Duhamel’s principle

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... 

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... 

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). 

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

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). 

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). 

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