Strains volumetric strain

Fig. 1.47 Stress-strain plots obtained from the uniaxial/triaxial compression tests of SiC-N specimens (tr —axial stress, —axial strain, Si—lateral strain, —volumetric strain), a without confining pressure b confining pressure of 350 MPa [33], With kind permission of Professor Brannon...

We first consider strain localization as discussed in Section 6.1. The material deformation action is assumed to be confined to planes that are thin in comparison to their spacing d. Let the thickness of the deformation region be given by h then the amount of local plastic shear strain in the deformation is approximately Ji djh)y, where y is the macroscale plastic shear strain in the shock process. In a planar shock wave in materials of low strength y e, where e = 1 — Po/P is the volumetric strain. On the micromechanical scale y, is accommodated by the motion of dislocations, or y, bN v(z). The average separation of mobile dislocations is simply L = Every time a disloca-... 

This is the approximation used in (7.53). Assume that the volumetric strain e is a function of t only i.e., that its spatial variation is negligible over the dimensions of the shear band. 

In (8.35) Y is the flow stress in simple tension (and may itself be a function of the temperature and strain rate) and is the critical volumetric strain at void coalescence (calculated within the model to equal 0.15 independent of material). Note that the ductile fragmentation energy depends directly on the fragment size s. With (8.35), (8.30) through (8.32) become, for ideal ductile spall fragmentation,... 

Bulk Modulus of oil, [3 = Volumetric stress/Volumetric strain... 

In a similar manner, if an isotropic body is subjected to hydrostatic pressure, p, i.e., ax = ay = 02 = -p, then the volumetric strain, the sum of the three normal or ewensional strains (the first-order approximation to the volume change), is... 

Compressive stress versus gas volumetric strain curve, for EPP foam of density 43 kg m (254). The loading... 

High-intensity vibration effect upon dissolved and molten polymers is relatively simple and can be obtained with the help of superimposition of high-frequency ultrasonic (US) vibrations. Periodic (cyclic) shear and volumetric strain of the medium is attained in this case without moving elements and due to elastic oscillations of US-vibrators of various design which function simultaneously as moulding elements (nozzles, cores) of extrusion heads. 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

Once p and 7r have been determined for a hydraulic and/or a chemical loading, then the volumetric strain can be computed according to t](p—atr)/G. 

A hydrostatic stress can be superposed, but it is caused only by elastic volumetric strain of the composite. The result in Eqn. (39) is, perhaps, not very useful since it is rare that a steady strain rate will be kinematically imposed. When both fiber and matrix creep, the steady solutions for a fixed stress in isothermal states are quite complex but can be computed by numerical inversion of Eqn. (39). The solution can, however, be given for the isothermal case where the fibers do not creep. (For non-fiber composites, this should be... 

The latter result indicates that the volumetric strains can be relaxed to some extent by matrix creep. This contrasts with the 3-D case where complete compatibility of strains precludes such relaxation. The extent to which the relaxation occurs has not yet been calculated. However, if it is assumed that the relaxation can be complete so that the matrix volumetric strain is zero, then the fiber stress tends towards aal3lf and, therefore, the composite strain approaches... 

Bulk modulus B V(AP/AV) The change in pressure divided by the volumetric strain. 

Figure 7 shows the trend in s, volumetric strain (Equation (5)), over 2.40xl0 yr. The data show the following physical changes (i) large volumetric expansion (s > 0) occurred in the... 

Figure 7 (a) Volumetric strain (szr.w) plotted against depth for soils on a marine terrace chronosequence on the... 

When Tj = I, no mobilization of j occurs, if Tj = 0, j has been completely mobilized. The concurrent volume change due to compaction or dilation by weathering and bioturbation is defined by the volumetric strain s such that... 

The corresponding volumetric strains are compared in Fig. 3, which records horizontal line profiles through the fault zone for the two numerical experiments. The difference is striking in the homogeneous cohesionless sandstone a minor contraction is demonstrated however, with clay in the fault, a strong dilatation results. This implies that incorporation of a clay... 

Fig, 3. Horizontal line profile of the volumetric strains across a fault in cohesionless homogeneous sand and across a clay smear in sand of identical properties. 

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... 

Fig. 5.14. Energy as a function of volumetric strain as computed using atomic-scale analysis in terms of embedded-atom potentials (courtesy of D. Pawaskar). The atomistic result is compared with the quadratic fit like that suggested in eqn (5.93).

The principle of these testers is that the specimen can be subjected to controlled stresses in two orthogonal directions (biaxial testers) or three orthogonal directions (triaxial testers). In the case of the triaxial testers, two of the orthogonal stresses are usually equal, normally generated by liquid pressure in a pressure chamber. The specimen is placed in a cylindrical rubber membrane and enclosed by rigid end cups. The specimen is consolidated isotropically, i.e. by the same pressure in all three directions which leads to volumetric strain but little or no shear strain. This is followed by anisotropic stress conditions, whereby a greater axial stress is imparted on the specimen by mechanical force through the end cups. In the evaluation of results it is assumed that the principal stresses act on horizontal and vertical planes, and Mohr circles can be easily drawn for the failure conditions. 

It is currently not well established which failure model is most appropriate for predicting failure of fluoropolymers that are monotonically loaded to failure. Commonly used approaches include the maximum principal stress, the maximum principal strain, the Mises stress, the Tresca stress, the Coulomb stress, the volumetric strain, the hydrostatic stress, and the chain stretch. In the chain stretch model, the failure is taken to occur when the molecular chain stretch, calculated fromPl... 

A direct comparison between these and other failure models has not been performed for fluoropolymers, but a recent study of UHMWPEl showed that, for UHMWPE, these models are very different. For example, it was shown that the chain stretch model is the most promising for predicting multiaxial deformation states, and that the hydrostatic stress, and the volumetric strain are not good predictors of failure. 

Pore water pressure changes in the vicinity of the tunnel excavation are a direct consequence of changes in the volumetric strain of the rock. Later, pore water pressure dissipations are a consequence of the transition flow towards a new equilibrium, which now has a modified boundary condition (the tunnel surface) in the vicinity. Therefore, fully coupled hydro-mechanical analyses are required to try to capture actual measurement. In fact, one-way... 

Figure 2 shows the calculated distribution of average volumetric strain rate for the assumed stress field. The location of six points (0= 30° to 0 = 69°) where fluid pressure changes were evaluated and interval P4 of borehole FEX 95.002 are also shown. Simulation results, as well as analytical solutions, reveal that a pore-pressure increase will only occur at the four points located in the contracted zone of compressive strain rate (0 > 45° in Figure 2). However, the P4 interval (0 = 14° in Figure 2) is located in the zone of extensional strain rate, and therefore no pore pressure increase can occur in at that location for the assumed stress field.

The intrinsic permeability k = k(e, volumetric strain or in the case of fractured rock it can be formulated as a function of the effective normal stress using e.g. the following definition of the void aperture... 

Figure 5. Identification of the yield points from different stress-strain criterion a) volumetric strain versus mean effective pressure and b) dissipated stress energy versus ratio q/p. ...

Where eJ is the elastic volumetric strain and efj components of the elastic deviatoric strain. Kq and Po are respectively the initial bulk modulus and shear modulus of undamaged material. The scalar variable d characterises the isotropic damage. The... 

The constitutive equations are expressed in terms of the rates of the following variables the effective isotropic stress p = J i the deviator stress q = V(3/2sijSij) the volumetric strain = s,i and the deviator strain = V(2/3e,je,), where Sjj and e, refer to the stress and strain deviator, respectively. Compression is taken positive. The material is assumed to be isotropic and its deformation can be decomposed into elastic and plastic parts. The chemical effects on the material behaviour are described in terms of the contaminant mass concentration c, the ratio of contaminant mass to total fluid mass. 

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