PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 125–133 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

128

4

where

2 0 R R   .

(10)

Ci G and Ci  are the values of i G and i  in the centre of the beam cross-section, i f

In formulae (8) and (9),

and i p are parameters which control the distribution of i G and i  . The beam length is l . The length of beam portion, 1 2 L L , is 1 l . A lengthwise crack representing a circular cylindrical surface of radius, 1 R , and length, a , is located in beam portion, 2 3 L L . The beam is loaded in torsion in cross-section, 2 L , so as the angle of twist,  , of this cross-section varies with time according to the following law: v t    , (11) where  v is a parameter that controls the variation. The beam is clamped in its two ends. It is obvious that the beam has one degree of static indeterminacy. The torsion moment, 1 L T , in the left-hand end of the beam is treated as a redundant unknown. In order to resolve the static indeterminacy, the following equation is written by using the fact that the angle of twist of the left-hand end of the beam is zero:     0 1 2 1 1 3 4 1 3      l l a R l a R L L L L   , (12) where 1 3 L L  is the shear strain at the surface of the beam in portion, 1 2 L L , and at the surface of the internal crack arm (the internal crack arm represents a beam of length, a , with a circular cross-section of radius, 1 R ), 3 4 L L  is the shear strain at the surface of the beam in portion, 3 4 L L . The shear strains are distributed along the radius according to the following laws: R R L L SL L 1 1 2 1 3    , (13) R R L L SL L 2 3 4 3 4    . (14) There are two unknowns, 1 3 L L  and 3 4 L L  , in equation (12). Two equations are written by considering the equilibrium of elementary forces in the cross-sections of the beam: R dR T R L L 2 0 2 1 1 3     , (15) R dR T L L R L L 2 0 2 3 4 2 3 4     , (16) where  is the shear stress in beam portion, 1 2 L L , and in the internal crack arm, 3 4 L L  is the shear stress in beam portion, 3 4 L L . The torsion moments, 1 3 L L T and 3 4 L L T , are obtained as

1 3 L L L T T 

,

(17) (18)

1

T T T L L L   1 3 4 .

In equation (18), T is the external torsion moment in cross-section, 2 L , of the beam ( T is unknown). A further one equation is written by expressing the angle of twist of cross-section, 2 L , of the beam as a