Issue 39
M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22
k 1 sin
E k ,
d 2 2
(9)
0
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
2
2 0 2 0
A L
x
1 4
2
s
dx
1 2
cos 2
L
2
A L
x dx
2
1 sin 2
1 2
L
(10)
2
2
2 0
A L
A L
x dx
2
1 2
2
sin 2
L
2
A
2
2 2
A L
x
L
2
0
dx
1 2
1
sin 2
2
L
A
1 2
L
dx dx L 2
The application of the substitution x x L 2
yields the differential
and leads to
2
A
2
2 2
A L
L
1 1 2
L
2
0
s
x dx
1
sin
2
4
2
A
1 2
L
2
A
2
2
2 0
L A 2
L
2
x dx
1
sin
2
2
A
1 2
L
(11)
2
A
2
2
L 2
L
A E
, 2
2
2
A
1 2
L
L A 2 2 4 2
A 2 2 4 , 2
E
2
L A 2 2 4 2
4
L A 2 2 2 2 4 4
A L A 2 2 2 2 2 4 4
where the factor 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 can be interpreted as a diminution factor and the argument
2
2
A L
x
2
0
s
dx
1 2
cos 2
L
233
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