671017.pdf

0

C−^1 :휀

C−^1 :휀p

휎 0

휎

f(휎)= 0

휎eq

휎yc

휎

Figure 1: The closest-point projection method.

volume.푆eqand푆yc, respectively, represent external actions
and structural resistance. The real stress field must satisfy
the yield criterion;휎=휎yc. Equivalent nodal forces of the
difference between the two stress fields, which is the driving
force of structure deformation, can be defined as unbalanced
force as

ΔU=∫

푉

B푇(휎eq−휎yc)푑푉=F−∫

푉

B푇휎푑푉. (10)

For a given load step, the unbalanced force is the driving
force of deformation in the FE iterations. The principle of
minimum PCE implies that elasto-plastic structures deform
tending to the state where PCE is minimized under certain
actions as

min퐸(휎eq,휎yc), ∀휎eq∈푆eq, 휎yc∈푆yc,

퐸(휎eq,휎yc)=∫

푉

1

2

(휎eq−휎yc):C:(휎eq−휎yc)푑푉.

(11)

Unbalanced force is the measurement between external
actions and structural resistance. The plastic complementary
energy퐸is a norm of unbalanced force. Once퐸reaches its
minimum value퐸min, the unbalanced force stays constant
and the structure suffers steady plastic flow until failure
occurs. Only if퐸min=0,thestructureisstable.
In elastoplastic FEM iteration, PCE and unbalanced force
canbecalculatedbythefollowingsteps.First,thenearest
stable stress field휎yc 1 can be achieved from an arbitrary
equilibrium stress field휎eq 1. Then, the nearest equilibrium
stress field휎eq 2 is also obtained from휎yc 1 .Finally,structure
stress state tends to the two closest stress fields휎and휎eq,
as shown inFigure 2. The iteration converges if the plastic
complementary energy is reaching a steady valueΔ퐸 0 .For
ideal model,Δ퐸 0 is equal to its minimum valueΔ퐸min.If
Δ퐸min=0, the structure remains stable and the stress field휎
is the real stress response. Otherwise, the structure is unstable
and PCE is a magnitude estimation of its global instability.
Unbalanced force reflects the structural failure behavior,
including failure position and pattern.

휎yc 1

휎yc 2

휎yc 3

휎eq 1

휎eq 2

휎eq 3

Seq

Syc

Emin

Figure 2: Demonstration of the elastoplastic iteration in FEM anal-

ysis.

Figure 3: Sketch map of the precrack specimen.

2.3. Explanation of Unbalanced Force in Viscoplastic Damage

Model.Stress state beyond the yielding surface is unaccept-

able for perfect elastoplasticity. It exists only in the iteration

process of FEM calculation. However, in viscoplasticity mod-

els, it is of explicit physical significance as the driving force of

visco-plasticity deformation. The Perzyna associative visco-

plasticity strain rate could be stated as [ 25 , 26 ]

휀vp푖푗̇ =

{

{

{

0, 푓 ≤ 0,

ΓvpΦ(푓)

휕푄

휕휎

,푓>0,

(12)

where휀vp푖푗̇ is the visco-plasticity strain rate.Γvpis the viscosity

parameter.푓=푓(휎,휅) = 0is the yield function.푄is

plastic potential function, and for associative flow rule,푄=

푓(휎).Φ(푓)is overstress function characterized by the yield

function, commonly expressed asΦ(푓) = (푓/푓 0 )푛.푓 0 is a

reference constant of the same dimension with yield function

푓.

According to Perzyna’s visco-plasticity theory, visco-

plasticity strain rate is in proportion to overstress function.

휕푓/휕휎represents the direction of휀푖푗̇vp,whileΦ(푓)reflects

the magnitude of휀̇

vp

푖푗. If the overstress function is restricted

to yield function, plastic flow occurs only on the condition