Volume deformations

Since simple shear is a constant volume deformation, the solution does not depend on coefficients of terms involving tr(various values of a are shown in Fig. 5.9. The solution for a grade zero material using Jaumann s stress rate (5.120) corresponds to = Ug = Ug = 0 so that a = -1, while the solution using Truesdell s stress rate (5.122) corresponds to = 0 and Ug = 1 so that a = 0. The shear... 

Relative volume deformations of initial sample at the end of the vacuum step and at the end of the atmosferic step, respectively. 

The volume deformations of concrete are shrinkage, which occurs under drying conditions, and creep, which is the additional deformation obtained under an applied stress. Creep does occur under saturated conditions (basic creep) but increases considerably under conditions of moisture loss. The picture is rather complicated in that creep is made up of a recoverable and irrecoverable portion on removal of the applied stress. 

Volume deformations are largely a function of the nature and quantity of the cement paste in the concrete and it has been shown [113] that studies on... 

It is difficult to prepare a definitive synopsis of the effect that the various classes of admixtures will have on the volume deformations of concrete because of the conflicting results between workers and different types of test methods used, but the points given below should enable some guidelines to be set. 

Volume deformations under drying conditions It is under conditions where moisture is lost from concrete that volume deformations under loaded or unloaded conditions occur to any magnitude. It is difficult to say what degree of relative humidity structural concrete will be subjected to in actual practice, but certainly for thin sections or near the surface of large sections, considerable interchange of water due to changing climatic conditions will occur. 

Although there are some anomalies in the literature, it is generally agreed that both types of volume deformation are a function of the same fundamental mechanism and that the influence of other factors such as admixtures will affect both shrinkage and creep in a similar manner. As outlined earlier, water-reducing admixtures can be used to obtain different effects on the plastic/hardened concrete and it is this factor, together with the admixture type, that is important in determining the effect on the volume deformations of concrete. 

It can be seen, therefore, that on typical paste shrinkage in the region of 4000 microstrain, that some 30-40% of this could be accounted for by the sulfate component. An important point to note is that the sulfate reaction has a negative influence on shrinkage and, therefore, acts as a restraint to creep and shrinkage as the reaction proceeds. It was noted in an earlier section that the addition of a water-reducing admixture delays ettringite reactions and could be a possible mechanism by which the volume deformations are increased. 

Water-reducing admixtures containing calcium chloride should not be used in concrete containing embedded metal or where volume deformations are important. 

Ligno sulfonate admixtures containing triethanolamine should not be used in situations sensitive to increased volume deformations. 

The minus sign in Eq. (5.9) is to account for the fact that Ad as defined above is usually negative. Thus, Poisson s ratio is normally a positive quantity, though there is nothing that prevents it from having a negative value. Eor constant volume deformations (such as in polymeric elastomers), v = 0.5, but for most metals, Poisson s ratio varies between 0.25 and 0.35. Values of Poisson s ratio for selected materials are presented in Appendix 7. 

For most solids, one can neglect the difference between Pp f (ap f/3 for an isotropic body) and the coefficient of thermal expansion at constant P is usually used. Therefore, we may use P and a without subscripts. Assuming that E and p are independent of temperature and ignoring the change in lateral dimensions during defonnation (i.e. we take the Poisson s ratio p = 0, because this simplification gives effects of only the second order of smallness), one can arrive at relations similar to Eqs. (17)—(21). To do this, it is necessary to replace in Eq. (16) the volume deformation e by e, the modulus K by E and a by p (see Fig. 1). For the simple deformation of a Hookean body the characteristic parameter r is also inversely dependent on strain, viz. r = 2PT/e and sinv = —2PT. It is interesting to note that... 

If one proceeds in this way, it does not cause problems in treating constant-volume deformations (i.e. dry network deformations), but it meets with obstacles in swelling type strains because the logarithmic term HvlnXxXyXz in Eq. (III-9) is absent in the simplified treatment of Eq. (IV-5). A full derivation according to the HFW approach but employing the series distribution Eq. (IV-4), is, however, not available. 

We can, of course, account for a possible front factor in the usual way by converting A into Ax, and v into Av(z8>/0 (compare Eq. III-ll). It might be pointed out that the logarithmic term in Eq. (IV-32) bears no relation to the logarithmic term which appears in the HFW treatment, but is absent in the JG treatment. In fact, Eq. (IV-32) is obtained on the basis of a simplified JG treatment, so that its validity is restricted to constant volume deformations (compare also Eq. III-8). In unidirectional extension or compression Eq. (IV-32) yields for the stress per strained cross-section ... 

Air-entraining admixtures, therefore, produce concrete which is more durable to conditions of freezing and thawing, particularly in the presence of de-icing salts, more resistance to sulphate attack, provides better protection to embedded reinforcement and is more tolerant of poor curing conditions. There appears to be no great difference in the way air-entrained concrete behaves in terms of compressive strength development and volume deformations. 

Because of the large ratio K/G, most studies of rubber elastic behavior are restricted to constant volume deformations, i.e., to Dey, and to the resulting deviatoric stress Dty. Although this restriction is widely made, it is often tacit. Because of its importance, we make it explicit here because it has important consequences for the current discussion. 

Figure 12. Intrinsic atomic stresses <7n, Vjj. <733 as determined from molecular dynamics simulation of tetrafunctional network model in uniaxial volume deformation and for corresponding melt. (After Ref. [19].)...

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