Computed multicomponent phase diagrams for hardenability (H) and HSLA steels with application to the prediction of microstructure and mechanical properties推荐.pdf

Chapter 6 Cagniard-de Hoop Technique The success of the integral transform method hangs on the evaluation of inversion integrals. It is not easy to find a suitable integration formula. If we cannot find any suitable formula, the inversion is left in its integral form and some numerical techniques must be applied for the evaluation. However, in the case of double inversion, such as Laplace and Fourier inversions, if we could convert the first Fourier inversion integral to a definition integral of Laplace transform, the next Laplace inversion can be carried out by inspection without evaluating its Laplace inversion integral. For example, when we have the double transformed function F ðn; sÞ and its Fourier inversion integral þ1 f ðx ; sÞ ¼ Z F ðn; sÞ expðinx Þd n ð6:1Þ 1 if we could apply some mathematical techniques to the integral and convert the integral to the definition form of Laplace transform, 1 f ðx ; sÞ ¼ Z Gðx ; tÞ expðstÞdt ð6:2Þ a i.e. the Laplace transform integral, the Laplace inversion is easily carried out by inspection and its inversion is given by f ðx ; tÞ ¼ Gðx ; tÞ ; t [a ¼ H ðt aÞGðx ; tÞ ð6:3Þ 0 ; t \a The most important and substantial part of the this technique is to convert the Fourier inversion integral to the form of the Laplace transform, þ1 1 Z F ðn; sÞ expðinx Þd n ¼ Z Gðx ; tÞ expðstÞdt ð6:4Þ 1 a K. Watanabe, Integral Transform Techniques for Green’s Function, 107 Lec