12 Multi-mode Model and Calculation Method for Fatigue Damage …
163
cycle of a deformable specimen supplemented by a code to calculate the damage
equation and changes of elasticity modulus.
To integrate the equation dψ
dN = Bψ
γ
/(1 − ψ
α
), where B either B LH or B VH ,
the damage function approximation was applied at the k-node of the computational
grid for given discrete values ψ
n
k at moments N
n and sought ψ
n+1
k
at moments N
n+1 .
To calculate the damage equation, the value α = 1 − γ was chosen for which by
analytic integration an explicit expression for ψ
n+1
k
(ψ
n
k , ,N
n
) can be obtained:
ψ
1−γ
/(1 − γ ) − ψ
2(1−γ )
/2/(1 − γ )
ψ
n+1
k
ψ
n
k
= B N|
N n+1
N n
.
With the denotations (ψ
n+1
k
)
1−γ
= x, q = 2(1 − γ )BN
n
+ (ψ
n
k )
1−γ
−
2(ψ
n
k )
2(1−γ ) and N
n
= N
n+1
− N
n the equation transforms to x
2
− 2x + q = 0
and its valid root x = 1 −
√
1 − q < 1. The analytical expression for the increment
of damage on the increment of the number of cycles N
n is derived from:
ψ
n+1
k
=
1 −
1 −
2(1 − γ )BN n + (ψ
n
k ) 1−γ − 2(ψ
n
k ) 2(1−γ )
1/(1−γ )
.
Increment value N
n is defined as follows. Based on the stress state calculation
data, the coefficient B = B LH , B VH is calculated for each node. After that, for each
node, the following values are calculated by
˜
N
n
k =
ψ
1−γ
/(1 − γ ) − ψ
2(1−γ )
/2/(1 − γ )
1
ψ
n
k
/B
corresponding to the number of cycles, at which in the node k from its current level
of damage and equivalent stress complete destruction will be achieved (damage is
equal to 1). If the damage level in the considered node is less than the threshold ψ 0
(threshold ψ 0 = 0.95 is selected), then the value for this node ˜
N
n
k is multiplied by a
factor of 0.5. Otherwise, it is multiplied by a factor of 1. Thus, the step of incrementing
the number of cycles for a given node is N
n
k = 0.5(1 + H (ψ
n
k − 0.95))) ˜
N
n
k . Of
all the N
n
k values, the smallest one is selected. The increment of the number of
loading cycles for the calculation of the entire specimen is N
n
= min
k
N
n
k . For
each node based on its current level of damage and equivalent voltage, a new level
of damage is found taking into account the calculated increment N
n .
All elements are sorted out, for each of them the most damaged node is searched
and according to its damage the mechanical properties of the element are adjusted:
λ(ψ
n
k ) = λ 0 (1 − κψ
n
k ), μ(ψ
n
k ) = μ 0 (1 − κψ
n
k ).
Those elements that belong to nodes with damage ψ = 1 are removed from
the calculation area and form a localized zone (crack-like) of completely destroyed
material. The calculation ends when the boundaries of a completely damaged region
exit to the specimen surface (macro destruction) or the evolution of this region stops.
163
cycle of a deformable specimen supplemented by a code to calculate the damage
equation and changes of elasticity modulus.
To integrate the equation dψ
dN = Bψ
γ
/(1 − ψ
α
), where B either B LH or B VH ,
the damage function approximation was applied at the k-node of the computational
grid for given discrete values ψ
n
k at moments N
n and sought ψ
n+1
k
at moments N
n+1 .
To calculate the damage equation, the value α = 1 − γ was chosen for which by
analytic integration an explicit expression for ψ
n+1
k
(ψ
n
k , ,N
n
) can be obtained:
ψ
1−γ
/(1 − γ ) − ψ
2(1−γ )
/2/(1 − γ )
ψ
n+1
k
ψ
n
k
= B N|
N n+1
N n
.
With the denotations (ψ
n+1
k
)
1−γ
= x, q = 2(1 − γ )BN
n
+ (ψ
n
k )
1−γ
−
2(ψ
n
k )
2(1−γ ) and N
n
= N
n+1
− N
n the equation transforms to x
2
− 2x + q = 0
and its valid root x = 1 −
√
1 − q < 1. The analytical expression for the increment
of damage on the increment of the number of cycles N
n is derived from:
ψ
n+1
k
=
1 −
1 −
2(1 − γ )BN n + (ψ
n
k ) 1−γ − 2(ψ
n
k ) 2(1−γ )
1/(1−γ )
.
Increment value N
n is defined as follows. Based on the stress state calculation
data, the coefficient B = B LH , B VH is calculated for each node. After that, for each
node, the following values are calculated by
˜
N
n
k =
ψ
1−γ
/(1 − γ ) − ψ
2(1−γ )
/2/(1 − γ )
1
ψ
n
k
/B
corresponding to the number of cycles, at which in the node k from its current level
of damage and equivalent stress complete destruction will be achieved (damage is
equal to 1). If the damage level in the considered node is less than the threshold ψ 0
(threshold ψ 0 = 0.95 is selected), then the value for this node ˜
N
n
k is multiplied by a
factor of 0.5. Otherwise, it is multiplied by a factor of 1. Thus, the step of incrementing
the number of cycles for a given node is N
n
k = 0.5(1 + H (ψ
n
k − 0.95))) ˜
N
n
k . Of
all the N
n
k values, the smallest one is selected. The increment of the number of
loading cycles for the calculation of the entire specimen is N
n
= min
k
N
n
k . For
each node based on its current level of damage and equivalent voltage, a new level
of damage is found taking into account the calculated increment N
n .
All elements are sorted out, for each of them the most damaged node is searched
and according to its damage the mechanical properties of the element are adjusted:
λ(ψ
n
k ) = λ 0 (1 − κψ
n
k ), μ(ψ
n
k ) = μ 0 (1 − κψ
n
k ).
Those elements that belong to nodes with damage ψ = 1 are removed from
the calculation area and form a localized zone (crack-like) of completely destroyed
material. The calculation ends when the boundaries of a completely damaged region
exit to the specimen surface (macro destruction) or the evolution of this region stops.