340
S. Zacek et al.
where w gp is the Gauss integration weight, N gp is a matrix arrangement of the
discontinuous enrichment functions used to compute the displacement jump vector,
and n t gp is the traction vector defined in Equation (2). Differentiating Equation (13)
with respect to the internal variables yields
∂F
int,{cohesive}
gp
∂ n−1 δ
gp
max
= w gp N
T
gp
∂ n t gp
∂ n−1 δ
gp
max
dA,
∂ n t gp
∂ n−1 δ
gp
max
=
⎧
⎪ ⎨
⎪ ⎩
0
if loading,
1
n−1 δ
gp
max
dt ∗
d n−1 δ
gp
max
−
t ∗
n−1 δ
gp
max
×
β 2 δ + (1 − β 2 )(δ · n)n
if unloading,
(14)
where t ∗ is defined in Equation (4) and
dt ∗
d n−1 δ max
is easily computed from Equation (3).
The right-hand sides of Equations (11) and (12) also contain the derivatives of the
internal variables with respect to the parameters from the previous load step. These
derivatives are simply stored as additional internal variables for each quadrature
point and initialized as
d 0 δ max
dη i
= 0. For subsequent steps, the components of the
vector are updated using
d n δ max
dη i
=
1
2 n δ (2β 2 n δ s
d n δ s
dη i
+ 2 n δ n
d n δ n
dη i
) if loading,
d n−1 δ max
dη i
if unloading,
(15)
where
d n δ gp
dη i
= N gp
d n U elem
dη i
.
(16)
In Equation (16),
d n U elem
dη i
can be solved using Equation (11). These updated internal
variable derivatives are then used in the sensitivity analysis at the end of the next
load step.
The last missing term is the partial derivative of the internal force with respect to
specific interface parameters. The sensitivity derivations presented in the remainder
of this section are specific to η i = σ c , leaving a summary of the derivations of the
sensitivity with respect to the critical displacement jumps δ ci (i = 1, 2, 3) for the
Appendix.
Again, the contributions from the linear elastic bulk elements to the partial
derivative vanish as the stress does not depend explicitly on the cohesive internal
strength. The partial derivative of Equation (13) with respect to σ c is
∂ n F
int,{cohesive}
elem
∂σ c
=
n gp
gp=1
w gp N
T
gp
∂ n t gp
∂σ c
dA.
(17)
S. Zacek et al.
where w gp is the Gauss integration weight, N gp is a matrix arrangement of the
discontinuous enrichment functions used to compute the displacement jump vector,
and n t gp is the traction vector defined in Equation (2). Differentiating Equation (13)
with respect to the internal variables yields
∂F
int,{cohesive}
gp
∂ n−1 δ
gp
max
= w gp N
T
gp
∂ n t gp
∂ n−1 δ
gp
max
dA,
∂ n t gp
∂ n−1 δ
gp
max
=
⎧
⎪ ⎨
⎪ ⎩
0
if loading,
1
n−1 δ
gp
max
dt ∗
d n−1 δ
gp
max
−
t ∗
n−1 δ
gp
max
×
β 2 δ + (1 − β 2 )(δ · n)n
if unloading,
(14)
where t ∗ is defined in Equation (4) and
dt ∗
d n−1 δ max
is easily computed from Equation (3).
The right-hand sides of Equations (11) and (12) also contain the derivatives of the
internal variables with respect to the parameters from the previous load step. These
derivatives are simply stored as additional internal variables for each quadrature
point and initialized as
d 0 δ max
dη i
= 0. For subsequent steps, the components of the
vector are updated using
d n δ max
dη i
=
1
2 n δ (2β 2 n δ s
d n δ s
dη i
+ 2 n δ n
d n δ n
dη i
) if loading,
d n−1 δ max
dη i
if unloading,
(15)
where
d n δ gp
dη i
= N gp
d n U elem
dη i
.
(16)
In Equation (16),
d n U elem
dη i
can be solved using Equation (11). These updated internal
variable derivatives are then used in the sensitivity analysis at the end of the next
load step.
The last missing term is the partial derivative of the internal force with respect to
specific interface parameters. The sensitivity derivations presented in the remainder
of this section are specific to η i = σ c , leaving a summary of the derivations of the
sensitivity with respect to the critical displacement jumps δ ci (i = 1, 2, 3) for the
Appendix.
Again, the contributions from the linear elastic bulk elements to the partial
derivative vanish as the stress does not depend explicitly on the cohesive internal
strength. The partial derivative of Equation (13) with respect to σ c is
∂ n F
int,{cohesive}
elem
∂σ c
=
n gp
gp=1
w gp N
T
gp
∂ n t gp
∂σ c
dA.
(17)