5.2 Prandtl Model
217
σ
n
p = σ
p − E λ
σ
p
|σ
p |
.
(5.86)
The sensitivity of σ
n
p = σ
n with respect to
n is denoted the algorithmic tangent
E a (thus dσ = E a d) and is computed from the product rule while noting that λ
depends implicitly on
n
∂ σ
n
p = E − E λ ∂
σ
p
|σ
p |
− E
σ
p
|σ
p |
∂ ((λ).
(5.87)
The first derivative term on the right-hand-side computes to zero since
∂
σ
p
|σ
p |
=
1
|σ
p |
E −
σ
p
|σ
p | 2
σ
p
|σ
p |
E ≡ 0.
(5.88)
It shall be noted that the corresponding tangent modulus (tensor) in more than one
dimension is different from zero. The second derivative term on the right-hand-side
computes from requiring satisfaction of (the yield condition) ∂ φ
n
= 0 for ongoing
plastic flow at the end of the time step, i.e.
Table 5.5 Algorithmic update for the specific Prandtl model
Input
n n−1
p
Trial Strain
p = n−1
p
Trial Stress
σ
p = −E [
p − n ]
Trial Yield
φ = |σ
p | − σ y
Loading Check IF φ < 0 THEN
λ = 0
ELSE
λ =
φ
E
ENDIF
Update Strain n
p =
p + λ
σ
p
|σ
p |
Update Stress σ n = E [ n − n
p ]
Tangent
E n
a = E − H 0 ((λ) E
Output
σ n n
p E n
a