MATERIAL MODELLING OF CAST IRON
4.3 STANDARD PLASTICITY MATERIAL MODEL
4.3.2 USER-DEVELOPED SUBROUTINE ISOTROPIC STRAIN HARDENING MATERIAL MODEL
The A B A Q U S standard program has intcrfaccs that allow new material m odels, including the A B A Q U S cast iron m odel, to be added using the Fortran programming language. T o im plem ent a material m odel, the Fortran based subroutine requires users to assign the stress and strain states, the Jacobian matrix o f the constitutive m odel and so on. The fo llo w in g section describes a procedure used by the present author to create a general isotropic strain hardening material m odel which used the von M ises yield criterion with temperature dependence. Plastic strain generated at elem ent data was used to determ ine the temperature dependent yield stress at current load condition using a linear interpolation from the data input between stress and plastic strain at various temperature conditions show n in Appendix 3. The Fortran coding for this user- developed material m odel is given in Appendix 4. By com paring the results o f applying this m odel with the standard A B A Q U S isotropic hardening material m odel, confidence in the accuracy and robustness o f the user-developed subroutine w as generated
Figure 4 . 1 1 sh ow s the flowchart for the material m odel with temperature dependence when the stress state is in the elastic region. Com parison betw een the current yield stress in tension or com pression and the von M ises stress from elem ent data is the key to determ ine whether the stress state is beyond the elastic region. Figure 4 .1 2 show s the flowchart when the von M ises stress from elem ent data is greater than the current yield stress. Returning the stresses to the yield surface is achieved using the backward Euler m ethod [61].
T o dem onstrate this m ethod, Figure 4.13 show s the first prediction due to the strain increment {A e}, resulting in the increase o f the stress state from A to B, i.e.
Chapter 4 M aterial M odelling o f Cast Iron
{<rB} = [D ]{A e} + { a A} (4 J 8 >
Sin ce the stress state {gb} at B is located outside the yield surface, the yield function (F) is m ore than zero and the correction o f the stress state from B to C is essential as shown in Figure 4 .1 4 . At this stage, the plastic strain increment {Aepl}can be calculated as follow s:
{A e pl} = A A ( - ^ - ) (4.19)
3{cr}
where AX is the plastic strain multiplier. The yield function (F) is a function o f the current yield surface ( a 0) and von M ises stress ( o e) defined as
F = a e -cr0( e pl) (4 -2 0 )
A s a result o f reducing the stress state using the backward Euler procedure, the stress state at C, {C c }, becom es:
{<tc}= {<tb} - [ D ] { A £ p'} (4-21) A pplying Equation 4 .1 9 yields :
{<Tc ) = ( a Bl- A A B[ D ] - ^ i <4 -22) d{(7} or 3F (4 23) {A<JC} ={(TC} - { ( T B} = -A A B[ D ] — — d{<r}
From the von M ises yield criterion [61], the plastic strain m ultiplier (AX) is equal to the where [D] is the elastic stiffness matrix defined in Figure 4.11 .
equivalent plastic strain A e plcalculated from A e pl}: { A e pl}.
A ccording to Equation 4 .2 2 , the plastic strain multiplier,AA,b, at B can be determined by firstly using the Taylor expansion as show n in Equation 4.24:
3 F T 3 f dcy , (4 -2 4 ) AF = Fc - FB = —— - { A g } + ° A £ pl = 0
c B 3 { a } d a 0 d £ p'
Chapter 4 Material Modelling o f Cast Iron
or
■c TJ , 3F d o
Fc = F b + --- {Acrr } + ---7 -A e p (425)
c B aio-} c d ( j 0 d e p' (4 -0 )
3F
W ith —— = -1 fro m E q u a tio n 4 .2 0 , AA = A e pl ,F C = 0 and {Acrc } from E q u atio n 4.23 , th e m a n ip u la tio n o f E q u a tio n 4.25 yields the value o f the p lastic strain m u ltip lie r (AXB) a t B:
Fb (4-26)
A /l
3{cr} 9{cr} 5 { e p }
F ro m th e flo w c h a rt in F ig u re 4 .1 2 , the n ew yield su rface fu n ctio n at C is again checked. If F c is g re a te r th an zero, th e se co n d co rrection is app lied to red u c e the stress state to the n e x t yield su rfa c e as sh o w n in th e fee d b a c k loop o f F ig u re 4.1 2 and in th e Gr<32 plane o f F ig u re 4.1 5 . T h e stress state at D , {cfo}, an d the p lastic strain m u ltip lie r at p oint C,
AXC, are o b ta in e d u sin g a sim ila r d eriv atio n as show n in E q u a tio n s 4 .2 7 and 4.28:
3F 5 {a } {<7D} = {a c } - AXC [D] , at C (4.27) A k r = C dF " m i dF | d a - (4 '28) 5{cr} 3{cr} 3 { e pl}
T he b ack lo o p p ro c e d u re sh o u ld be rep e a te d until the cu rren t yield su rface function is red u ced to zero. A fte r th e yield c riterio n is satisfied, the Jaco b ian m atrix [D t] is updated as sp e cifie d in th e flo w c h a rt o f F ig u re 4.12.
Chapter 4 M aterial M odelling o f Cast Iron
T o d eriv e th is new Ja co b ia n m atrix , the stress state in cre m e n t {Aa} and th e g e n e ra lise d
p lastic strain m u ltip lie r (AX) are in trod uced. E q uation 4 .2 9 show s that the stress state
in crem en t {Aa} can b e d e riv e d fro m the stiffn ess m atrix [D], the total strain increm ent {Ae} an d th e p la stic strain in crem en t {Aepl}:
{A a} = [D][{Ae} - { A epl}] R e p la cin g {Aep1} fro m E q u a tio n 4 .1 9 gives:
0 F {Act} = [D][{Ae} - AA——-]
d{o}
(4.29)
(4.30)
T h e g e n e ra lise d p la stic strain m u ltip lie r (AX) can be c a lc u la te d from the yield su rface in cre m e n t (AF) in E q u a tio n 4 .2 4 by su b stitu tin g A e pl = AA as sh ow n in E q u atio n 4.31:
3 F r 5F d o .
AF = - --- { A a } + - --- — ^ A A = 0
3{a}
3 a „ 3 e pl (4.31)5F
R e p la cin g --- = - 1 an d su b stitu tin g E q u a tio n 4 .3 0 into E q u a tio n 4.31 give:
d o n T hu s, J F _ t
3{a}
5F [D ][{ A e } -A A ——-]3{a}
d o de AA = 0 AA = - 3F3{a}
[D]{Ae} 3F [D] 3 F + do,3{a}
3{a} 3{ep'}
(4.32)
(4.33)
A p p ly in g (AX) fro m E q u a tio n 4.33 into E q u a tio n 4 .3 0 gives:
{Act} = [D ] {Ae} 3F 3{cr} [D] {A e } 3 { a } 3 { a } 3 { e p'} 3F 3{a} (4.34) 99
M a n ip u la tin g E q u a tio n 4 .3 4 gives:
Chapter 4 Material Modelling o f Cast Iron
{ A d } = [D] [I]- dF 3F 5 { d } 9{a} [D] 5{ct} 3{cr} 3 { e p } {Ae} (4.35) yj T h e refo re, th e n ew Ja co b ia n m atrix [D t] is:
/ 5F 3 F t [D , ] = [D] [I]- 3{fj} 3 { a } [D] 5{<t} 5{cr} 3 { £ p } (4.36) 4 .3 .3 IN V E S T IG A T IO N O F IS O T R O P IC H A R D E N IN G M A T E R IA L M O D E L S T h e p u rp o se o f th is sectio n is to in v estig ate and valid ate the iso tro p ic h arden ing m aterial m o d el d e v e lo p e d by the au th o r u sin g the abo v e theory. T o this end, co m p arison b etw een th e sta n d a rd an d u se r-d e v elo p e d su b ro u tin e iso tro p ic m aterial m odels w as c arried o ut u sin g th e F E m o del d e fin e d in S ection 4.1.1. B oth m aterial m o d els used the m o n o to n ic stre ss-stra in c u rv e in te n sio n fro m th e c a st iron m aterial tests, C h ap ter 3. T he full d etails o f m aterial p ro p ertie s u sed are su m m arised in A p p e n d ix 3. T he top o f the g aug e len g th w as su b je c ted to a te n sile lo ad up to 90 M P a at a c o n sta n t tem p eratu re o f 300 °C.
F ig u re 4 .1 6 sh o w s th e resp o n se o f th ese m aterial m o dels co m p a red to the m ono to nic stre ss-stra in in p u t data. It can b e seen that both m aterial m o d els after y ield in g follow e x actly th e e x p e rim e n ta l m o n o to n ic curve. T h u s the u s e r-d e v e lo p e d iso tro p ic h ard enin g m aterial m o d el has b een v a lid a te d a g a in st the stan d ard A B A Q U S m aterial m odel.
Chapter 4 M aterial M odelling o f Cast Iron