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MODELLING OF INELASTIC BEHAVIOUR OF MATERIALS
Evolution of elastic range accountig for
strength differential effect andmicro-shear banding
Ryszard B. Pcherski
Institute of Fundamental Technological Research,
Polish Academy of Sciences, Warsaw
Agadir, July 08-12, 2013
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Professor Piotr Perzyna (1931-2013)
passed away June 22, 2013
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelastic behaviour of materials 2
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Acknowledgement
The part of these lectures originated during my
visits in Metz as le professeur invit de lEcole
Nationale dIngnieurs de Metz (National
Engineering School of Metz) within the years(2009 2012). I appreciate the work and
discussions with professor Alexis Rusinek on
dynamic behaviour of materials.
Udine, July 16-20, 2012 R.B. Pcherski, Inelastic flow and failure ofmetallic solids
3
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Invitation to Poland
Warsaw Krakow
Ryszard B. PCHERSKI
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Departments of the Institute of Fundamental
Technological Research, Polish Academy of Science:
Physical Acoustics
Computational Science
Mechanics of Materials Div. Applied Plasticity
Strength of MaterialsIntelligent Technologies
Theory of Continuous Media
Ultrasound
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1. Introduction
experimental motivation Strength Differential Effect
observations main assumptions the notion of material effort
2. Energy-based limit condtions for materials
with asymmetry of elastic range isotropic solids examples of aplications
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Part I. Asymmetry of elastic range
List of contents
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
10
3. Elasto-plasticity theory accounting for
paraboloid Burzyski yield surface numerical approach examples of experimental verification
4. Identification methodolgy
List of contents
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1. Inntroduction
experimental motivation
2. Phenomenological description of inelastic flow
viscoplasticity theory accounting for micro-shearbanding
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List of contents
Part II. Phenomenological description
of micro-shear banding
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
12
3. Phenomenological approach to the evolution
of yield surface (elastic range)
discussion of a new methodology
examples
4. Concluding remarks
List of contents
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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.
Agadir, July 08-12, 2013 13R.B. Pcherski, Modelling of inelasticbehaviour of materials
1. Introduction
experimental motivation
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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 definiteeffect of pressure on the strain hardening curve.
(citation from C.D. WilsonA Critical Reexamination of Classical Metal Plasticity,
J. Appl. Mech., 2002, 69, 63-68)
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1. Introduction
experimental motivation
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W.A. Spitzig et al. [1975],
O. Richmond, W.A. Spitzig [1980],
W.A. Spitzig, O. Richmond [1984].
4330 steel
2 1I c aI
Drucker-Prager
yield condition
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1. Introduction
experimental motivation
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C. D. Wilson:
A Critical Reexamination
of Classical Metal
Plasticity, J. Appl. Mech.,
2002, 69, 63-68
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
Agadir, July 08-12, 2013 16R.B. Pcherski, Modelling of inelasticbehaviour of materials
1. Introduction
experimental motivation
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Bai, T. Wierzbicki, A new model of metal
plasticity and fracture with pressure and
Lode angle, Int. J. Plasticity, 24, 2008.
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1. Introduction
experimental motivation
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Explanation of the Lode angle concept
Agadir, July 08-12, 2013 18R.B. Pcherski, Modelling of inelasticbehaviour of materials
3 det( ) ,J s1
( )3
trs 1
Lode angle expressed by means of the second
and third invariants of stress deviators:
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
19
Explanation of the Lode angle concept
Lode angle dependence
of the cross-section of
the yield surface given
at the octahedral plane,
cf. M. Nowak et al.
EngineeringTransactions,
59, 273-281, [2011].
S. K. I y e r, C. J. L i s s e n d e n, Multiaxial constitutive
model accounting for the strength-differential in
Inconel 718, Int J Plast 19, 2055-2081 (2003).
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[mm]
Investigations of notched round
bar specimens - A60 steel
V2 V4 V6
U1 U2 R2
[mm]
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1. Introductionexperimental motivation
T. Fr, A. Rusinek, R. Pcherski [2010]
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The larger value ofhydrostatic stress,
the larger deviation of
the load predicted from
the theory.
5.60 2.3 1.7 3.8 7.6 4.5
U1 U2 V2 V4 V6
smoothR2
differenceinvalueofmax
experimentalandsimulated
load[kN]
1. Introductionexperimental motivation
max hydrostaticstress [GPa]
1.181.021.01 0.860.82
0.780.24
0
10
20
30
40
50
60
70
0 0,1 0,2 0,3 0,4 0,5
Geometry: V4.
Quasi static tension: 0.001 1/sAbaqus simulation: standard, axi-sym. el.
V4 num. modellingV4 experiment
load
[kN]
displacement [mm]
2J
The effects of stress concentration in the notched round bar specimens
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T.Fr, A. Rusinek, R. Pcherski [2010]
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1. IntroductionSDE - Strength Differential Effect
SDE is observed in many materials, e.g.:
- geological materials
- high strength steels & hard deformable alloys- ultra fine grained and nano metals
- cast iron
- polymers0
1 2 3
0
, 1, ( , , )f J J J k
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1. IntroductionSDE - Strength Differential Effect
1. D.C. Drucker [1972], W.A. Spitzig et al. [1975], O. Richmond,W.A. Spitzig [1980], W.A. Spitzig, O. Richmond [1984].
2. P. S. Theocaris, Failure criteria for isotropic bodies revisited,
Engineering Fracture Mechanics, 51, 239264, 1995, collectedthe data of the first basic experiments for different materials:
- G.I. Taylor & H. Quinney [1931] - copper and mild steel: .
- R.C. Grassi & I. Cornet [1949]; L.F. Coffin, Jr. [1950] graycast-iron: .
- various polymers: .
3
1.3
1.3
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SDE - Strength Differential Effect
Coincidence of the experimental results with the modified
yield locus (P.S. Theocaris [1995]).
1.3
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SDE - Strength Differential Effect
Coincidence of the experimental results with the
SDE modified yield locus (P.S. Theocaris [1995]).
3
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SDE - Strength
Differential
Effect
The yield locus forvarious polymers
revealing the SDE
(P.S. Theocaris [1995]).
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1.3
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The following observations can be drawn from
the discussed experimental investigations:
it appears that in some cases the commonly usedHuber-Mises yield condition is not sufficient for
adequate description of material behaviour,
the asymmetry of elastic range (SDE) should be taken
into account, the initial anisotropy can play important role in the
adequate simulations of material behaviour.
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1. Introductionobservations
1 I t d ti
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1. Introduction observations
Asymmetry of elastic range for isotropic materialsis related with:
pressure sensitivity of the limit state SDE is observed
influence of the Lode angle on the limit state limitsurface becomes asymmetric
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.
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
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1. Introduction main assumption
The novelty of proposed approach is based on
the hypothesis that certain portions of elasticenergy density accumulated in the deformed
body can be applied to define the measure of
material effort.
1 I t d t i
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1. Introductrion
the notion of material effort
Consider deformable continuous body. The material point ofthe body corresponds to the Representative Volume Element
(RVE) of the considered condensed matter.
External loading applied to the deformed body changes on
the atomic level the relative positions of the constituents of
matter.
This produces variation of the energy of the system and
results in the change of chemical bonding strength.
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
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The notion of material effort
Material effort known since long in German literature onmechanics asAnstrengung, in Russian as napryazhennostand
in Polish as wytenie has beenused rather intuitevely.
It can be defined more precisely as a state of material
point of the loaded deformable body related with the
change of chemical bonding strength in the RVE of
the condensed matter.
Agadir, July 08-12, 2013 31R.B. Pcherski, Modelling of inelasticbehaviour of materials
1 Introductrion
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
32
1. Introductrionthe notion of material effort physical interpretation
(K. Nalepka [2012c])
Illustration of the EM
observations of thechange of relative posi-
tions of the constituents
of the matter in the
aluminum oxide-metal
interface as an exampleof the studies how the
change of the atomic
structure of the boundary
influences the energy of
atomic interactions (cf.
K. Nalepka and R. B.
Pcherski [2009]. [2010]).
1 I t d t i
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
33
Illustration of the EM
observations of the change ofrelative positions of the
constituents of the matter in
the aluminum oxide-metal
interface. An example of the
studies how the change of theatomic structure of the
boundary influences the
change of the symmetry of Cu
structure: cubic tertragonal
(cf. K. Kowalczyk-Gajewska etal. [2003], K. Nalepka and R.
B. Pcherski [2009], [2010]).
1. Introductrionthe notion of material effort physical interpretation
(K. Nalepka [2012c])
tetragonal
1 I t d t i
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
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1. Introductrionthe notion of material effort physical interpretation
Tertragonal
deformation pathsand the change of
interaction energy
In the Cu component
of the interface
calculated with use theoriginally specified
atomic model
by K. Nalepka
[2012a], [2012b].
1 Introductrion
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
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1. Introductrionthe notion of material effort physical interpretation
The change of the
interaction energyIn the Ni component
of the interface Nialuminum oxide
calculated with use
the originallyspecified atomic
model by K. Nalepka
and R.B. Pcherski[2010].
1 I t d t i
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
36
1. Introductrionthe notion of material effort physical interpretation
The interaction energy
changes for tetragonaldeformation paths of Cu
constrained in the
interface Al2O3-Cu with
use of the proposed
atomic model. Thecomparison with
the results known in the
literature K. Nalepka
[2012a], [2012b].
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Measure of material effort
A measure of material effort is required to assess the
distance of the considered state of stress from the
postulated surface of limit states.
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials
37
1 2 3, ,
An example of the yieldsurface considered in the
paper: M. Nowak et al.,
EngineeringTransactions,
59, 273-281, [2011].
limit stateYield surce of Inconel 718
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Measure of material effort
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1 2 3( , , )
limit state
A limit value of the given
measure of material effortdefines the limit state of the
considered material.
The limit state can be related
with the limit of linearelasticity,onset of yielding,
failure etc.
The limit state is commonlyrelated with the notion of the
strength of material.
3. Energy-based limit condtions for materials
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3. Energy based limit condtions for materials
with asymmetry of elastic range.
Energy as a multilevel scalar quantity can be assumed
as apropriate 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 ofdistortion (in private letter to William
Thomson (Lord Kelvin)[1856]).
Agadir, July 08-12, 2013 39R.B. Pcherski, Modelling of inelasticbehaviour of materials
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Density of elastic energy
2 2 2
1 2 1 3 2 3
1( ) ( ) ( )
12f
G
f v
2
1 2 3
1 2( )
6v
E
Agadir, July 08-12, 2013 40R.B. Pcherski, Modelling of inelasticbehaviour of materials
distortion volume change
shear modulus , Young modulus, Poisson ratioG E
Decomposition of elastic energy density for isotropic solids
derived by Stokes [1855] and Helmholtz [1907]:
Th h th i f i bl li it
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W. BURZYSKI: Study on Material Effort Hypotheses,Lww, 1928 (in Polish) ; English translation: EngineeringTransactions, vol. 57, No. 3-4, 185-215, 2009.
1 2 3
3 3
(
,
) vf
pp
p K=
f
v
density of elastic energy of distortion
density of elastic energy of volume change
Agadir, July 08-12, 2013 41R.B. Pcherski, Modelling of inelasticbehaviour of materials
The hypothesis of variable limit energy
of volume change and distortion
Ueber die Anstrengungshypothesen,Schweizerische Bauzeitung, 94, 259-162,1929.
The hypothesis of variable limit energy
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2 23 Y YT CY Y Y
Huber Mises Hencky
condition
. , 0, 0;
. 0, 0, ;
. , 0,
I II III
I II I
T
II
I II I
Y
Y
Y YI
C
I
I
II
III
Agadir, July 08-12, 2013 42R.B. Pcherski, Modelling of inelasticbehaviour of materials
The hypothesis of variable limit energy
of volume change and distortion
( , , ) ( , , )T CY Y Y
K
I II III
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43
Basic modes of stress realized in experiments
. Uniaxial tension , 0, 0;
. Uniaxial compression 0, 0, ;
.Pure shear , 0,
I II III
I I
T
Y
C
Y
Y
I III
I II I YII
I
II
III
1 2 3
1 2 3
1 2 3
1
. Biaxial uniform tension: , , 0;
. Biaxial uniform compression: 0, , ;
. Triaxial uniform tension: , , ;
. Triaxial uniform compression:
TT TT
Y Y
CC CC
Y Y
TTT TTT TTT
Y Y Y
IV
V
VI
VII 2 3, ,CCC CCC CCC
Y Y Y
The Burzyski hypothesis expressed in the
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2
2
2 2 2
1 2 2 3 3 1
21 2 3
1 2 3
6
1
0
3
( (
(
( ) ) )
)
)(
Y
Y
T C
Y Y
T C
Y Y
C T T C
Y Y Y Y
The Burzyski hypothesis expressed in the
form of quadric in the principal stress space
The different relations between the parameters
determine the separate limit conditions.
, ,T CY Y Y
Discussion of some special cases
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45
Discussion of some special cases
ofBurzyskis quadric
W. Burzyski [1928], p. 115, Fig. 64 and 65.
2
ellipse or circle
3 T CY Y Y
11
equivalent stress pressure coordinatesf
2 12f f
G
p
Discussion of some special cases
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46
Discussion of some special cases
ofBurzyskis quadric
W. Burzyski [1928], p. 115, Fig. 66 and 67.
3 T CY Y Y
1 1parabola
trace of the half of a cylinder
equivalent stress pressure coordinatesf
p
Discussion of some special cases
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47
Discussion of some special cases
ofBurzyskis quadric
W. Burzynski [1928], p. 116, Fig. 68 and 69.
233
T C T C Y Y Y Y
YT CY Y
hyperbola
2
3
T CY Y
Y T CY Y
trace of the half of a cone
equivalent stress pressure coordinatespf
Ilustration of some criteria resulting from
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Huber-Mises-Hencky cylinder
Burzynski-Drucker-Prager cone
Burzynski-Torreparaboloid
ellipse
m
e
YC
T
Y 3
C
Y 3
YT
3 T CY Y Y 3T C
Y Y Y
23
T C
Y Y Y
Ilustration of some criteria resulting from
Burzyskis hypothesis
SDE
SDE Strength Differential EffectC
Y
T
Y
k
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, Some applicationof Burzyski yield condition in metal plasticity, Material andDesign, 2011Agadir, July 08-12, 2013
2
3
T CY Y
Y T CY Y
Criteria resulting from Burzyskis hypothesis
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 49
Criteria resulting from Burzyski s hypothesis
in the space of principal axes
1 2 3, ,
3 T CY Y Y- elipsoid of revolution (circular)
3 T CY Y Y - paraboloid of revolution (circular)
23
T CY Y
Y T C
Y Y
- hyperboloid of revolution of two
sheets (one of sheets is considered)
2
3
T CY Y
Y T CY Y
- cone of revolution of two sheets
(one of sheets is considered only)
The symmetry axis:1 2 3
An extension of Burzyski hypothesis of material
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
50/134
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 50
R.B. Pcherski et al., Arch. Met. Mat. [2011]M. Nowak et al., Engineering Transactions, [2011]
An extension of Burzyski hypothesis of materialeffort accounting for the effect of Lode angle
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
51/134
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 51
R.B. Pcherski et al., Arch. Met. Mat. [2011]M. Nowak et al., Engineering Transactions, [2011]
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
52/134
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 52
R.B. Pcherski et al., Arch. Met. Mat. [2011]M. Nowak et al., Engineering Transactions, [2011]
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
53/134
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 53
R.B. Pcherski et al., Arch. Met. Mat. [2011]M. Nowak et al., Engineering Transactions, [2011]
Some more detail references
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
54/134
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelasticbehaviour of materials 54
Some more detail references
Nalepka K., Symmetry-based approach to parametrization of
embedded-atom-method interatomic potentials,
Computational Materials Science, 56, 100-107, 2012.Nalepka K., Efficient approach to metal/metal oxide interfaces within
variable charge model, European Physical Journal B, 85,1-12, 2012.
K. Nalepka, R.B.Pcherski, Modelling of the interatomic interactionsin the Copper crystal applied in the structure (111)Cu/(0001) Al2O3,
Archives of Metallurgy and Materials, vol. 54, pp. 511-522, (2009).
K. Nalepka and R.B. Pcherski, The Strength of the interfacial bondin the ceramic matrix composites Al2O3-Ni, Mechanics and Control,
29, pp. 132-137, (2010).
R.B. Pcherski, P. Szeptyski, M. Nowak,An extension of Burzynskihypothesis of material effort accounting for the third invariant of
stress tensor,Archives of Metallurgy and Materials, 56, 503-508, 2011.M. Nowak, J. Ostrowska-Maciejewska, R.B. Pcherski, P. Szeptyski,Yield criterion accounting for the third invariant of stress tensor deviator,
Engineering Transactions, 59, 273-281, 2011.
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
55/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 55
3. Elasto-plasticity theory accounting for
paraboloid Burzyski yield surface numerical approach examples of experimental verification
4. Identification methodolgy
Examples of the applications of paraboloid
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
56/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 56
Examples of the applications of paraboloid
Burzyski limit criterion
T. Fr, Z.L. Kowalewski, R.B. Pcherski, A. Rusinek,Applications of Burzyskifailure criteria.Part I. Isotropic materials with asymmetry of elastic range,
Engineering Transactions, 58, 3-13, 2010,
Fr, T., Nowak, Z., Perzyna, P., Pecherski, R.B. Identification ofthe model describing viscoplastic behaviour of high strength
metals, Inverse Problems in Science and Engineering,
19, 17-30, 2011.
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
57/134
grey cast iron
historical dataconfirmation of
Burzynski
paraboloid criterion
Teresa Fr, (with use ofMATHEMATICA),
PhD thesis, [2013], Univ.
Lorraine, Metz -
supervisors:
R. Pecherski & A. Rusinek
Agadir, July 08-12, 2013 57R.B. Pcherski, Modelling of inelastic behaviour of materials
3T C
Y Y Y
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
58/134
mild steel & copper
historical dataconfirmation ofBurzynski
paraboloid criterion
Teresa Fr, (with use ofMATHEMATICA),
PhD thesis, [2013], Univ.
Lorraine, Metz - supervisors:
R. Pecherski & A. Rusinek
Agadir, July 08-12, 2013 58R.B. Pcherski, Modelling of inelastic behaviour of materials
3T C
Y Y Y
[1926]
[1935]
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
59/134
polymers
historical dataconfirmation of
Burzynski
paraboloid criterion
Teresa Fr, (with use ofMATHEMATICA),
PhD thesis, [2013], Univ.
Lorraine, Metz, supervisors:
R. Pecherski & A. Rusinek
Agadir, July 08-12, 2013 59R.B. Pcherski, Modelling of inelastic behaviour of materials
3T C
Y Y Y
The application of Burzyski yield condition
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
60/134
y y
for nano-materials (T. Fr et al. [2009])The section of the
Burzyski paraboloid
yield condition
Coulomb-Mohrcondition
Agadir, July 08-12, 2013 60R.B. Pcherski, Modelling of inelastic
behaviour of materials
C.A. Schuh,
A.C. Lund,
Atomistic
basis for the
plastic yield
criterion of
metallic
glass,Nature
Materials, 2,
449-452,
2003.
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
61/134
glassy metals
Lund&Schuh [2003]confirmation of
Burzynski
paraboloid criterion
Teresa Fr, (with use ofMATHEMATICA),
PhD thesis, [2013], Univ.
Lorraine, Metz, supervisors:
R. Pecherski & A. Rusinek
Agadir, July 08-12, 2013 61R.B. Pcherski, Modelling of inelastic behaviour of materials
3T C
Y Y Y
Limit surface for Al2O3 foam
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
62/134
Limit surface for Al2O3 foam
Teresa Fr with use of MATHEMATICA
Agadir, July 08-12, 2013 62R.B. Pcherski, Modelling of inelastic
behaviour of materials
3 T CY Y Y
Burzyski paraboloid yield criterion
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
63/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 63
Burzyski paraboloid yield criterion
for the polycarbonate
T. Fr et al. Engineering Transactions [2010]
Burzyski yield paraboloid criterion for the
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
64/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 64
metal matrix composite: 6061 +2Zr + 20 Al2O3
T. Fr et al. Engineering Transactions [2010]
Burzyski yield paraboloid criterion for the
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
65/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 65
metal matrix composite: 75 Cr + 25 Al2O3
T. Fr et al. Engineering Transactions [2010]
Some criteria resulting from Burzyskis hypothesis
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
66/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 66
Some criteria resulting from Burzyski s hypothesis
in the space of principal axes
2
3
T CY Y
Y T CY Y
The one sheet of circular hyperboloid can appear also
a versatile possibility to approximate the experimental
data.
3. Elasto-plasticity theory accounting for
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
67/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials 67
p y y g
paraboloid Burzyski yield surface numerical approach examples of experimental verification
4. Identification methodolgy
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, Someapplication of Burzyskiyield condition in metalplasticity, Material and Design, [2011].
Elasto-plasticity with paraboloid
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
68/134
p y p
Burzyski yield condition
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Agadir, July 08-12, 2013
Return mapping integration algorithm
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
69/134
Return mapping integration algorithm
Agadir, July 08-12, 2013
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Solution of elasto-plasticity problem
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
70/134
y
( , , ) 0
0
C
m e Y
p q
e m
m p e qp
C
Y
2
' '11 12 21
2' '
222
3 3(1 ) 2 6
9( )
q t tt t t
e e e
q t t
t t
e e
GKG KGK KC G C C
GC
' 1 1 I 1 1
Stiffness matrix:
Newton- Raphson
qtee
p
t
mm
G
K
3
G.Vadillo, J. Fernandez-Saez, R.B.
Pcherski, Some application of Burzyskiyield ocndition in metal plasticity, Material
and Design, 2011
The UMAT was programmed
and implemented in Abaqus
by Marcin Nowak, IPPT
Agadir, July 08-12, 2013 70R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.M h fi t
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
71/134
Mesh refinement
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Agadir, July 08-12, 2013 71R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.E i t l d t f Wil [2002]
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
72/134
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Experimental data of Wilson [2002]
Agadir, July 08-12, 2013 72R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.C f t ti l l ti ith i t
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
73/134
Confrontation calculations with experiment
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Experimental data of Wilson [2002]
Agadir, July 08-12, 2013 73
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.
C f t ti f l l ti ith i t
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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Confrontation of calculations with experiment
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Experimental data of Wilson [2002]
Agadir, July 08-12, 2013 74
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
75/134
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Confrontation of calculations with experiment
Experimental data of Wilson [2002]
Agadir, July 08-12, 2013 75
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of notched specimens.Confrontation calculations with experiment
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
76/134
Confrontation calculations with experiment
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Agadir, July 08-12, 2013 76
R.B. Pcherski, Modelling of inelastic
behaviour of materials
S. K. I y e r, C. J. L i s s e n d e n, Multiaxial constitutive
model accounting for the strength-differential in
Inconel 718, Int J Plast 19, 2055-2081 (2003).
FEM calculations of notched specimens.Confrontation of calculations with experiment
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
77/134
Agadir, July 08-12, 2013
R.B. Pcherski, Modelling of inelastic
behaviour of materials 77
Confrontation of calculations with experiment
S. K. I y e r, C. J. L i s s e n d e n, Multiaxial constitutive
model accounting for the strength-differential in
Inconel 718, Int J Plast 19, 2055-2081 (2003).
Comparison of empirically proposed
ellipsoidal yield surface with rotational
paraboloid resulting from energy-basedBurzyski hypothesis of material effort(R.B. Pcherski et al. Arch. Met. Mat. [2011]
FEM calculations of deformation of InconelMaterial characteristics
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
78/134
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Material characteristicsExperimental data ofS. K. I y e r, C. J. L i s s e n d e n [2003]
Agadir, July 08-12, 2013 78
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of deformation of Inconel.Confrontation of experiment with calculations
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
79/134
Confrontation of experiment with calculations
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Experimental data of
S. K. I y e r, C. J. L i s s e n d e n [2003]
Agadir, July 08-12, 2013 79
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of deformation of Inconel.Confrontation of experiment with calculations
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
80/134
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Confrontation of experiment with calculations
Experimental data ofS. K. I y e r, C. J. L i s s e n d e n [2003]
Agadir, July 08-12, 2013 80
R.B. Pcherski, Modelling of inelastic
behaviour of materials
FEM calculations of deformation of Inconel.Confrontation of experiment with calculations
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
81/134
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]
Confrontation of experiment with calculations
Experimental data of
S. K. I y e r, C. J. L i s s e n d e n [2003]
Agadir, July 08-12, 2013 81
R.B. Pcherski, Modelling of inelastic
behaviour of materials
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
82/134
Agadir, July 08-12, 2013
R.B. Pcherski, Modelling of inelastic
behaviour of materials 82
5. Identification methodolgy
The idea of shear-compression test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
83/134
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 metallicmaterials with symmetric
elastic range under
quasi-static and dynamic
conditions.
3 ,S C C TY Y YHuber-Mises condition
Agadir, July 08-12, 2013 83
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Analytical study of the shear-compression test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
84/134
y y p
(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 0y xu u y
sin( ) cos( ) for x x y w
h
xy
P
A
B
D
C
* w
cos( ) sin( ) 0 for xy yy y w
0 on ABCDzet
' ' ' '0 on BCCB and ADD Axet
Agadir, July 08-12, 2013 84
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Analytical study of the shear-compression test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
85/134
2 2
2 22 4 3 cos( ): ( )3 3 2 4sin ( ) cos ( )
eq xy yyh
2 2
cos( )
4sin ( ) cos ( )yy w
2 2
2 sin( )
4sin ( ) cos ( )xy
w
yyzz
2 2 2 23 3 3: (4 ) cos( ) 4sin ( ) cos ( )2 4 2
eq xy yy
P
D tS S
y y p
xy
w
h
xy
P
A
B
D
C
*
Agadir, July 08-12, 2013 85
R.B. Pcherski, Modelling of inelastic
behaviour of materials
State of stress and strain for w
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
86/134
000
00
00
xy
xy
000
0
0
p
p
xy
xy
23 3: (2 ) 32 2
eq xy xy
' '
w
t
Agadir, July 08-12, 2013 86
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Analytical results
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
87/134
1 2
0
( ) exp ( )qp
eqe k kP
D t
1
1
( )eq
k h
2 2
1
3( ) cos( ) 4sin ( ) cos ( )
2k
22 2
3 cos( )( )
2 4sin ( ) cos ( )k
h
xy
P
A
B
D
C
*
Agadir, July 08-12, 2013 87
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Experimental investigations in the lab ofthe Division of Applied Plasticity IPPT
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
88/134
the Division of Applied Plasticity, IPPT
shearcompress ion
specimen
(SCS)
dimensions:
L= 20.0 mm
D = 7.0 mm
w = 2.0 mm
t = 1.0 mm
= 45
h = 1.42w
wh
t
L
D
Agadir, July 08-12, 2013 88
R.B. Pcherski, Modelling of inelastic
behaviour of materials
AlMg 5%SiC Composite
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
89/134
Agadir, July 08-12, 2013 89
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Experimental results
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
90/134
Agadir, July 08-12, 2013 90
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Numerical simulation of the shecompression test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
91/134
Assumptions:
element: C3D8
friction: 0.0001
vertical displacement: 1.0 mm
number of elements: 11540
number of nodes: 13871
ABAQUS Standard (Marcin Nowak [2011])
Agadir, July 08-12, 2013 91
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Approximation of material characteristic
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
92/134
eqeqeq
C
DCBA exp1
19.3018
21.264
02.2124.235
D
C
MPaBMPaA
Agadir, July 08-12, 2013 92
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Results of numerical simulation versus
experiment
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
93/134
experiment
Agadir, July 08-12, 2013 93
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Paraboloidal criterion of Burzyski
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
94/134
T
Y
C
Yk
C
Y
T
YS
- yield limit in compression
- yield limit in tension
- yield limit in shear
1 :3
1m
3( : )
2eq
' '
2 2 21 3 1 9 1 4 02
C
m m eq Y k k k
3 C TY YSfor paraboloid of revolution
Agadir, July 08-12, 2013 94
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Identification of the strength differential factorkby means of FEM simulation of compression test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
95/134
by means of FEM simulation of compression test
1
3
S C
Yk
235.24MPaCY
For AlMg 5%SiC:
15.1k
126.64MPaS
C
Y
T
Y
k
Agadir, July 08-12, 2013 95
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Aluminum alloy PA6
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
96/134
Skad 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
( )p N
eq eqA B
6.070
220
NMPaB
MPaA
y
Agadir, July 08-12, 2013 96
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Experimental results obtained in the lab of the
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Division of Applied Plasticity, IPPT
Agadir, July 08-12, 2013 97
R.B. Pcherski, Modelling of inelastic
behaviour of materials
1
Analytical results
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1 2( ) exp ( )
p
eq eqo
P
k k D g
1
1
( )
p
eqk h
2 2
1
3( ) cos( ) 4sin ( ) cos ( )
2k
22 2
3 cos( )( )2 4sin ( ) cos ( )
k
0.85(1 0.2 )p
eq eq
o
P
D g
h
p
eqRittel et al.
(2002)
Vural et al.
(2010)
Agadir, July 08-12, 2013 98
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Experiment versus analytical results
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99/134
Agadir, July 08-12, 2013 99
R.B. Pcherski, Modelling of inelastic
behaviour of materials
Analysis of the effect of Lode angle
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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2
3
3
2
271
eq
J
jkikij SSSJ313
kkp3
1
eqq
Agadir, July 08-12, 2013 100
R.B. Pcherski, Modelling of inelastic
behaviour of materials
The influence of the slit angle
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101/134
Agadir, July 08-12, 2013 101R.B. Pcherski, Modelling of inelastic
behaviour of materials
= 45
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Agadir, July 08-12, 2013 102R.B. Pcherski, Modelling of inelastic
behaviour of materials
Literature
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
103/134
W. BURZYSKI: Study on Material Effort Hypotheses, Lww, 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 Orientationin Shear-Compression Specimen (SCS), Experimental Mechanics,
2010
G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, Some application of
Burzyski yield ocndition in metal plasticity, Material and Design,2011
Agadir, July 08-12, 2013 103R.B. Pcherski, Modelling of inelastic
behaviour of materials
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1. Inntroduction
experimental motivation
2. Phenomenological description of inelastic flow viscoplasticity theory accounting for micro-shear
banding
Agadir, July 08-12, 2013 104R.B. Pcherski, Modelling of inelastic behaviour of materials
Part II. Phenomenological description
of micro-shear banding
Outline
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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1. Shear banding:
- one of the mechanisms of plastic flow in
metallic materials
- the dominant mechanism of plastic deformation of
ufg, nano-metals and glassy metals.
2. Phenomenological description.
3. Identification of the model for ufg and nano-
crystalline Fe.
4. Concluding remarks.
Agadir, July 08-12, 2013 105R.B. Pcherski, Modelling of inelastic
behaviour of materials
Shear banding
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First observations of shear bands - F. Adcock, The internal
mechanism of cold-work and recrystallization in Cupro-Nickel, J. Inst.
Metals, 27, 1922.
Description of shear banding as the mechanism competing with
dislocation glide; shear banding contribution function -
R.B. Pecherski, Archives of Mechanics [1992], [1997]; Acta
Mechanica [1998]; Technische Mechanik [1998]. Necessity of the description of shear banding contribution in
deformation of ufg and nano-metals, hard deformable materials,
amorphous materials (glassy metals), e.g.: L. Anand, C. Su,A theory
for amorphous viscoplastic materials undergoing finite deformations,
with application to metallic glasses, JMPS, 53, 1362-1396, 2005.
Z. Nowak, P. Perzyna, R.B. Pecherski, Description of viscoplastic flow
accounting for shear banding, Arch. Metall. Mat., 52, 2007.
SBf
Agadir, July 08-12, 2013 106R.B. Pcherski, Modelling of inelastic behaviour of materials
Physical motivation
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SEM image revealing the
deformation and fracture
behaviour of the bulk
glassy Co43Fe20Ta55B31.5alloy rod with a diameter
of 2 mm deformed up to a
true strain of 0.021at698K. Shear bands and
shear deformation
induced fracture are
observed on thespecimen surface.
Akihisa Inoue et al., nature materials, 2, 661-663, 2003.
Agadir, July 08-12, 2013 107R.B. Pcherski, Modelling of inelastic behaviour of materials
Physical motivation
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108/134
An illumination-mode
atomic force microscopeimage of a Berkovich
indentation made in a
Zr-based metallic glass,
illustrating the formationof shear bands on the
surface around the
indent.
J-J. Kim, Y. Choi, S. Suresh and A.S. Argon, Science, 295, 654, 2002.
C.A. Schuh and T.G. Nieh, J. Mater. Res.,19, 46-57, 2004.
Agadir, July 08-12, 2013 108R.B. Pcherski, Modelling of inelastic behaviour of materials
Observations of shear banding - ufg Fe
i t ti hi h t i t d f ti
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d = 980 nm d = 268 nm
quasistatic high-strain rate deformations
Change in deformation mode of ultrafine grained consolidated iron under uniaxial
compression: (a) uniform low-rate deformation with d = 980 nm; (b) non-uniform
low-rate deformation with d = 268 nm and (c) non-uniform high-rate deformation
with d = 268 nm (Jia, Ramesh and Ma [2003]) .
d = 980 nm d = 268 nm
crystallographic slip shear banding
d = 268 nm
Agadir, July 08-12, 2013 109R.B. Pcherski, Modelling of inelastic behaviour of materials
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Evolution and development of shear bands in 268 nm-Fe. Observations of shear bands
at the same location at different nominal strain levels: (a) 3.7%; (b) 7.8%. Loading axis isvertical. Note the development of new shear bands, the broadening of existing shear
bands, and the propagation of a shear band tip (Jia, Ramesh and Ma [2003]).
Agadir, July 08-12, 2013 110R.B. Pcherski, Modelling of inelastic behaviour of materials
Experimental results of Jia, Ramesh and Ma,Acta Materialia, 51 (2003)
Deformation of nano - and ufg metals
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p , , , ( )
SBf - shear banding
contributionfunction
Z. Nowak, P. Perzyna
R.B. Pecherski,
Archives of Metallurgy and
Materials 52, 2007 freepdf available on the Journal
website.
Typical stress-strain curves
obtained for the consolidated
iron under quasistatic and
high-strain-rate uniaxial
compression.
Agadir, July 08-12, 2013 111R.B. Pcherski, Modelling of inelastic behaviour of materials
Multiscale hierarchy of shear bands inpolycrystalline metals
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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p y y
Homogeneous deformation by the crystallographic slip replacedby heterogeneous and localized one produced by micro-shearbands leads to significant reduction of global hardeningrate.
This phenomena may be enhanced and stabilized by the
subsequent (e.g. cyclic) changes of the strain path duringthe deformation process KOBO method
The controlled cyclic strain path changesenable also the refinement of the materialmicrostructure.
A. Korbel and W. Bochniak [2004]
Agadir, July 08-12, 2013 112R.B. Pcherski, Modelling of inelastic behaviour of materials
Multiscale hierarchy of shear bands in
polycrystalline metals
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Trace ofthe cluster
of MSB
CSB
[Dziado, 1993]
(courtesy of profesor Andrzej Korbel)
polycrystalline metals
Micro-shear band - long and verythin (ca. 0.1 ) sheet-like regionof concentrated and intensiveplastic shear crossing grainboundaries without deviation
Particular MSB operates only onceand develops fully in very shorttime
m
Agadir, July 08-12, 2013 113R.B. Pcherski, Modelling of inelastic behaviour of materials
Multiscale hierarchy of shear bands inpolycrystalline metals
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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polycrystalline metals
Position of shearbands depends onthe scheme ofloading duringdeformation. In rolled
sheet - inclined byabout 35o to therolling plane andorthogonal to the
specimen lateral face
[A. Korbel and W. Bochniak, 2004]
Traces of
the clusters
of MSB
Agadir, July 08-12, 2013 114R.B. Pcherski, Modelling of inelastic
behaviour of materials
Multiscale hierarchy of shear banding in
polycrystals
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Trace ofmsb cluster
CSB
[Dziado, 1993] (The micrograph provided by A. Korbel)
polycrystals
msb micro-shear bandsCSB coarse slip band
msb
msb clusters
shear bands
Agadir, July 08-12, 2013 115R.B. Pcherski, Modelling of inelastic
behaviour of materials
Literature
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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1. R.B. Pcherski, Modelling of large plasticdeformations based on the mechanism of micro-shearbanding. Physical foundations and theoreticaldescription, Arch. Mech. (1992).
2. R.B. Pcherski, Macroscopic measure of the rate ofdeformation produced by micro-shear banding, Arch.Mech. (1997).
3. R.B. Pcherski, Macroscopic effects of micro-shearbanding in plasticity of metals, Acta Mechanica (1998)
Agadir, July 08-12, 2013 116R.B. Pcherski, Modelling of inelastic
behaviour of materials
Multiscale hierarchy of shear bands
Schematic illustration
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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of multi-level hierarchy
of micro-shear bands:
a) polycrystalline RVE
with the increasing
zone of shear banding,
b) cluster of active
micro-shear bands,
c) a single micro-shear
band (comp. of CSB)
R.B. Pcherski, Macroscopic measure of the rate of deformationproduced by micro-shear banding, Arch. Mech. [1997]
Agadir, July 08-12, 2013 117R.B. Pcherski, Modelling of inelastic
behaviour of materials
Some relations of the model
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118/134
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials
118
System of active micro-shear bandsas surface of strong discontinuity
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Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelastic behaviour of materials 119
R.B. Pcherski, Macroscopic effects of micro-shear banding in plasticity of metals,Acta Mechanica (1998)
Macroscopic measure of the rate of deformationaccounting for shear banding
R.B. Pcherski, Arch. Mech. 49, (1997)
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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, 2
p p p
S SB
pS
SB
p
SB
p p
SB SB
SB
rateof plastic deformation by slip
rate of plastic deformation by shear banding
d d d shear strain ra
df instantaneous shear banding contribution
d
te
D D D
D
D
D D
(1 )
S SB
SB s
ff
SBSBf
Agadir, July 08-12, 2013 120R.B. Pcherski, Modelling of inelastic behaviour of materials
Contribution function of shear banding
identification with use of the channel-die test
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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1. R.B. Pcherski, Continuummechanics description of
plastic flow produced bymicro-shear bands,Technische Mechanik 1998
2. Z. Nowak, R.B. Pcherski,Plastic strain in metals ...
II. Numerical identificationand verification of plasticflow law, Arch. Mech. 2002
)(1
)(33bae
fFf o
MSMS
Agadir, July 08-12, 2013 121R.B. Pcherski, Modelling of inelastic behaviour of materials
Account for the change of deformation path (Lode angle )K. Kowalczyk Gajewska, R.B. Pcherski (2005)
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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( ( ))1
oMS
ff a b
epp
pp
D
Ddet
2
36)cos(
))cos(1()(
Agadir, July 08-12, 2013 122R.B. Pcherski, Modelling of inelastic behaviour of materials
Plane strain compression(experimental data: Anand et al.)
numerical simulations: K Kowalczyk-Gajewska&R B Pcherski [2009]
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numerical simulations: K. Kowalczyk-Gajewska&R.B.Pcherski [2009]
Agadir, July 08-12, 2013 R.B. Pcherski, Modelling of inelastic behaviour of materials 123
Simple compression
(experimental data: Anand et al.)
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numerical simulations: K. Kowalczyk-Gajewska&R.B.Pcherski [2009]
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials
124
Simple shear
(experimental data: Anand et al.)
numerical simulations: K Kowalczyk Gajewska&R B Pcherski [2009]
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numerical simulations: K. Kowalczyk-Gajewska&R.B.Pcherski [2009]
Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic
behaviour of materials
125
V
VsViscoplastic flow law
accounting for shear
b di i li i f
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Representative
Volume Element
traversed by shear
bands
VSB
banding in application for
ufg metals(Z. Nowak, P. Perzyna, R.B. Pecherski,Arch. Metall. Materials, 2007)
S SB
s SBV V V
SBSBf
Volume fraction
of shear banding
Inst. contr.
of shear
banding
Agadir, July 08-12, 2013 126R.B. Pcherski, Modelling of inelastic behaviour of materials
Balance of plastic deformation power in RVE
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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,
, ( ),
(1 )
yield stren
(
gth at sh
(1 )(
ea
1 ) ,
r
) 01
s SB
s s s SB SB SB SB SB
V SBs SB V SB SB V SB
V
s S S SBB B
P P P
P k V P k V V P k V
Vk k f f k f f
k k
k
f
V
for kf
assumption - no hardening
0
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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12
0
2 2 2
0
1
1 (1 )(1
( ) , 0
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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Viscoplasticity model accounting
for SDE
G
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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1
2 2 2
,
( )
1 3( 1) 9( 1) 4
2
p
G G
T
Y
m m e
G G
G F
F
D
=
Agadir, July 08-12, 2013 130R.B. Pcherski, Modelling of inelastic behaviour of materials
Viscoplasticity model accounting
for micro-damage
G
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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1
0
0
2 2
,
( )
, 1(1 ) (1 )(1 )
, , the void growht threshold stress
= 1, [ , ), 3 (1 )(1 )
1 3( 1) 9( 1) 4
2
p
G G
T
Y
VSB tr SB SB
p
tr tr
T V
Y s SB SB
m m
G G
G F
Fg
f k f f
k K
A
k f f
F
D
2
e=
Agadir, July 08-12, 2013 131R.B. Pcherski, Modelling of inelastic behaviour of materials
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3 1 3 0x10vp s
SBf
Dynamic compression
Agadir, July 08-12, 2013 132R.B. Pcherski, Modelling of inelastic
behaviour of materials
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Agadir, July 08-12, 2013 133R.B. Pcherski, Modelling of inelastic
behaviour of materials
Concluding remarks
1. The novelty of presented approach consists in derivation of energy-based
7/22/2019 Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and micro-shear banding. In memoriam Piotr Perzyna
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hypothesis ofmaterial effortwell-founded on the multiscale analysis of
deformation processes and assuming a measure of material effortas a
definite part of elastic energy density on macroscopic level of continuousbody related with critical strain processes.
2. Studying the literature of the subject one can observe that the yield or
failure criteria, which are obtained in the rigorous way from the energy-
based material effort hypothesis proposed originally by (Burzyski, 1928)were later rediscovered again and again for different ranges of empirical
parameters independently by many researchers (R. Pecherski, Engng.
Trans. 2008).
3. Recent studies show that an adequate prediction of dynamic fracture
processes should account for physical mechanisms activated on different
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