Page 261 - Marine Structural Design

Chapter I2 A Theory of Nonlinear Finite Element Analysis 237

By considering the increments of nodal displacement from time t-dt to time t, measured in the
local coordinate system at time t-dt, we obtain:
sinp = (duZ2 -du,,)/~,
Cos p = (L+.d, + dux, - du,,)/L,
sin e = (duyz - du,, )/L~ (12.62)
case = LJL,
where,

(12.63)

is the distance between nodes 1 and 2 at time t-dt. Furthermore, the angle a is calculated
as:

(12.64)

12.5 Appendix A: Stress-Based Plasticity Constitutive Equations

12.5.1 General
This appendix is written based on a Japanese book authored by Yagawa and Miyazaki (1985).
When the formulation presented in this chapter was made, the author had been inspired by this
book and Yamada (1968). The objective of this Appendix is to describe the basics of plasticity
that may be useful to help understand the mathematical formulation presented in the main
body of this chapter.
In the uni-axial tensile test, when the stress is small, the material behavior is elastic. The
proportional constant E is Young's modulus. If the load is released, the stress will become 0,
and the material will return to its original condition. On the other hand, when the stress
exceeds a limit, permanent deformation may occur. The permanent deformation is called
plastic deformation.
Figure A.l shows a typical stress-strain diagram of metallic materials. The material is in
elastic behavior range until the yield point A, and the stresso and strains are in proportion.
This proportion relationship is called Hook's law. After going over point A, the gradient of the
stress-strain curve decreases, and the gradient, H', is called tangent modulus. If unloading
occurs at point B, the stress will decrease along with B+C, which is parallel to OA. The
residual strain is called plastic strain, E' . On the other hand, the recovered strain
corresponding with CB' is called elastic strain, se. The total strain is the sum of the elastic
strain and plastic strain.

s=se+sp (A-1)
Figure A.2 shows the relationship between stress and plastic strain. The gradient H' in this
stress - plastic strain curve is called strain-hardening rate. Referring to Figs. A.l, A.2, the
following relationship may be obtained.