Residual Strength and Fatigue Lifetime of ... - Solid Mechanics

S
S
y(
t ) y(
t )
( t )
2 1
12 2
(4.1)
tcyc
y(
t ) y(
t )
( t )
3 2
23 3
(4.2)
tcyc
where tcyc t2
t1
t3
t2
is the time of each cycle.
Figure 4.2: Schematic presentation of the cycle jump method, after Cojocaru et al. (2006).
The parameter qy is introduced as the maximum relative error to control the accuracy of the
simulation by the following criterion:
S
jump
( t
3
t
y,
jump ) S
S ( t )
23
3
23
( t3
)
q
y
where qy is the maximum allowed relative error, t y,
jump the number of jumped cycles (assuming
tcyc=1) and S jump is the estimated slope after the jump for the state variable y, using linear
extrapolation given by
S ( t ) S ( t )
( ) ( )
(4.4)
23 3 12 2
S jump t3
t y,
jump S23
t3
t y,
jump
tcyc
The introduced criterion ensures that the slope of the increment of the variable y after the cycle
jump is “close enough” to its slope before the jump. qy is specified by the user for each state
parameter such as deflection or material properties. From Equations (4.3) and (4.4) the allowed
jump for each extrapolated parameter is determined by
66
S
S
23
12
y(
t3)
y(
t2)
( t3)
tcyc
y(
t2)
y(
t1)
( t2)
t
cyc
(4.3)