Uniaxial modulus

By analogy we can define a uniaxial modulus of compression K(q>) and given that the volume fraction is inversely related to the total volume we get... 

Providing different centrifuge speeds produces a sediment with an equilibrium height he and evaluation of the uniaxial modulus of com-... 

Substituting into the expression for the uniaxial modulus and retaining only the most significant terms we get... 

It has been concluded that in fiber reinforced composites the aspect ratio L/D of the fiber is one of the most important parameters in controlling the uniaxial modulus and ultimate tensile strength (Kardos and Raisoni, 1975). 

From this it is clear that the strain-dependent decrease of the uniaxial modulus with increasing tensile strain can be obtained readily by differentiation ... 

The early work on properties of molecular composites was done by Hwang et al. (1983a, b). They performed experimental studies of solution processing of films based on poly(p-phenylene benzobisthiazole) with both poly(2,5 (6 ) benzimidazole (ABPBI) and poly(2,5 (6 ) benzothiazole) (ABPBT). The films were shown to possess very high modulus and strength values, which improve upon heat treatment. The uniaxial modulus of highly oriented molecular composites was shown to follow the linear rule of mixtures. 

The viscosity Hq is related to the uniaxial modulus that appears in models using an elastic constitutive equation [54] if a sample is placed in a cylinder (so that no lateral strain is possible) and compressed with a piston (assumed to be porous so that the liquid can escape), the strain rate produced by the applied stress, a, is = o /Hq, (See ref. 64, p. 86, for an analysis of this problem for an elastic or viscoelastic material.) The pressure distribution given by Eq. 72 is illustrated in Fig. 12 for several values of the parameter a as a increases, a greater portion of the interior of the gel experiences high pressure and correspondingly slow contraction. As indicated by Eq. 73, this situation is favored by large samples with low permeability. 

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

Figure 2-26 Uniaxial Loading at 45° to the 1-Direction Then, by use of the modulus transformation relations in Equation (2.97)...

In a uniaxial tension test to determine the elastic modulus of the composite material, E, the stress and strain states will be assumed to be macroscopically uniform in consonance with the basic presumption that the composite material is macroscopically Isotropic and homogene-ous. However, on a microscopic scSeTBotFTfhe sfre and strain states will be nonuniform. In the uniaxial tension test,... 

For a lower bound on the apparent Young s modulus, E, load the basic uniaxial test specimen with normal stress on the ends. The internal stress field that satisfies this loading and the stress equations of equilibrium is... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

In the uniaxially oriented sheets of PET, it has been concluded that the Young s modulus in the draw direction does not correlate with the amorphous orientation fa or with xa "VP2(0)> 1r as might have been expected on the Prevorsek model37). There is, however, an excellent correlation between the modulus and x,rans,rans as shown in Fig. 15. It has therefore been concluded 29) that the modulus in drawn PET depends primarily on the molecular chains which are in the extended trans conformation, irrespective of whether these chains are in a crystalline or amorphous environment. It appears that in the glassy state such trans sequences could act to reinforce the structure much as fibres in a fibre composite. 

Fig. 15. Modulus versus x,ril smin. for uniaxially oriented PET. Circles, single stage draw triangles two-stage draw open symbols, draw temperature 80 °C full symbols, draw temperature 85 °C. Reproduced from Polymer by permission of the publishers Butterworth Co (Publishers) Ltd. (C)...

Table IV. Shear modulus and Tg of polyurethane networks prepared from bulk reactios (29). G(298K) - shear modulus at 298K, (a) from uniaxial...

It is not necessary to know the bulk modulus to convert E to G. If the transverse strain, , of a specimen is determined during a uniaxial tensile test in addition to the extensional or longitudinal strain e their ratio, called F oisson s ratio, v can be used ... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

For elastomers, factorizability holds out to large strains (57,58). For glassy and crystalline polymers the data confirm what would be expected from stress relaxation—beyond the linear range the creep depends on the stress level. In some cases, factorizability holds over only limited ranges of stress or time scale. One way of describing this nonlinear behavior in uniaxial tensile creep, especially for high modulus/low creep polymers, is by a power... 

The material properties used in the simulations pertain to a new X70/X80 steel with an acicular ferrite microstructure and a uniaxial stress-strain curve described by er, =tr0(l + / )", where ep is the plastic strain, tr0 = 595 MPa is the yield stress, e0=ff0l E the yield strain, and n = 0.059 the work hardening coefficient. The Poisson s ratio is 0.3 and Young s modulus 201.88 OPa. The system s temperature is 0 = 300 K. We assume the hydrogen lattice diffusion coefficient at this temperature to be D = 1.271x10 m2/s. The partial molar volume of hydrogen in solid solution is... 

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