Theory Implementation in ANSYS Experience · Theory Implementation in ANSYS. Experience. ......
Transcript of Theory Implementation in ANSYS Experience · Theory Implementation in ANSYS. Experience. ......
Use of Zarka‘s Method at FHL
Theory
Implementation in ANSYS
Experience
Hartwig HübelAndreas Kretzschmar
(FH Lausitz)
Cracow March 11, 2003
Background, History
in nuclear industry: elastic-plastic response required for life assessment under cyclic loading:
strain range ∆ε cyclic accumulated (ratchet-) strain εacc
displacements at end of life
20 years ago:only rough estimates available as „simplified“ elastic-plastic methods (knock-down factors), without capturing effects of:
individual geometry of structure kind of loading (thermal, displacement-, force-controlled, ...)
hardening of material
3
Background, History (ctd.)
publication of some examples obtained by Zarka with Zarka‘s method revealed:
however: method remained largely obscure:
superior quality of results (approximations almost exact) obtained with little numerical effort (few lin. elastic analyses)
description of theory appeared to be not complete some assumptions seemed to be heuristically motivated
by end of seventies Zarka‘s method came up, claiming it was able to account for these effects
why that ambiguous?
when advantageous?question:
some applications outside Zarka‘s team showed poor results
4
Outline of Zarka‘s methodmaterial model:
- yield surface must exist- hardening required:
» purely kinematic
σ´ - space
simplified method:
⇒“infinite” ratcheting ruled out:limit state = ES or PS, associated with ”finite” ratcheting
only partial information: - no evolution of σ, ε with cycles - only post-shakedown quantities - loss in accuracy (Vp, σ, ε)
only reduced effort:- direct method- few elastic analyses sufficient
+ some “local” calculations
- temperature dependence:only size of yield surface, i.e. σy (not E, ν, Et)
» linear or multilinear
σ
ε
σyEt
5
Specialisation, Modification
Zarka-method
specialisation,
modifications
simplified theory of plastic zones (VFZT)
- moderate levels of strain (e.g. no deep drawing)- isotropic material - no strength differential effects (tensile = compressive)- linear and bilinear kinematic hardening (bilin., trilin. σ−ε)- Mises yield surface- temperature-dependent yield stress- mainly cyclic loading between 2 extreme states of loading - logic of iterative improvement:
identifying Vp, estimating TIV Y
6
Reformulation of “Exact” Plastic Theory
residual state can be obtained by 1 mea with modif. elastic materialdata (E*) and modif. loading (ε0), if Vp and Y are known
idea of Zarka-Method: estimation of Vp and Y
yesεel-pl = σel-pl/Eεel-pl = σel-pl/E + ξ/Cexact(el-pl)
loadappliedin el. zone Ve (=V-Vp)in plastic zone Vpkind of
analysis
yesεf.el = σf.el/Eεf.el = σf.el/Efictitious elastic (f.el)
initial strainε0 inVp
=> ε*= ρ•(1/E)=> ε*= ρ•(1/E*) + ε0with 1/E*=1/E+1/Cand ε0=Y/C
modif. elast.(mea)
noεel-pl-εf.el =(σel-pl-σf.el)/Eε* ρ
εel-pl-εf.el =(σel-pl-σf.el)/E + ξ/Cε* ρ ξ = ρ+Y
differencedef
here: - uniaxial stress state- bilinear σ−ε-diagram (kinem. hardening)- monotonic loading
σ
εEEtσy
σ
εpl
C
7
Mon. Loading: Estimation of Yhere: bilinear σ−ε-diagram (lin. kinem. hardening)
σ´ is unknown; estimation not possible, since center ξ (i.e. εpl) also unknown
Y is unknown; but estimation possible, since center σ´f.el is known
Mises yield condition:f(σ´,ξ,σy) f(Y,σ´f.el,σy)
Y = ξ−(σ´−σ´f.el)ρ´
def
projection
multiaxial stress: Vp and Y unknown; local estimation of Y sufficient, since field effects of Y are relieved compared to σ (=heuristic)
σ´ - space:
σ´
ξ=C·εpl*.. .
Y – space:
Y
*σ´f.el. ..
8
uniaxial stress: Y exactly known by projection; only Vp unknown
numerical effort required: few linear analyses (fictitious elastic and modified elastic) local calculations
iterative improvement of Vp and Y
Mon. Loading: Algorithmfictitious elastic analysis (f.el) of max. loading σf.el
estimate extension of plastic zone Vp Vp
modify elastic parameters:Ε, ν → E* , ν*
modify loading:real loading → initial strain ε0
estimate TIV Y Y
modified elastic analysis (mea) ρ
elastic-plastic solution by superposition, e.g. σel-pl=σf.el + ρ σel-pl,εel-pl,
displace.etc.
9
Mon. Loading: Iterative Improvement
iterative improvement of Vp:
→
≤>−
VeVp
yplel
v σσ
iterative improvement of Y:
with from previous iterationplelij
−σ
Y – space:
Y
*σ´f.el.
∗Y*=-ρ’
projectionwith from previous iteration
ijρ
10
iteration of Vp partly necessary exact results obtained after maximum 1 iteration
σ
ε
EEtσy
u
• hand calculation ⇒ analytical solution• e.g. for specific configuration (A1/A2, l1/l2; Et/E):
Example 1: Mon. Loading: 2 Bars in Series
1
1.5
2
2.5
3
1 2 3σf.el / σy
withoutiteration
1 iterationexact
εel-pl
εf.el
11
σ
εE
Etσy
• hand calculation ⇒ analytical solution; for specific Et/E:
after 1 iteration close to exact results
∆w0
Example 2a: Mon. Loading: Beam of ideal I-Profile
withoutiteration
1
1.2
1.4
1.6
1.8
2.0
2.2
2.4
1 1.5 2 2.5 3 3.5σf.el / σy
1 iteration
exact
εel-pl
εf.el
12
∆w0 ε
σ
EEt=0σy
hand calculation possible for Et=0 (via initial stress):
exact
1 iteration
withoutiteration
top surface
middle surface
mid
leng
th
clam
ped
plastic zone
1
1.5
2
2.5
3
3.5
1 1.25 1.5 1.75 2 2.25
σf.el / σy
exact 1 iteration
withoutiteration
εel-pl
εf.el
Example 2b: Mon. Loading: Beam of rect. Cross Section
Implementation in FE-codes
extension for specific structures (trusses, beams):- trilinear σ – ε- increm. projection in ES, if > 2 states of loading
implementation in any commercial FE code (ANSYS, ABAQUS, ADINA, ...) usually not easy:
- access to elastic material properties and application of initial strain usually inconsistent:E element, ε0 Gauss or nodal point
- modification of elastic material properties and of loading (initial strain) required during iterative solution
* ** *
Ve
Vp- exact definition of Vp somewhat arbitrary
(Vp, if σv>σy in > 50% of Gauss-points )
implementation at FHL in ANSYS :for general structures (1D, 2D, 3D):
- bilinear σ – ε- T-dependent σy- mon. + cyclic loading (2 states of loading)
ε
σ
E
EtT1
σy,2Et
T2
σy,1
13
Ways of Implementationin ANSYS
1. VEFZOT: APDL-Macro (=ANSYS parameter language; slow)initial strain simulated by:* thermal strain at nodal points* orthotropic coefficient of thermal expansion ( deviatoric)* orthotropic directions changing from el. to el. sometimes poor qual.
2. Userpl.f: FORTRAN user subroutine (compiling and linking required)initial strain simulated by:* inelastic strain at Gauss points* formally equilibrium iterations performed, although not necessary
3. IS-Macros (APDL-Macro) or Ustress.f (FORTRAN-subroutine):initial strain simulated by initial stress at Gauss points
4. Particular implementation for trusses and bending beams:initial strain simulated by ts and ∆t across section
14
Example 3: Mon. Loading: Hollowed Plate
axial strain
σ
εE
Etσy
displace-ment
+ Miseseffective stress
15
exact VFZTplastic zone
plastic zone
exact VFZT
Example 4: Mon. Loading: Crack in a Plate
σ
ε
EEt
monotonic loading
crack
16
Mises effect. stress:
fictitious elastic
VFZT, 3. mea evolutive
plastic zone
Cyclic Loading: ES: Estimation of Y
elastic shakedown (ES): if Ω exists at each point of structure
Y
*σ´f.el
min
*σ´f.el
max
Ω.∗Y*=-ρ’
Y
*σ´f.elmin
*σ´f.elmax
Ω
.∗Y*=-ρ’
Y
*σ´f.el
min
*σ´f.elmax
Ω∗Y*=-ρ’.
Y – space
*σ´f.el
min
*σ´f.el
maxΩ
load histogram:
timemax
mincyclic load
const. loadtime
Vp, if Y*=-ρ‘ is outside Ω:
17
Example 5: ES: 2 Bars Parallel
F
T
timeF
T
020406080
100120140
0 500 1000 1500 2000 2500no. of cycles
εel-pl/εy
numerical effort required to achieve ES: exact: 1000 cy. * 10 loadsteps * 5 iterations
~ ca. 50 000 linear elast. anal. VFZT: 2 f.el. + 2 mea = 4 linear elast. anal.
02468
101214
0 5 10 15 20 25no. of cycles
εel-pl/εy
exact:
σ
ε
EEtσy
εy
elastic shakedown
εacc
identical results
18
Example 6: ES: Thermal Stratification
σ
ε
EEt
εy
p
∆T
time
≈ elastic shakedown
0 100 200no. of cycles
10
5
0
hoop εpεy
VFZT:good approxim. with 1 mea (i.e. 3 linear elastic analyses)
shakedown
evolutive
hoop εacc
fictitious elastic analysis
evolutiveanalysis
1 2 3 4 5 6 7 meaVFZT
∆Tcoldhot*
p
19
Cyclic Loading: PS: Estimation of ∆Y
range values: Vp, if ∆σv > 2σy:
load histogram:
timemax
mincyclic load
const. loadtime
Y – space
*σ´f.el
min
*σ´f.el
max
*σ´f.el
min
*σ´f.el
max
∆Y *σ´f.el
min
*σ´f.elmax
∆Y
plastic shakedown (PS): if there is nointersection at least at 1 point of structure
σ
ε
εmean
∆ε
*εacc
20
Cyclic Loading: PS: Estimation of Ymean
mean values: Vp, if σv,min > σy or σv,max > σy:
in Ve,∆:projection of Y*± ½ ∆Y on max (or on min, if closer):
*σ´f.el
min
*σ´f.el
max
Ymean
in Vp,∆:
*σ´f.el
min
*σ´f.elmax
½ ∆Y½ ∆Y
Ymean
∗Y*=-ρ’mean
21
Example 7: PS: Beam under N, ∆w0
∆w0
N
N
w0
time
≈
plastic shakedown
σ
εE
Etσy
∆ε
εacc
fictitious elasticanalysis
evolutive analysis
1 2 3 4 5 6 7 meaVFZT
22
Example 7 (ctd.): PS: Beam: ∆ε
23
VFZT:
∆w0
N
N
w0
time≈
plastic shake-down
fictitious elastic
evolutive: 15 cycles
1 mea
2 mea
3 mea
Example 7 (ctd.): PS: Beam: εacc
24
∆w0
N
N
*w0
time≈
plasticshakedown
fictitious elastic
evolutive: 15 cycles
2 mea
3 mea
5 mea
1 mea
VFZT:
4 mea
Example 8: PS: Cylindrical Shell under Axial
Force and TemperatureCL
T
N
radius
axial ∆ε
axial εacc
fictitious elastic analysis
evolutiveanalysis
1 2 3 4 5 6 7 meaVFZT
σ
εE
Et
Et/E=0.03ν=0.3
N
T
time
≈
σNf.el/σy=0.8
σTf.el/σy=2.2 (PS)
25
fictitious elastic Zarka’s method
evolutive (100 cycles)
1 mea 2 mea 3 mea
Example 8 (ctd.): PS: Cylindrical Shell: axial ∆ε
CL
T
N
radius
Application: Ratcheting-Interaction Diagrams
which combination of load parameters yields ES, PS, R for specific configuration of structure and loading? by shakedown theorems no post-shakedown quantities
load 1
load
2
E
ES
PS R
Zarka‘s method: ES, PS known a priori R not possible: subregions ES2, PS2
post-shakedown quantities (∆ε, εacc)
load 1
load
2
E
ES1
PS1
ES2
PS2
26
εel-pl
εy
without iteration
exact after max.1 iteration
0
5
0 1 2 3σt/σy0
2
1
0 1
E
Sa
SbSc
P
exact with-out iteration
exact with1 iteration
σtσy
σpσy
hand calculation for any configuration of loading (T, F) and hard. param. (Et/E):
exact after maximum 1 iteration
Example 9: Ratch-Inter: 2 Bars Parallel
F
T F
T
time
σ
ε
EEtσy
εy
σp/σy = 0.2
σp/σy = 0.5
27
0
1
2
3
4
5
0 1 2 3 4
εy1εel-pl
σ σt y1
0,05
0,2
0,5
1,0
σp σy1
σ
ε
σy1
σy2
g
e
b f h
a
c
d
j
k
l
σpσy1
σtσy1
0 0,5 1
2
1
0
E
S
P
Example 10: Ratch-Inter: 2 Bars Parallel (Trilinear)
28
TP
0 0.2 10.4 0.6 0.8
1
2
3
4
0
σt/σy
σp/σy
0 iteration
1 iteration2 iterations(≈exact) Pc
Sc
Pb
ESa
Sb
close to exact after 2 iterations
0
0.5
1
1.5
0 1 2 3 4
σpσy
= 0.4
0 iteration
εel-pl
εy
σt/σy
1 iteration
2 iterations
≈ exact withoutiteration
exact
p
∆Ttime
p ∆T
Example 11: Ratch-Inter:Bree Problem
hand calculation for any configuration of loading (T, P) and hard. param. (Et/E):
29
Example 12: Ratch-Inter:Beam under N, ∆w0
accumulated strain:
0 0,5 1,00,00
0,50
1,00
1,50
2,00
2,50
3,00
3,50
10,0-12,0
8,0-10,0
6,0-8,0
4,0-6,0
2,0-4,0
0,0-2,0
εacc/εy:
σw0σy
f.el
σN /σyf.el
∆w0
N w0
N
*time≈
max. load level
30
1.Benchmark CUT-FHL:Thick-walled Cylinder
thick walled cylinder under internal pressure
p
material: lin. kin. hardeningσ
εE
Etσy
εy
mon. loading, load level σv
f.el/σy=1.665 timep
combination ν Et/EA 0.49 0.001B 0.49 0.1C 0.3 0.001D 0.3 0.1
31
1.Benchmark CUT-FHL (ctd.):Thick-walled Cylinder
results:combinations A, B, D: all residual stress- and strain-components after 6 mea
close to exact values (difference <1%);combination C (Et/E=0.001, ν=0.3):
res.
stra
in
radius
exactVFZT
inner outer
hoopaxialradial
res.
stre
ss [M
pa]
radiusinner outer
hoop
axialradial
no improvement with more mea loss in accuracy: why?
32
2.Benchmark CUT-FHL: Hertz Contact
lin. kin. hardeningEt/E = 0.1
σ
εEEtσy
Hertz contact (plane stress)
q (parabolic, regularized)
stationary cycl. loading (ES);load level σv
f.el/σy=1.388 timeq
33
2.Benchmark CUT-FHL (ctd.): Hertz Contact
results: residual stresses along vertical section:
sect
ion
top
x
y
1 mea 7 mea 30 mea
top
ρx
ρy
evolutive
slow approxi-mation to exact solution:why ?
34
3.Benchmark CUT-FHL: Bree Problem
cycl. loading (ES, PS)
p
∆Ttime
thin-walled cylinder under internal pressure and radial temperature gradient
p ∆T
lin. kin. hardening:Et/E = 0.1,T-dependent yield stress
σ
εE
Etσy1
εy1
σy2
40.2Pb
41.91.91.2σt /σy
0.80.80.20.7σp /σy
PcScSbSacombi-nation
f.el
f.el
35
3.Benchmark CUT-FHL (ctd.): Bree Problem
results for combination Pc:
0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 mea
[%]εacc,hoop
εacc,axial
evolutive analysis
VFZT
fictitious elastic analysis
- accum. state at minimum load level:- range values = exact after 2 mea
σy= 200 at inner and outer surface for min. load level (no temperature),σy= 161.1 at inner surface and 238.9 at outer surface for max. load level
(fully developed T-gradient)
εaccreasonable, but not superior: why?
36
Nozzle:T.
σ´ - space
path of fictitious elastic stress
coolantmedium
Further Development: Non-radial Loading
3-Bar Problem: T1 T2 T3T1
time
T2T3
Moving Load:
37
SummaryZarka’s Method (and it’s modifications by FHL) is a simplified method to provide post-shakedown quantities in ES and PS by direct means(∆ε for fatigue, εacc for ratcheting-assessment, displacements)
examples show in case of radial loading:- good approximation to exact solution- at small numerical effort (few linear elastic analyses)- in particular if directional redistribution is not strongly developed- range-values (PS) usually better than mean values
is implemented in commercial FE-program ANSYS for monotonic and cyclic loading
benchmarks between CUT and FHL raised some questions
38