Elastic range

The thermoelastic law, valid only within the elastic range of isotropic and homogeneus materials, relates the peak to peak temperature changes to the peak to peak amplitude of the periodic change in the sum of principal stresses. 

The SPATE technique is based on measurement of the thermoelastic effect. Within the elastic range, a body subjected to tensile or compressive stresses experiences a reversible conversion between mechanical and thermal energy. Provided adiabatic conditions are maintained, the relationship between the reversible temperature change and the corresponding change in the sum of the principal stresses is linear and indipendent of the load frequency. 

In a correcdy designed and assembled compound cylinder the stresses are all within the elastic range however, if the interference is too large the... 

As a pipeline is heated, strains of such a magnitude are iaduced iato it as to accommodate the thermal expansion of the pipe caused by temperature. In the elastic range, these strains are proportional to the stresses. Above the yield stress, the internal strains stiU absorb the thermal expansions, but the stress, g computed from strain 2 by elastic theory, is a fictitious stress. The actual stress is and it depends on the shape of the stress-strain curve. Failure, however, does not occur until is reached which corresponds to a fictitious stress of many times the yield stress. 

Different instmmentation is needed for the deterrnination of meaningful modulus values over wide viscosity and elasticity ranges. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Within the elastic range, loading applied along nonspecific crystallographic directions results in propagation of both longitudinal and shear waves which may be of considerable amplitude [80C01],... 

To describe properties of solids in the nonlinear elastic strain state, a set of higher-order constitutive relations must be employed. In continuum elasticity theory, the notation typically employed differs from typical high pressure science notations. In the present section it is more appropriate to use conventional elasticity notation as far as possible. Accordingly, the following notation is employed for studies within the elastic range t = stress, t] = finite strain, with both taken positive in tension. 

Conner has recently extended the longitudinal stress loading investigations of vitreous silica to shear loading, and shown that within the accepted elastic range the materials deformation properties are strongly influenced by shear [88C02]. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

In this chapter physical properties of solids at finite strain within their purely elastic ranges will be investigated. Although the strain levels of a few percent are small relative to the total compressions of typical shock-compression studies, they are large compared to those typically encountered in higher-order elastic property investigations. 

Experimental studies within the elastic range have been performed on monocrystalline AI2O3 (sapphire) and the nonpiezoelectric z-cut of quartz. Experiments are performed with a circuit devised by Ingram [68G05] in which a low-loss coaxial cable is used for both application of the potential and monitoring the current. As shown in Fig. 4.7, at an applied potential difference of a few kilovolts, a current of about 1 mA is produced at a compression of several percent. 

It appears that the observed breakdown must be explained in terms of the transient behavior of stress-induced defects even though the stresses are well within the nominal elastic range. In lithium niobate [77G06] and aluminum oxide [68G05] the extent of the breakdown appears to be strongly influenced by residual strains. In the vicinity of the threshold stress, dielectric relaxation associated with defects may have a significant effect on current observed in the short interval preceding breakdown. 

Kennedy and Benedick [67K02, 68K03] were successful in carrying out difficult Hall effect measurements in germanium samples explosively loaded at the upper end of the elastic range. Nevertheless, the measurements did not provide sufficient information to develop a physical interpretation. 

The piezoelectric constant studies are perhaps the most unique of the shock studies in the elastic range. The various investigations on quartz and lithium niobate represent perhaps the most detailed investigation ever conducted on shock-compressed matter. The direct measurement of the piezoelectric polarization at large strain has resulted in perhaps the most precise determinations of the linear constants for quartz and lithium niobate by any technique. The direct nature of the shock measurements is in sharp contrast to the ultrasonic studies in which the piezoelectric constants are determined indirectly as changes in wavespeed for various electrical boundary conditions. 

The piezoelectric response investigation also provides direct evidence that significant inelastic deformation and defect generation can occur well within the elastic range as determined by the Hugoniot elastic limit. In quartz, the Hugoniot elastic limit is 6 GPa, but there is clear evidence for strong nonideal mechanical and electrical effects between 2.5 and 6 GPa. The unusual dielectric breakdown phenomenon that occurs at 800 MPa under certain... 

In this book those ferroelectric solids that respond to shock compression in a purely piezoelectric mode such as lithium niobate and PVDF are considered piezoelectrics. As was the case for piezoelectrics, the pioneering work in this area was carried out by Neilson [57A01]. Unlike piezoelectrics, our knowledge of the response of ferroelectric solids to shock compression is in sharp contrast to that of piezoelectric solids. The electrical properties of several piezoelectric crystals are known in quantitative detail within the elastic range and semiquantitatively in the high stress range. The electrical responses of ferroelectrics are poorly characterized under shock compression and it is difficult to determine properties as such. It is not certain that the relative contributions of dominant physical phenomena have been correctly identified, and detailed, quantitative materials descriptions are not available. 

The concept of a welt defined elastic range to large strain is not realistic. The concept of well defined stress at which mechanical yielding occurs leading to well defined elastic-inelastic conditions is not realistic. Actually, such conclusions could well be anticipated from strength studies at atmospheric pressure, but there has been little explicit reason to consider the nonideal effects from the mechanical-response shock studies. 

Consider fibers that all have the same strength and are relatively brittle in comparison to the matrix as studied by Kelly and Davies [3-26]. Moreover, both the fibers and matrix are active only in the linear elastic range (stage 1 in Figure 3-46). If the composite material has more than a certain minimum volume fraction of fibers, V, the ultimate strength is achieved when the fibers are strained to correspond to their maximum (ultimate) stress. That is, in terms of strains. 

Elastic Collapse Pressure Formula. The minimum collapse pressure of the elastic range of collapse is calculated by... 

Material behavior have many classifications. Examples are (1) creep, and relaxation behavior with a primary load environment of high or moderate temperatures (2) fatigue, viscoelastic, and elastic range vibration or impact (3) fluidlike flow, as a solid to a gas, which is a very high velocity or hypervelocity impact and (4) crack propagation and environmental embrittlement, as well as ductile and brittle fractures. 

Poison s ratio It is the proportion of lateral strain to longitudinal strain under conditions of uniform longitudinal stress within the proportional or elastic limit. When the material s deformation is within the elastic range it results in a lateral to longitudinal strain that will always be constant. In mathematical terms, Poisson s ratio is the diameter of the test specimen before and after elongation divided by the length of the specimen before and after elongation. Poisson s ratio will have more than one value if the material is not isotropic... 

Different materials can be used such as nylon, polyester (TS), and epoxy, but TS polyurethane (PUR) is predominantly used. Almost no other plastic has the range of properties of PUR. Modulus of elasticity range in bending is 200 to 1,400 MPa (29,000-203,000 psi) and heat resistance from 90 to over 200°C (122-392°F). The higher values are for chopped glass-fiber-reinforced RIM (RRIM). 

The linear visco-elastic range ends when the elastic modulus G starts to fall off with the further increase of the strain amplitude. This value is called the critical amplitude yi This is the maximum amplitude that can be used for non-destructive dynamic oscillation measurements... 

Figure. 9 Visco-elastic range of pectin gels Strain frequency sweep...

Gels made from citrus pectin have a small linear visco-elastic range. To puncture the gels takes a big but only short time effort. The gel breaks into small lumps immediately. In the... 

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