. The authors also considered lateral expansion of the composite tow, which can lead to gaps and/or laps between adjacent tows. For constant temperature and loading conditions, their analysis can be integrated exactly to give the expression developed by Wang and Gutowski [27]. In fact, the expression for lateral expansion was used to fit tow compression data to determine the temperature dependent non-Newtonian viscosity and the power law exponent of the fiber-matrix mixture. 

Calculated from d = (4vspMj,/itNA)i 2 with the Avogadro constant NA b Calculated from the fitting to osmotic pressure and osmotic compressibility data (cf. Sect. 3) e Calculated from intrinsic viscosity or sedimentation coefficient data d Obtained from crystallographic data... 

Figure 4 compares osmotic compressibility data for isotropic schizophyllan-water solutions [63] with the scaled particle theory. The ratios of the z-average to the weight-average molecular weights of these schizophyllan samples are ca. 1.2. The solid curves, calculated with d taken to be 1.52 nm and other molecular parameters (Lc, v, and c ) estimated from Mw and the wormlike chain parameters in Table 1, are seen to come close to the data points for all samples. 

The values of the Born exponent, n, for various crystal structures are estimated from compressibility data. The values recommended for use with various ion configurations are shown in Table 7.6. 

Even if strong A -way methods are used to reduce image noise, compress data, and improve data cube visualization, weak multiway methods are more often used as they facilitate classification using classical single-point spectra. 

Assumes that the driving force for reactions between metals is a function of two factors a negative one, arising from the difference in chemical potential, A y of electrons associated with each metal atom, and a positive one that is the difference in the electron density, Anws, at the boundaries of Wigner-Seitz tvpe cells surrounding each atom. Values of for the metals are approximated by the electronic work functions n ws is estimated from compressibility data. The atomic concentrations in the alloy must be included in the calculation. ... 

Fig. 14. A plot of swelling pressure and unilateral compression data according to Eq. (III-34) for a poly-p-nitrophenylmethacrylate network in nitrobenzene. The slope yields % and the intercept Bv. Assuming v to be known from chemical analysis, B is found to be 0.5 0.1 [Van de Kraats, Winkeler, Potters and...

Kraats, E. J. van de, M. A. M. Winkeler, J. M. Potters, and W. Prins Polymer network characterization by means of swelling pressure and unilateral compression data. Rec. Trav. Chim. Pays-Bas (in press). 

---