Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22 233   E k k d 2 2 0 , 1 sin        (9) For the description of real geometric dimensions the consideration of the arbitrary amplitude A and an arbitrary length L as two independent parameters is necessary. The arc length of a sine can then be achieved by considering the complete elliptic integral of the second kind. Therefore the relation x x 2 2 sin cos 1   between the squared sine and cosine with the same arguments has to be applied. Additionally, the lower integration limit is set to 0 and the upper integration limit to 2  , respectively. Carrying out the substitution x x L 2    one quarter of the arc length can be calculated by     A x s dx L L A x dx L L A A x dx L L L A A x L dx L L A L 2 2 2 0 2 2 2 0 2 2 2 2 0 2 2 2 2 2 0 1 1 2 cos 2 4 1 2 1 sin 2 1 2 2 sin 2 2 1 2 1 sin 2 1 2                                                                                                        (10) The application of the substitution x x L 2    yields the differential   dx dx L 2    and leads to             A A L L s x dx L A L A L L A x dx A L A L L A E A L L A A E L A 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 sin 4 2 1 2 2 1 sin 2 1 2 2 , 2 2 1 2 4 4 , 2 4 4                                                                                     (11) where the factor L A 2 2 2 2 4 4    can be interpreted as a diminution factor and the argument A L A 2 2 2 2 2 4 4    can be determined as the elliptic modulus k . The arc length of one complete ondulation s can be calculated by simply multiplying the repeating quarter-sequences with 4 yielding A x s dx L L 2 2 2 0 1 2 cos 2                   