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This lecture I have delivered on February 9, 2012 at the Ecole Nationale d'Ingenieurs de Metz as visiting professor from the Institute of Fundamental Technological Research, Polish Academy of Sciences, IPPT PAN, Warsaw, Poland.
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PLASTICITY
Problems related with asymmetry of elastic
range: polymer & metallic materials
Ryszard B. Pęcherski
Institute of Fundamental Technological Research,
Polish Academy of Sciences, Warsaw
&
AGH University of Science and Technology, Krakow
February 9, 2012
1. Introduction
2. Physical motivation
3. Experimental foundations
4. The Burzyński hypothesis of material effort
accounting for asymmetry of elastic range
4. Experimental specification of asymmetry
of elastic range
5. Conclusions
Introduction
Asymmetry of elastic range for isotropic materials
is related with:
- pressure sensitivity of the limit state,
- influence of the Lode angle on the limit state,
- dependence of the limit state on both:
pressure and Lode angle.
limit state – limit of linear elasticity or yield limit.
Lode angle – related with the third invariant of
stress deviator.
Explanation of the Lode angle concept
From: Y. Bai, T. Wierzbicki, A new model of
metal plasticity and fracture with pressure and
Lode angle, Int. J. Plasticity, 24, 1071-1096, 2008
Physical motivation
Two experimental observations about
metals by P.W. Bridgman [1947]:
no influence of hydrostatic pressure on yielding,
incompressibility for plastic straining,
became the basic tenets of classical metal plasticity.
Percy Williams
Bridgman
(1882 – 1961)
1946 Nobel Prize
Bridgman, P.W., 1947,"The Effect of Hydrostatic Pressure on the Fracture of Brittle
Substances," Journal of Applied Physics, Vol. 18, p. 246.
Physical motivation
P.W. Bridgman: Studies in Large Plastic Flow
and Fracture with Special Emphasis on the
Effects of Hydrostatic Pressure, [1952], p. 64:
By the time the last series of measurements was being
made under the arsenal contract, however, skill in
making the measurements had so increased, and
probably also the homogeneity of the material of the
specimens had also increased because of care in
preparation, that it was possible to establish a definite
effect of pressure on the strain hardening curve.
Experimental foundations
W.A. Spitzig et al. [1975],
O. Richmond, W.A. Spitzig [1980],
W.A. Spitzig, O. Richmond [1984].
4330 steel
2 1
31
1Y
I c aI
c ap
a c
Drucker-Prager
yield condition
C. D. Wilson:
A Critical Reexamination
of Classical Metal
Plasticity, J. Appl. Mech.,
2002, 69, 63-68
Experimental foundations triaxial state of stress in notched
specimen:
results of
tension test
alluminium alloy: 2024-T351 Al
Yield criteria:
result of calculations with use
of J2 theory
Experimental foundations
From: Y. Bai, T. Wierzbicki [2008]
New plasticity model is required
Material effort
Material effort is the state of material point of
continuous body produced by the increase of internal
forces (stresses) and resulting in the change of the
strength of chemical bonds in the RVE of condensed
matter under investigation.
A measure of material effort is
required to estimate the distance
of considered state of stress from
the postulated surface
of limit states. Y. Bai, T. Wierzbicki [2008]
Energy-based hypotheses of material effort
Energy as a multilevel scalar quantity can be assumed
as the universal measure of the change of the
strength of chemical bonds – material effort.
E. Beltrami [1885] – the density of total elastic energy.
M.T. Huber [1904] - the density of elastic energy of
distortion.
J.C. Maxwell [1936] - the density of elastic energy of
distortion (in private letter 1856).
Density of elastic energy
2 2 2
1 2 1 3 2 3
1
12f
f v
2
1 2 3
1 2
6v
E
shear modulus , Young modulus , Poisson ratioG E
volumetric
change
distortion
Distribution of elastic energy density for isotropic solids
derived by Stokes [1855] and Helmholtz [1907]:
Elastic energy density for moderate strains
2
2 2 2
1 2 3 1 2 3
ε - infinitesimal strain
2
22 2 2
1 2 3 1 2 3
Lame constant
,2
ln , 1,2,3
1 1 2
i i
e e e e e e
e i
E
L. Anand, On H. Hencky’s approximate strain-energy function for moderate
deformations, Journal of Applied Mechanics (trans. ASME), 46, 1979, 78-82.
W. BURZYŃSKI: Study on Material Effort Hypotheses, Lwów,
1928 (in Polish) ; English translation: Engineering Transactions,
vol. 57, No. 3-4, 185-215, 2009.
2
1 2
2
3
2 2
2
, 0, 0;
0, 0,
,
6
;
0
,
,
,3 3
39 3 0
3
,
I II III
I II III
I I
C T
eq f Y
I III
Y
T
Y
C
Y
S S
C TC T C TY YY Y Y Y
S
C TY Y
eq
S
vf
G
Energy based hypothesis of material effort
pp
p p
p K
f
v
density of elastic energy of distortion
density of elastic energy of volume change
I II III
W. T. BURZYŃSKI (1900-1970)
„Study on Material Effort Hypotheses”,
Lwów 1928 – PhD thesis (in Polish).
„Ueber die Anstrengungshypothesen”,
Schweizerische Bauzeitung, 94, 259-162,1929.
changevolumeofenergyelasticofdensity
distortionofenergyelasticofdensity
v
f
2 2
2 2
0 0
' '23
39 3 0
3
0 1
13
,
, :
e m m
f m v cr
m
T C T CC T T CY Y Y YY Y Y Y
etr
σ σ σ
2 2
0 0
0
3
T C
Y Y
Huber Mises Hencky
condition
Huber-Mises-Hencky cylinder
Burzynski-Drucker-Prager cone
Burzynski-Torreparaboloid
ellipse
m
e
YC
T
Y3
C
Y3
YT
0 3 T C
Y Y 0 3 T C
Y Y
2
03 T C
Y Y
Criteria resulting from Burzyński’s hypothesis
SDE
SDE – Strength Differential Effect C
Y
T
Y
k
G.Vadillo, J. Fernandez-Saez, R.B. Pęcherski, Some application
of Burzyński yield condition in metal plasticity, Material and
Design, 2011
Experimental specification
Coincidence of the experimental results with the modified yield locus
for metallic materials (P.S. Theocaris [1988]).
1.3C
Y
T
Y
k
Limit surface for steel and cupper – historical data
according to Burzynski criterion
Teresa Frąś with use of Mathcad 15
3 s r ck k k
Experimental specification
The yield locus for various polymers revealing the SDE for
(P.S. Theocaris [1988]).
1.3C
Y
T
Y
k
Limit surface for polymers – historical data
according to Burzynski criterion
3 s r ck k k
Teresa Frąś with use of Mathcad 15
Limit surface for grey cast iron – historical data
according to Burzynski criterion
Teresa Frąś with use of Mathcad 15
3 s r ck k k
Limit surface for Al2O3 foam
23 s r ck k k
Teresa Frąś with use of Mathcad 15
Extension of Burzyński hypothesis accounting
for Lode angle effect
Extension of Burzyński hypothesis accounting
for Lode angle effect
The idea of shear-compression test
D. Rittel, S. Lee, G. Ravichandran, Exp. Mech. [2002]
Originally the idea of the
applications of SCS
was used for obtainning
the stress-strain
characteristics of metallic
materials with symmetric
elastic range under
quasi-static and dynamic
conditions.
3 ,S C C T
Y Y Y Huber-Mises condition
Analytical study of the shear-compression test
(M.Vural, A. Molinari, N. Bhattacharya, Analysis of slot orientation
in shear-compression specimen (SCS), Exp. Mech., 2010)
Boundary conditiongs:
displacements
tractions
0, 0 for 0 y xu u y
sin( ) cos( ) for x x y w
h
x y
P
A
B
D
C
*
w
cos( ) sin( ) 0 for xy yy y w
0 on ABCDze t
' ' ' '0 on BCCB and ADDAxe t
22
22
cossin4
cos
2
3
3
4:
3
2
hyyxyeq
εε
22 cossin4
cos
wyy
22 cossin4
sin2
wxy
yyzz
tD
Pyyxyeq
2222 cossin4cos
2
34
4
3:
2
3SS
Analytical study of the shear-compression test
x y
w
h
x y
P
A
B
D
C
*
State of stress and strain for w << t
000
00
00
xy
xy
ε
ε
ε
000
0
0
p
p
xy
xy
σ
23 3: 2 3
2 2eq xy xy ' '
σ σ
w
t
210
exp p
eeq qk kP
D t
1
1eq
k h
22
1 cossin4cos2
3k
222
cossin4
cos
2
3
k
Analytical results
h
x y
P
A
B
D
C
*
Experimental investigations in the lab of
the Division of Applied Plasticity, IPPT
shear
compression
specimen
(SCS)
dimensions: L= 20.0 mm D = 7.0 mm w = 2.0 mm t = 1.0 mm = 45 h = 1.42w
w
h
t
L
D
AlMg 5%SiC Composite
Experimental results
Load – displacement curve
Numerical simulation of the shear-compression test
Assumptions:
element: C3D8
friction: 0.0001
vertical displacement: 1.0 mm
number of elements: 11540
number of nodes: 13871
ABAQUS Standard
Approximation of material characteristic
eqeqeq
C
DCBA exp1
19.3018
21.264
02.21
24.235
D
C
MPaB
MPaA
Results of numerical simulation versus experiment
Paraboloidal criterion of Burzyński
T
Y
C
Yk
C
YT
Y
S
- yield limit in compression
- yield limit in tension
- yield limit in shear
1σ :3
1m
3:
2eq ' '
σ σ
2 2 21
3 1 9 1 4 02
C
m m eq Yk k k
3 C TY YSfor paraboloid of revolution
Solution of elasto-plasticity problem
0
0,,
C
Y
qepmp
m
q
e
p
C
Yem
tt
t
e
q
t
e
t
t
e
t
t
e
t
e
q
CG
CKG
CKGG
GKCK
' '
222
2
'
21
'
12
2
11
9
33621
σσ
σ11σI11ε
σ '
Stiffness matrix:
Newton- Raphson
q
t
ee
p
t
mm
G
K
3
G.Vadillo, J. Fernandez-Saez, R.B.
Pęcherski, Some application of Burzyński
yield ocndition in metal plasticity, Material
and Design, 2011
The UMAT was programmed
and implemented in Abaqus
by Marcin Nowak, IPPT
G.Vadillo, J. Fernandez-Saez, R.B.
Pęcherski, Some application of Burzyński
yield ocndition in metal plasticity, Material
and Design, 2011
Identification of the strength differential factor k
by means of FEM simulation of compression test
1
3
S C
Yk
235.24MPaC
Y
For AlMg 5%SiC:
15.1k
126.64MPaS
C
Y
T
Y
k
The preliminary results
Skład chemiczny:
Si 0.2 -0.8
Fe 3.5 - 0.7
Cu 0.4 - 4.5
Mn 0.4 -1.0
Mg < 1.0
Cr < 0.1
Zn < 0.25
Ti + Zr Al < 0.25
Np
eqeq BA
6.0
70
220
N
MPaB
MPaA
Aluminum alloy – PA6
Experimental results obtained in the lab of the
Division of Applied Plasticity, IPPT
o
p
eqeqgD
Pkk
21 exp
hk
p
eq
1
1
22
1 cossin4cos2
3k
222
cossin4
cos
2
3
k
o
p
eqeqgD
P
2.0185.0
h
p
eq
Analytical results
Rittel et al.
(2002)
Vural et al.
(2010)
Effect of Lode angle
2
3
3
2
271
eq
J
jkikij SSSJ3
13
kkp 3
1
eqq
The influence of the slit angle θ
= 45
Literature
W. BURZYŃSKI: Study on Material Effort Hypotheses, Lwów, 1928
(in Polish) ; English translation: Engineering Transactions, vol. 57,
No. 3-4, 185-215, 2009.
D.Rittel, S. Lee, G. Ravichandran, A Shear-Compression Specimen
for Large Strain Testing, Experimental Mechanics, 2002
M.Vural, A. Molinari, N. Bhattacharya, Analysis of Slot Orientation
in Shear-Compression Specimen (SCS), Experimental Mechanics,
2010
G.Vadillo, J. Fernandez-Saez, R.B. Pęcherski, Some application of
Burzyński yield ocndition in metal plasticity, Material and Design,
2011
Literature
R.B. Pęcherski, P. Szeptyński, M. Nowak, An extension of Burzyński
hypothesis of material effort accounting for the third invariant of stress
Tensor, Archives of Metallurgy and Materials, 56, 503-508, 2011.
M. Nowak, Ostrowska-Maciejewska, R. Pęcherski, P. Szeptyński, Yield
criterion accounting for the third invariant of stress tensor deviator.,
Part. I. Proposition of the yield criterion based on the concept of influence
functions, Engineering Transactions, 59, No. 4, 2011.
Thank you for your attention
MARIA SKŁODOWSKA CURIE (1867-1934)
born in Warsaw, Poland