Tensile Electroforming Model - Electrically Assisted Forming - page 134

Table 6.4 Conventional
room and elevated
temperature model
coefficients for Mg AZ31B

Constant Value

457.72

K RT

− 1.9529

K 1

K 2 500.68

931.06

K 3

− 0.01

K 4

n RT 0.1818

− 0.0004

n 1

0.1909

n 2

n 3 − 0.0009

0.2713

n 4

0.00008

s 1

s 2 − 0.0357

3.1162

s 3

Thus,

K m + 1 ε n m + 1

K m ε n L , m ,total e ε L , m ,total s m A o , m

L , m + 1,total e ε L,m + 1,total s m + 1 A o , m + 1

ln

− ln

(6.58)

− ε L , m + ε L , m + 1 = 0

or

K m
K m + 1

ε n L , m ,total ε n m + 1

ln

+ ln − ln + ε L , m ,total s m

L , m + 1,total

(6.59)

A o , m
A o , m + 1

− ε L , m + 1,total s m + 1 + ln − ε L , m + ε L , m + 1 = 0

Finally,

K M
K M + 1

A O , M
A O , M + 1

ε I − 1

LN

+ LN + N M LN

L , M ,ACC + ε L , M

ε I − 1

− N M + 1 LN

L , M + 1,ACC + ε L , M + 1

(6.60)

+ ε L , M ( S M − 1 ) − ε L , M + 1 ( 1 − S M + 1 ) + ε I − 1

L , M ,ACC S M

− ε I − 1

L , M + 1,ACC S M + 1 = 0

Using Eq. ( 6.60 ), this leads to a system of m − 1 equations for m = { 1 ... ( m − 1 ) }
where m is the number of nodes/elements.
It should be noted that the terms A o , m and A o , m + 1 change independently from
each other from the time step i to i + 1, thus simulating a varying cross-sectional
area along the specimen length.
This leads to a system of m − 1 implicit nonlinear equations with m unknowns.
Thus, the final condition required is