Volumetric deformation

Next, let us determine the multiplicity Av of volumetric deformation via expression... 

Here Av is a multiplicity of the volumetric deformation of Flory ball. 

Let us assume that the chains in w—ball aren t intertwined, every among them occupies the isolated volume equal to Rmd/m. Then the multiplicity of the volumetric deformation of Flory ball into m-ball will be equal to... 

With taking into account the connection (47) the force can be determined via the multiplicities of linear and volumetric deformation of m—ball... 

This work of the break is created by the work of the w—ball deformation at some critical value of the multiplicity of volumetric deformation Av. That is why by equating a work of... 

Needed for the estimation of (/ value of critical multiplicity of volumetric deformation... 

Figure 2. Volumetric deformation of the bone matrix under maximum load. Values are in microstrain.

While most of the translational constraints to local motion disappear at Tg, a volumetric deformation under pressure does not invoke translational constraints. The reduction of B at the glass transition is therefore much less drastic than the reductions of E and G. Consequently, while all three moduli are of the same order of magnitude below Tg, E and G are of the same order of magnitude but B is much larger than both E and G above Tg. 

A flexible boundary cubical triaxial test is another commonly used test for compression studies (Kamath et al., 1993 Li and Puri, 1996). A picture of a triaxial compression tester is shown in Figure 8. It allows not only the application of the three principal stresses independently, but also constant monitoring of the volumetric deformation and deformations in three principal directions. In a triaxial compression test, the specimen is at an initial isotropic state of stress then the three pressure lines apply the same pressure at the same rate to all six faces thus pressure is the same in all three directions (i.e., cti = 02 = 03). 

The effect of volumetric deformation is that permeability decreases slowly as for instance in point 74.2. Clearly, the hydrological effect was not enough to explain the permeability decrease for point 74.2. 

If intrinsic permeability decreases strongly (maximum of 4 orders of magnitude near the drift) due to volumetric deformation induced by compression, the dried zone becomes elongated horizontally but it is flattened in the vertical direction. The volume of the dried zone is smaller than in the case of constant permeability so the drying due to vapor flow is less efficient. This can be due to the lower gas phase permeability. 

The terms and are the volumetric deformations induced by net mean stress and suction changes at both macro and microstructural levels respectively. 

Test results and model predictions are presented in Figure 6. Measured (and modelled) volumetric deformation is represented as a function of the microstructural suction since the macrostructure can be considered in an almost saturated condition from the beginning of the test. The proposed model seems to provide an adequate framework to explain the observed behaviour. 

In this particular case, a special attention was devoted to the coupling between the thermal and the hydraulic problems. In fact, a simple linear elastic model was assumed for the mechanical behaviour, although a coupling between intrinsic permeability and void ratio (and therefore volumetric deformation) has been considered. 

Figure 1. Chemo-elastic volumetric deformation E, due to an increase in concentration c.

Figure 4. Experimental data and model prediction for the volumetric response of Sarnia clay permeated with ethanol under 160 kPa a) effective vertical stress vs concentration, b) and c) volumetric deformation vs concentration.

An analysis of Eq. (41) shows, that under any deformation of the spherical conformational volume of polymeric star, to which the values of = j =. d and = correspond, into the conformational ellipsoid, the multiplicity of the volumetric deformation is decreased (a, ibat means the compaction of the conformational volume of polymeric star. 

Deformation of the m-ball at its transition from the spherical (imstrained) conformational state into the ellipsoid with the semi-axes X. let s express via the multiplicities of the linear A, and volumetric deformation via ratios ... 

Parameters A,- are the multiplication factors of a linear deformation of the Flory ball along the corresponding axis s of multiplication factor of the volumetric deformation. For a polymeric chain into the ideal solution the all = 1 and X = 1. Under any deformations of the F/ory ball its conformational volume is decreased, that is why in the real solution X, < 1. 

Here, as before, the parameter is the multiplicity of the volumetric deformation of the /M-ball into the ideal solution and melt = 1, into the real concentrated solutions < 1. 

The multiplicities of the linear and volumetric deformation are not undefined and are connected via the ratio [24 ... 

Here = n ig the multiplicity of the volumetric deformation of conformational volume in real adsorbed layer connected with the multiplicities of the linear deformations via the ratio similar to Eq. (9). 

At the transfer of the polymeric chain from the ideal into the real solution its conformational volume is deformed with the transformation of the spherical Flory ball into the conformational ellipsoid elongated or flattened along the axis connecting the begin and the end of a chain 26, that leads to decrease of the conformational volume and accordingly to the Eq. (8) to decrease of X. at any deformations of the Flory ball X became less than the one. That this why the effects related with the notions hardness of the polymeric chain and the thermodynamic quality of the solvent can be quantitatively estimated via the multiplicity of the volumetric deformation X <. The indicated effects are visualized in the adsorption layer weaker than in the solution firstly, because the conformational volume in the adsorption layer equal to //2, is less, than in solution. This increases the elastic properties of the conformational volume of polymeric chain and thereafter increases the deformation woik. Moreover, the concentrated adsorption layer corresponding to the quasi-plateau on the adsorption isotherm is more near to the ideal than the diluted real solution. That is why under other equal conditions X > X. This means, that the adsorption of polymer from the real solution is more than ftom the ideal one. 

If volumetric deformation of the soil particles due to stationary water and air pressures is also taken into account, equation 6 must be modified with Biot s constant (Biot 1957) as follows. 

If the material is isotropic, the response can be given separately for the deviatoric and volumetric deformations as follows ... 

By substituting (5.54) into (5.51), we finally obtain a seepage equation that includes the volumetric deformation of the solid phase tr Z) as follows ... 

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58)

total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

Assumption 3 Both the clay crystal (i.e., solid part) and the water are incompressible. The permeability can change due to the volumetric deformation Agy during a consolidation procedure, since an amount of external water and/or the interlayer distance between clay minerals changes. This assumption is represented as... 

The parameter a is considered by some to be the true hardness or the hardness due to volumetric deformation processes in the absence of surface effects. 

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