Ductile Piping Fracture Mechanics (csni84-97)

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Such a spin chain (a magnetised whisker) was used in earlier experiments testing the existence of the B-A effect. 38 CHAPTER 3. 2: Explicit realisation of the magnetic (electric) cylindrical solenoid by means of a linear chain of magnetic (electric) dipoles. , [16]): Be = r~ [r~ r(mr) - m] + 8; mo (r). 3 Sometimes in the physics literature another representation of B is used [16,59]: 1 [3 411" 3 B'ffi="3 "2r(mr)-m ] - -mo (r). r r 3 This difference arises for the following reason [60]. If we identify a magnetic dipole with an electric current flowing in an infinitely small circular coil, then the vector potential is given by 1 cr -3 me= (me ~ f(r X X r), j)dV.

44) :rx Ql/2 (z2 +~: + X2) . 44) that for p > d the argument y = (p2 + Z2 + x 2 )/2ax of the Legendre function Q-l/2 always exceeds 1 for all x in the interval 0 S; x S; a. This means (as the cut of the Legendre functions coincides with the interval (-1, 1» that the function Q' and all its derivatives are continuous functions of z for p > a. For p < a y acquires the value 1 for z = 0, x = p. In this case the function Q' and its derivatives may possess singularities. 45). In fact, the argument of the (Z2 + x 2 + a 2 )/2ax of the Ql/2 always exceeds 1 for all p < a.

DIvA + -£pc -8¢ = O. 46) contain x and t in the combination x - vt. 47) 4rr --p, £ where 82 Ll. = 81i;2 + 82 8y2 . Let us now introduce the elliptic coordinates v, () : where a Ii; a cosh v cos () , fj a sinh v sin (), R sinhvo ' = tanhvo = J. 46) reduces to the form 8(P'-R) = 3.. coshvosinhvo 8(v-vo). - cos 2(} R cosh 21/0 The values 1/ > Vo and v < I/o correspond to points lying outside and inside the solenoid, respectively. In the coordinates 1/ and () the charge and current densities are given by p = sinh 21/0 sin () cR cosh 2vo - cos 2(} l'(3) - -- ) sinh 2vo cos () R cosh 2vo - cos 2(} - « 0 « 01/-1/0 1/ - I/o ) , ) , )1' sinh 2vo sin () « ) 0 v - I/o .

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