Reference deformation

By looking at the pair of values (T, f ) for a specified reference temperature Tr(f, and reference deformation rate satisfy Eq. 4.9, one can... 

By reference to the outline periodic table shown on p. (i) we see that the metals and non-metals occupy fairly distinct regions of the table. The metals can be further sub-divided into (a) soft metals, which are easily deformed and commonly used in moulding, for example, aluminium, lead, mercury, (b) the engineering metals, for example iron, manganese and chromium, many of which are transition elements, and (c) the light metals which have low densities and are found in Groups lA and IIA. 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

The interaction potential (R) describes both bonding and non-bon ding in teraction s. Th e bon dm g interactions arc u snally form u -lated as a strain energy that is zero at some ideal configuration of the atoms and describe how the energy increases as the ideal con-figu ration is deform ed. Don d in g in teraction s ii su ally refer to atom s in the following relationships ... 

Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

An unsteady ain-flow unbalance that alternates between inlets can set up an alternating thrust pattern which can be very damagiag to beariags designed for low thrust load. Mechanical vibration and elastic deformation problems and diagnostic techniques for stmctural iaadequacies ia fan design are discussed ia Reference 16. 

I, have a fairly broad distribution of bubble sizes and can therefore maintain spherical bubbles with significantly less Hquid. Empirically, foams with greater than about 5% Hquid tend to have bubbles that are stiH approximately spherical, and are referred to as wet foams. Such is the case for the bubbles toward the bottom of the foam shown in Figure 1. Nevertheless, it is important to note that even in the case of these wet foams, some of the bubbles are deformed, if only by a small amount. 

Hot pressing to produce substantial texture and magnetic anisotropy via plastic deformation is accompHshed by a process referred to as... 

Creep. The phenomenon of creep refers to time-dependent deformation. In practice, at least for most metals and ceramics, the creep behavior becomes important at high temperatures and thus sets a limit on the maximum appHcation temperature. In general, this limit increases with the melting point of a material. An approximate limit can be estimated to He at about half of the Kelvin melting temperature. The basic governing equation of steady-state creep can be written as foUows ... 

Some tests, while undergoing deformation, are usually referred to as static in that they are performed at slow speeds or low cycles. Examples of these tests are stretch modulus, ultimate tensile, and elongation to break, ie, a measure of total energy capabiUties or mpture phenomena. 

Wax usually refers to a substance that is a plastic solid at ambient temperature and that, on being subjected to moderately elevated temperatures, becomes a low viscosity hquid. Because it is plastic, wax usually deforms under pressure without the appHcation of heat. The chemical composition of waxes is complex all of the products have relatively wide molecular weight profiles, with the functionaUty ranging from products that contain mainly normal alkanes to those that are mixtures of hydrocarbons and reactive functional species. 

Most wrought alloys are provided in conditions that have been strengthened by various amounts of cold work or heat treatment. Cold worked tempers are the result of cold rolling or drawing by prescribed amounts of plastic deformation from the annealed condition. Alloys that respond to strengthening by heat treatment are referred to as precipitation or age hardenable. Cold worked conditions can also be thermally treated at relatively low temperatures to affect a slight decrease in strength (stress rehef annealed) to benefit other properties, such as corrosion resistance and formabiUty. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

If the work assumption is made, i.e., if it is assumed that the external work done by surface and body forces on a finite region in the reference configuration of a body undergoing homogeneous closed cycles of deformation is nonnegative, then an inequality may be deduced paralleling (5.37) by arguments essentially the same as those of Section 5.2.4. 

Kinematical relations in large deformations are given here for reference. Most of the material is well known, and may be extracted or deduced from the comprehensive expositions of Truesdell and Toupin [19], Truesdell and Noll [20], or other texts in continuum mechanics, where further details may be found. 

The polar decomposition (A. 13) implies that the deformation may be viewed as two successive deformations, the first being a pure stretch from the reference configuration into an unrotated configuration, and the second being a... 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

The components of strain ej- relative to the unrotated spatial configuration are shifted to components of strain relative to the reference configuration by the stretch U, or to components of strain Cy relative to the current spatial configuration by the rotation R. The tensors E, e, and e all are measures of the same irrotational part of the deformation, but with components relative to different configurations. 

It is evident from their definitions that /, and hence d and w depend on the instantaneous rate of deformation of the current configuration. On the other hand, F and hence U and R relate the current configuration to the reference configuration. In order to find relations for d and w in terms of material derivatives of U and R, the material derivative of (A. 13) may be inserted into (A. 10)... 

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