Duhamels principle

The basic principles are taken from Zwillinger (1989). Duhamel s principle enables solutions for surface conditions being functions of time to be calculated from solutions with permanent surface conditions. Although this principle is most easily derived through the use of Laplace transforms, more conventional demonstrations, not repeated here, can be found in Sneddon (1957) or Carslaw and Jaeger (1959). 

Note that, in spite of the time variable being t, surface conditions now depend only on parameter t. Duhamel s principle states that if C(x,t,t) can be calculated, which should usually be an easy task due to time-independent surface conditions, the solution writes 

An example can be found in the section dealing with the diffusion of radiogenic isotope out of a sphere. 

Duhamel s principle can be extended to cases of surface conditions being functions of both time and space variables and to variable source and sink terms as well (Zwillinger, 1989). 

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In order to apply the Duhamel principle, we must retrieve the solution v(r, t) for constant v(a,t)=vs(t) = vs from equation (8.6.10). Upon multiplication by r/a, the solution reads... 

In fact, since the unsteady-state transport equation for forced convection is linear, it is possible in principle to derive solutions for time-dependent boundary conditions, starting from the available step response solutions, by applying the superposition (Duhamel) theorem. If the applied current density varies with time as i(t), then the local surface concentration at any time c0(x, t) is given by... 

The same derivation as that used for the accumulation of radiogenic isotope in a slab would lead to the solution but we will take advantage of this case to fully develop an application of Duhamel s principle (Appendix 8C). The assumption of zero initial and surface concentration of the radiogenic isotope is equivalent to... 

We then let surface value us(t) vary with t according to equation (8.6.18). Using Duhamel s principle (Appendix 8C), we find that the solution for the time variable... 

Duhamel s principle can be confirmed from the fact that it yields the nonhomogeneons equation starting from the solntion itself, as follows ... 

The last relation expresses the principle of superposition, due to Duhamel, stating that the response of a system to a sum of perturbations x + y is equal to the sum of the responses to individual and separated perturbations x and y. The summation being a commutative operation, it also states that, despite the order of imposition of the perturbations to the system, the response will be the same. 

Book iv of B is on chemical principles. In 1668 Duhamel met Boyle in England and two chapters in the second edition (1669) of B contain experiments on the elasticity of air and refer to Boyle s work on it. C refers to many modern authors, including Boyle, Bacon, Gilbert, Willis, Hooke, Gassendi, Descartes, Pascal, Galileo, Torricelli, Guericke, Tachenius, and Erasmus Bartholinus. Boyle quotes Du Hamel s praise in C of his own Experiments and Observations touching Coldy 1665. Duhamel was the first Secretary of the Academic Royale des Sciences (1666-97). 

The integral in Equation (4.3) is called a Duhamel integral, and it is a useful illustration of the consequences of the Boltzmann superposition principle to evaluate the response for a number of simple loading programmes. Recalling the development that leads to Equation (5.2) it can be seen that the Duhamel integral is most simply evaluated by treating it as the summation of a number of response terms. Consider two specific cases ... 

The Boltzman superposition principle (or integral) is applicable to stress analysis problems in two and three-dimensions where the stress or strain input varies with time, but first the approach will be introduced in this section only for one-dimensional or a uniaxial representation of the stress-strain (constitutive) relation. The superposition integral is also sometimes referred to as Duhamel s integral (see W.T. Thompson, Laplace Transforms, Prentice Hall, 1960). 

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