Volume integral representation

Remark. The Stratton-Chu formulas are surface-integral representations for the electromagnetic ffelds and are valid for homogeneous particles. For inhomogeneous particles, a volume-integral representation for the electric held can be derived. For this purpose, we consider the nonmagnetic domains Dg and Di = Mi = 1), rewrite the Maxwell equations as... 

In 1976 Suprynowicz et al. [139] showed that differentiation of the integral equation (1) with respect to pressure gives a general integral representation of the overall retention volume ... 

In these conditions the quantum statistical path integral representation of quantum propagator, see Eq. (2.10) with Eq. (2.21), and Volume 1/ Chapter 4 of the present five-volume set, and takes the form... 

This expression represents the third level of the path integrals representation, after the classical quantum mechanical picture (see Volume I of the present five-volume work)... 

Edwards has also noted the strong mathematical analogies between the functional integral representation of the polymer excluded volume problem and questions associated with electronic structure in disordered systems. "" Thus a detailed discussion of this formal approach is of more than just academic interest. In the polymer case, approximation may be guided by probabilistic arguments, whereas in the disordered system analogous mathematical approximations rest upon less intuitive grounds. 

Strain-Induced Dilatation. An alternative view of yield in polymers comes from the fact that a tensile strain induces a hydrostatic tension in the material and a corresponding increase in the sample volume. This in turn translates to an increase in the free volume, which increases the polymer mobility and effectively lowers the glass-transition temperature (Tg) of the polymer (alternatively it can be looked upon as increasing the free volume to the value it would have at the normal measured Tg). The increased mobility results in a lowering of the yield stress. Rnauss and Emri (35) used an integral representation of nonlinear viscoelasticity with a state-dependent variable related to free volume to model the yield behavior, with the free volume a function of temperature, time, and stress history. This model uses the concept of reduced time (see VISCOELASTICITY), where application of a tensile stress causes a volume dilatation and consequently causes the material time scale to change by a shift factor related to the magnitude of the applied stress. Yield occurs because the free-volume shift factor causes the molecular mobility to increase in such a way that yield can occur. 

In the material (Lagrangian) representation of continuum mechanics a representative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector = ( i, 2. 3)-In this -space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (ri, r2, r3>. The coordinates ri, r2, r3 which gives the current position of the particle are called the spatial coordinates. Let V (r, t) be any scalar, vector or tensor function of time and position and V(t) a material volume. We may then define a variable f (t) as the volume integral [2] ... 

Differential equation [379] can be converted into an integral representation by integrating over the th volume element (with volume vj) to get... 

The volume integral equation method relies on the integral representation... 

In the graphical representation of the integral shown above, a line represents the Mayer function f r.p between two particles and j. The coordinates are represented by open circles that are labelled, unless it is integrated over the volume of the system, when the circle representing it is blackened and the label erased. The black circle in the above graph represents an integration over the coordinates of particle 3, and is not labelled. The coefficient of is the sum of tln-ee tenns represented graphically as... 

The design of smart materials and adaptive stmctures has required the development of constitutive equations that describe the temperature, stress, strain, and percentage of martensite volume transformation of a shape-memory alloy. These equations can be integrated with similar constitutive equations for composite materials to make possible the quantitative design of stmctures having embedded sensors and actuators for vibration control. The constitutive equations for one-dimensional systems as well as a three-dimensional representation have been developed (7). 

For volumes V) and Vn we consider the integrals as sums over small volumes, A Vj, defined by the motion of their corresponding surface elements, AS). Figure 3.3 provides a (two-dimensional) geometric representation of this process in which AS, moves over AS... 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

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