The Time-dependent Function

Equation 4.8 is the equation of a straight line of the form y = b + mx. Thus, the term (log a) represents the y intercept and (m) the slope. Applying logarithmic laws, Equation 4.8 can be written as  

Equation 4.10 is the power law that can be used to model creep as a time-dependent function. However, it is expected that the initial slope corresponding to the first creep stage is different from the longer term slope. As the first creep stage takes place in less than an hour, this equation can be corrected for a better fit by subtracting a constant b equivalent to the instantaneous strain. The corrected equation can be written as  

The linearity of the creep data presented on a log-log scale is consistent with the behaviour represented by the power law equation. 

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To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

Recall from the examples of partial fraction expansion that the polynomial Q(s) in the numerator, or the zeros, affects only the coefficients of the solution y(t), but not the time dependent functions. That is why for qualitative discussions, we focus only on the poles. 

A variety of forms have been suggested for the time-dependent functions in (6). A useful simplification is to assume that V t) (we drop the spin subscript to indicate the same form of the equation can be applied to both and V f) can be factored as a scalar, dependent on k, and a time-dependent function independent of k, i.e. 

Consequently, the time-dependent function coefficient can be calculated. 

For immobile particles A the density of traps ne enters into solution in such a way that the kinetic law of mass action holds formally (the concentration decay is proportional to the product of two concentrations) but replaces the constant reaction rate (the coefficient of this product) for the time-dependent function. Therefore, an exactly solvable problem of the bi-molecular A + B -y B reaction gives us an idea of the generalization of the... 

In this review, we begin with a treatment of the functional theory employing as basis the maximum entropy principle for the determination of the density matrix of equilibrium ensembles of any system. This naturally leads to the time-dependent functional theory which will be based on the TD-density matrix which obeys the von Neumann equation of motion. In this way, we present a unified formulation of the functional theory of a condensed matter system for both equilibrium and non-equilibrium situations, which we hope will give the reader a complete picture of the functional approach to many-body interacting systems of interest to condensed matter physics and chemistry. 

Here, x = r,p includes the positions r and momenta p of all particles. The time dependent function is the propagator of the system. For such... 

This calculation is performed here for the relaxation times T ( ) and T ( ) (the time constants of the time-dependent functions corresponding to (P,(cosi )) and (FJ(cos i )e )). The differential-difference equations for these are... 

The coefficients of the time-dependent functions are now expanded to first order in the small parameters Dvq2 and yDrq2. This gives Eq. (10.4.37). 

The time-dependent function (r,0 is here replaced by the steady-state function 0(r). The remainder of this chapter is concerned wuth the solution of Eqs. (5.21) and (5.22) for various specifications of the source S and of the geometric configuration of the medium in which the one-velocity neutrons are diffusing. 

This case obviously corresponds to the critical system, since the time-dependent function in the expression (9.70) now vanishes. Thus a second important fact to be obtained from the pulsed neutron-beam experiment is the critical dimensions of a given geometry. Another fact to be learned from the experiment is the diffusion coefficient of the mixture and the absorption cross section of the fuel, since for small specimens... 

A capital I is used to distinguish the time-dependent function (1-2) from the time-independent function (1-1). 

We can equate the time-dependent function in Eq. (2.68) to a superposition of many oscillating functions, aU of the foimexp —Et / H) but with a range of energies ... 

The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number rij in Eq. (3.50) either as the particular excited state occupied by oscillator j, or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can stiU be described by Eqs. (3.44) and (3.45) if the amplitude factor is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than with the matrix elements of the position operator Q. These matrix elements play a central role in the quantum theory of absorption and emission, as we ll discuss in Chap. 5. 

We can equate the time-dependent function on the right side of Eq. (3.56) to a frequency-dependent function... 

The time dependent function u describes the excitation of the truck due to the road roughness. The road roughness normally is described by a function having the distance from a particular point of the road to a fixed starting point as its... 

Evaluation of this expression requires first an evaluation of the time-dependent functions Qi t) and Q2 t) which are defined in terms of integrals over J u) as follows... 

To simplify the notation, explicit dependence of the different functions on time is generally omitted in this section. The time-dependent functions are F(f), f(f), p(f), s(t) and Ps i) (together with their first and second time derivatives) for the extended system, and F(f), r(r), p(f), s t), ps t), y (f) mAT t) (together with their first and second time derivatives) for the real system. The dot overscripts indicate differentiation with respect to the extended-system time i for the extended-system variables, and with respect to the real-system time t for the real-system variables. 

For the purposes of the current investigation, durability of FRP composites is characterized by the results of accelerated aging experiments. Predictions for FRP-composite tensile modulus and tensile strength for both CFRP and GFRP composites in this analysis are modeled with an Arrhenius rate relationship derived from results for wet lay-up CFRP composites immersed in deionized water at 23 °C for approximately 2 years (Karbhari and Abanilla, 2007). The time-dependent functions for tensile modulus and tensile strength for FRP composites are as follows ... 

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