Small deformation approximation

Though the stress at time t in memory fluids is expected to depend on the history of the deformation, the dependence is stronger for recent deformations than for ancient ones. In other words, these fluids exhibit fading memory (23). The slow flow and small deformation approximations have been used to establish constitutive equations for memory fluids. In the slow flow approximation (23), a sequence of deformation histories is assumed in which each history differs from a reference history in that the time scale is slowed by a... 

In the small deformation approximation, it is assumed that the deformations undergone by the material are small, at least in the recent past. Approximations of different orders can be developed. The approximation of first order for an incompressible fluid is given by Boltzmann s equation of linear viscoelasticity,... 

For the distorted structure E the helix axis will have small components in the x direction as before, and therefore in the small deformation approximation we have... 

The height of the interface over the reference flat interface is described by a 2D scalar field M(rn) (see Figure 2.4 for the definitions). The force Fj , defined in Equation 2.3 is easily evaluated up to second order in the small-deformation approximation [34] ... 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

The approximate transferability of fuzzy fragment density matrices, and the associated technical, computational aspects of the idempotency constraints of assembled density matrices, as well as the conditions for adjustability and additivity of fragment density matrices are discussed in Section 4, whereas in Section 5, an algorithm for small deformations of electron densities are reviewed. The Summary in Section 6 is followed by an extensive list of relevant references. 

Recall from Section 5.1.3.1 that thermal expansion is the change in dimensions of a solid due to heating or cooling. As a result, a thermaUy-induced strain can result when the material is heated, as dictated by Eq. (5.29). For small deformations at constant pressure, we can thus approximate the strain as... 

Even this form is probably only an approximation (267). More complicated expressions appear to be necessary to fit data on the same sample for different types of deformation (268). For sufficiently small deformations the tensile modulus becomes... 

In the present paper we extend our analysis of the experimental results obtained from this small deformation regime and we show that the result found by Reissner for the deformation of shallow spherical caps represents an excellent analytical approximation for the interpretation of the measurements. This result is varified by finite element modelling (FEM) and by experimental variation of the force probe geometry and radius as well as wall thickness of the studied capsules. This result is also applicable for other capsule deformation measurements, since it is independent of the specific Young s modulus. Furthermore, we report on speed dependent measurements that indicate the glassy nature of PAH/PSS multilayers. 

Due to the relatively small deformation in neutral complexes between uncharged molecules the frozen geometry approximation yields fairly correct results. In this simplified approach the geometries of the interacting subsystems are kept constant and only the internal degrees of freedom are optimized. 

It can easily be shown that Cy is a symmetrical tensor, that is ey = ep. For very small deformations, the partial derivatives dujdxj in Eq. (4.21) can be considered infinitesimal quantities of first order, so the components of the ey tensor can be approximately expressed as... 

One complication is that the boundary conditions (4-264)-(4-266) must be applied at the bubble surface, which is both unknown [that is, specified in terms of functions R(t) and fn(9,tangent unit vectors n and t, that appear in the boundary conditions are also functions of the bubble shape. In this analysis, we use the small-deformation limit s 1 to simplify the problem by using the method of domain perturbations that was introduced earlier in this chapter. First, we note that the unit normal and tangent vectors can be approximated for small e in the forms... 

As expected, this is identical to the 0(8) approximation obtained for the small-deformation limit in the preceding part of this subsection, namely Eq. (6-239). 

As pointed out by Shannon [15], who established the information theory as an autonomous mathematical discipline, the basic problem of communication is that of reproducing at one point (receiver, output), exactly or approximately, a message sent at another point (source, input). The free (isolated) constituent atoms, defining the promolecule , can be viewed as the molecular message source. The information contained in the probability distributions of this reference state is mostly preserved in the molecule, the molecular message receiver. Indeed, the bonded (chemical) AIM are known to be only slightly perturbed in their valence shell relative to their free analogs. However, these small deformations in the electron distribution, due to... 

The variance-covariance matrix <(Xe-jCo)(jCe-A o) > is usually contaminated by contributions from experimental error. Indeed, for small deformations and poor experimental data, it could be dominated by these experimental uncertainties. On the other hand, for very large deformations the quadratic energy dependence cannot be expected to hold. Once the harmonic approximation breaks down, the above equations would need to be replaced by more complicated expressions involving a larger number of unknown (anharmonic) force constants. 

The elastic behavior upon applied shear stress is primarily typical in the case of solids. The nature of elasticity is in the reversibility of small deformations of interatomic (or intermolecular) bonds. In the limit of small deformations the potential energy curve is approximated by a quadratic parabola, which corresponds to a linear t(y) dependence. Elasticity modulus of solids depends on the type of interactions. For molecular crystals it is 109 N m 2, while for metals and covalent crystals it is 1011 N m"2 or higher. The value of elasticity modulus is only weakly dependent (or nearly independent) on temperature. 

Crazing is an important source of toughness in mbber-modified thermoplastics. A craze can be described as a layer of polymer a nanometer to a few micrometers thick, which has undergone plastic deformation approximately in the direction normal to the craze plane as a response to tension applied in this direction [Kambour, 1986]. Crazing occurs without lateral contraction. As a result, the polymer volume fraction in the craze is proportional to 1/, where is the draw ratio in the craze. The reduction in density occurs on such a small scale that the refractive index is markedly reduced, which accounts for the reflectivity of the craze [Kramer, 1983]. 

Hooke s law is a good approximation for small deformations only. For larger deformations, one may add additional terms (cubic, quartic, etc.), or substitute a Morse potential for stretching. In general, simple potential functions are used when possible, and more complicated ones when necessary. Sufficiently complicated functions will reproduce any desired properties, but with additional labor. Additionally, the more parameters that are added, the more the results become obscured and removed from an intuitive understanding. 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

Many rubber products are normally subjected to fairly small deformations, rarely exceeding 25% in extension or compression or 75% in simple shear. A good approximation for the corresponding stresses can then be obtained by conventional elastic analysis assuming linear relationships. One particularly... 

Another possibility, perhaps more appropriate in the case of a continuous phase-separated network, is that the retractive force of the gel arises from a minimization of interfacial energy. For example, if the gel structure is imagined to be cellular, consisting of approximately spherical holes in the polymer-rich phase entrapping the dilute phase, then small deformations at constant volume (illustrated in two dimensions in Fig. 8) would result in an increase in interfacial area and, thus, an increase in free energy. The retractive force in such a... 

For sufficiently small deformation gradients the coordinates of the stress tensor may be approximated by hnearfunctiorrs of the coordinates of the strain tensor. This geometrical and physical hnearization leads to a generahzation of Hooke s law, well known for isotropic bodies. It takes the form... 

The formulation of Hooke s law rests on the assumption of infinitesimally small deformations. Its apphcation to the simple model of a mass connected with a spring results in a hnear force law and to the well known harmonic oscillation. Investigating even with very modest means the behavior of a real system of this sort shows that the limits of accuracy of this simple description are quite narrow indeed. A more general and accurate description will have to be a nonlinear one. This, in fact mrns out to be tme for all material properties, e.g. dielectric properties and the simple relation (4.2) is valid only for small fields and is an approximation in the same way as Hooke s law (3.51). If we are looking close enough we find that all phenomena aetually are nonlinear, which means that the response of even simple systems to an external influence cannot be precisely described by a direct proportionaUty. 

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