Page 255 - Marine Structural Design

Chapter I2 A l%w~ Nonlinear Finite Element Analysis 23 1
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Here, we introduce a kinematic hardening rate Hi, and an isotropic hardening rate H:i for the
full plastic cross-sections, which are defined by:

H:k ={~/aa>T{da}/dlP (12.14)
and
H:i = dg,/dlp (12.15)
Similarly, a strain-rate hardening rate H:, for the full plastic cross-section is defined as:

H:, = dg,/dgp ( 1 2.1 6)
{:r
With these definitions, the consistency condition Eq. (12.13) may be rewritten as:

df = - {do}-(H:k +Hii)dZP - H:,dgP =O (1 2.17)

The subscript "s" in Eqs. (12.14) thru (12.17) indicates generalized values related to the full
beam cross-section. To avoid confusion, the kinematic and isotropic material-hardening rate
obtained from uni-axial tests is denoted by Hi and H,! , respectively.

Introducing a linear interpolation for dlP , we obtain:
dFp = $(+d,)&it + g(l(1- 6)dt (12.18)

where B is a parameter, which will be taken as 1/2 in the numerical examples.
From Eq. (12.18), we find:
gP(,+dr) = [dFp - (1 - B&dtl/(&ft) (1 2.19)
Then the increment of the equivalent plastic strain-rate dgP may be estimated as:

dgP = ;(+dl) - 24) = [dZp - g(ldt]/(&ft) (12.20)
For simplicity, in the following equations, the subscript "t" will be omitted.
Considering Eq. (12.20), the consistency condition Eq. (12.17) is rewritten in the form:

df={af/do)T {da}-(Hlk +H:i + H:,/(8dt))dEP + [H:,/BFP = 0 (12.21)
in the following, the hardening rates H:,,Hikand Hi, in Eq. (12.21) will be discussed.
The isotropic hardening rate for the cross-section is evaluated as follows. Following Ueda and
Fujikubo, (1986) and Fujikubo et al, (1991), the increments of the generalized stress due to
isotropic hardening are estimated to be:
(da} = [H:i]{d~p} (12.22)
and the matrix [H:i] is obtained by first deriving a relationship between the stress increments
and the plastic strain increments for points and after integrating those stresses over the cross-
section. If we use the Von Mises yield criteria and neglect the interaction between shear stress
and axial stresses Eq. (12.23) is obtained: