Dilation uniaxial

Some efforts have been undertaken to study propellant dilatation in multiaxial stress fields and even in small motor configurations. Farris (25) has conducted limited investigations along this line and has made approximate correlations between uniaxial and multiaxial tests. 

Generally applicable mathematical representations of the dilational behavior of propellants have not been developed, as might be expected however, Fishman and Rinde (29) have derived empirical expressions for the formulations which they studied. These relationships give reasonable description of uniaxial behavior over wide ranges of strain, time, and temperature for several testing modes. Equation 1 is representative of one of the generalized expressions for the polyurethane and polybutadiene formulations studied. 

A general description of the fundamental relationships governing the dynamic response of linear viscoelastic materials may be found in several sources (28, 37, 93). In general, sinusoidally applied strains (stresses) result in sinusoidal stresses (strains) that are out of phase. Measurements may be made under uniaxial, shear, or dilational loading conditions, and the resultant complex moduli or compliance and loss-phase angle are computed. Rotating radius vectors are usually taken to represent the... 

For uniaxial (hexagonal) symmetry the 6 strain components are subdivided in two (invariant) one-dimensional subsets (indicated by the superscript a, and subscripts 1 and 2 for the volume dilatation and the axial deformation, respectively), and two different two-dimensional subsets, indicated by y for deformations in the (hexagonal) plane, and by e for skew deformations. These modes are also depicted in fig. 3. In this case, the magnetostriction can be expressed as... 

A new ultrasonic technique for studying dewetting and cumulative internal damage in solid propints has been reported (Refs 17 20). This technique yields volume-dilatation data on proplnt in tension, and on damage in uniaxial compression and shear strain fields. Estimates of vacuole size and number density arising from dewetting can be made, as well as can the time dependent void growth at constant strain be observed... 

We now discuss the basic features of the RG flow. This amounts to giving a physical interpretation of our general discussion of the dilatation group, and it explains qualitatively the characteristic features like universality, power laws, and scaling observed in a critical system. For a given class of systems (fluids, uniaxial ferromagnets, polymer solutions, etc.) we envisage the space of all Hamiltonians to be parameterized by a microscopic scale and dimensionless... 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

The shear component of the applied stress appears to be the major factor in causing yielding. The uniaxial tensile stress in a conventional stress-strain experiment can be resolved into a shear stress and a dilational (negative compressive) stress normal to the parallel sides of test specimens ofthe type shown in Fig. 11-20. Yielding occurs when the shear strain energy reaches a critical value that depends on the material, according to the von Mises yield criterion, which applies fairly well to polymers. 

Negative pressure specifically. With subscripts c, e, i, m, P, ST, TH, oo craze traction, Mises equivalent, one of three principal stresses, maximum level of craze traction where cavitation in PB begins, negative pressure in particle, negative pressure due to one of three principal stresses, negative pressure due to thermal mismatch, uniaxial applied stress at the borders With subscripts xx, yy, zz etc. for components of the local stress tensor Ratio of slope of the falling to the rising part of the traction cavitation law Craze dilatation Time constant... 

As mentioned above, the shear response of some viscoelastic materials can differ from the dilatational response. Thus, while the response in shear is viscoelastic, the dilatational response is elastic. It is clear that in these conditions, the analysis of the multiaxial problems differs from that of the uniaxial and simple shear cases. 

By studying the sample dilatation versus strain in uniaxial tension creep tests. Buck-nail is able to determine the operative niechanism in each system. Fracture mechanics is used to evaluate the toughness parameters of the various systems. 

In a uniaxial tensile test the volume dilatation simply be-... 

The above morphological characterization helps us to understand the influence of composition on the dilatation of the materials under uniaxial tension. 

In a majority of cases, a body under stress experiences neither pure shear nor pure dilatation. Generally, a mixture of both occurs. Such a situation is exemplified by uniaxial loading which, of course, may be tensile or compressive. Here a test specimen is loaded axially resulting in a change in length, AL. The axial strain, e, is related to the applied stress in an elastic deformation by Hooke s law ... 

It is useful to note that eqs. (4.19) and (4.20) for uniaxial tension (or compression), in addition to being applicable to uniaxial strain deformation, as stated above, are also applicable to the case of dilatation responding to negative pressure where the basic symmetry of the deformation is maintained. In that case, however, ffii is replaced with tensile stress (negative pressure), n is replaced with e, the dilatation, and Eq, Young s modulus, is replaced with the bulk modulus K(). Moreover, a must be replaced by fi, which represents the reciprocal of the critical athermal cavitation dilatation. 

Ideally, crazing is a form of plastic deformation of a linear-chain glassy polymer whereby under a tensile stress a slender polymer layer undergoes a uniaxial planar dilatational transformation ej producing a uniaxial strain c in the direction of the tensile stress, in proportion to the volume fraction c of transformed polymer, i.e.,... 

This response, which is unique to glassy polymers, is possible, in spite of the very substantial magnitudes of ej of the order of 3 4, because of the entangled nature of polymer molecules in the initial precursor state where the uniaxially dilated polymer matter, with substantially reduced density, remains fully load-bearing, in the form of stretched elastomer nano-fibrils. 

On the contrary, in the case of laboratory investigations on rock specimens under uniaxial or triaxial load, volume changes in the source play an important role. Dilatancy can be explained as volume expansion caused by tensile opening. In contrast to the fault-plane solution method, the more complex moment tensor method is capable of describing sources with volumetric components like tensile cracks, deviatoric sources like shear cracks, or a mixture of both source types. The volumetric source components can be easily obtained using the isotropic part, or one-third of the moment tensor trace. With the moment tensor method, the source mechanisms are estimated in a least-squares inversion calculation from amplitudes of the first motion as well as from full waveforms of P and S waves. This method requires additional knowledge about the transfer function of the medium (the so-called Green s function) and sensor response. 

Both methods have been applied in the field of AE in rocks. Some examples of the application of the fault-plane solution method are given by Sondergeld and Estey [1982], Kuwahara et al. [1985], Zang et al. [1998]. Because of an insufficient number of sensors these authors investigated only polarities of P-wave first motion to classify AE events recorded during uniaxial loading of rock specimens into shear and tensile types. A source with a imiform compressional polarity pattern was interpreted as a tensile t) e and one with eompressional as well as dilatational first motions as a shear type. It was not possible to determine the source orientation or to quantify the volume ehange of the mieroeracks. 

Uniaxial Extension. A rubber strip of original length Lo is stretched uni-axially to a length L, as illustrated in Figure 1. The stretch and elongation are AL = L — Lq and k = LILo, respectively. The strain e (also known as the relative deformation, linear dilation, or extension) and the elongation or extension ratio X are related by... 

Fig. 5 Volume dilatation for a rubber specimen undergoing a uniaxial tensile experiment [62], The volume change remainslimited over a wide strain range...

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