Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and...

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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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

Agadir, July 08-12, 2013 9R.B. Pcherski, Modelling of inelastic behaviour of materials

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


List of contents

Part II. Phenomenological description

of micro-shear banding


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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



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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)


1. Introduction



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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


1. Introduction



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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


1. Introduction



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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.


1. Introduction



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Explanation of the Lode angle concept


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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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]


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


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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25

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]).


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.


27

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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29

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.


30


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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.


1 Introductrion


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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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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]).


(K. Nalepka [2012c])

tetragonal

1 I t d t i


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34


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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35


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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36


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.


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


38

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]).



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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


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


The hypothesis of variable limit energy

of volume change and distortion

Ueber die Anstrengungshypothesen,Schweizerische Bauzeitung, 94, 259-162,1929.



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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



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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44

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


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



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46


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



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47


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


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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


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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


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Some more detail references


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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.


55/134

Agadir, July 08-12, 2013R.B. Pcherski, Modelling of inelastic

behaviour of materials 55




Examples of the applications of paraboloid


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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.


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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


3T C

Y Y Y


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mild steel & copper

historical dataconfirmation ofBurzynski




Lorraine, Metz - supervisors:



3T C

Y Y Y

[1926]

[1935]


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polymers

historical dataconfirmation of

Burzynski




Lorraine, Metz, supervisors:



3T C

Y Y Y

The application of Burzyski yield condition


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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.


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glassy metals

Lund&Schuh [2003]confirmation of

Burzynski




Lorraine, Metz, supervisors:



3T C

Y Y Y

Limit surface for Al2O3 foam


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Limit surface for Al2O3 foam

Teresa Fr with use of MATHEMATICA



3 T CY Y Y

Burzyski paraboloid yield criterion


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Burzyski paraboloid yield criterion

for the polycarbonate

T. Fr et al. Engineering Transactions [2010]

Burzyski yield paraboloid criterion for the


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metal matrix composite: 6061 +2Zr + 20 Al2O3


Burzyski yield paraboloid criterion for the


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metal matrix composite: 75 Cr + 25 Al2O3


Some criteria resulting from Burzyskis hypothesis


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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.



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p y y g



G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, Someapplication of Burzyskiyield condition in metalplasticity, Material and Design, [2011].

Elasto-plasticity with paraboloid


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p y p

Burzyski yield condition

G.Vadillo, J. Fernandez-Saez, R.B. Pcherski, [2011]


Return mapping integration algorithm


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Return mapping integration algorithm



Solution of elasto-plasticity problem


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



FEM calculations of notched specimens.M h fi t


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Mesh refinement




FEM calculations of notched specimens.E i t l d t f Wil [2002]


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Experimental data of Wilson [2002]



FEM calculations of notched specimens.C f t ti l l ti ith i t


73/134

Confrontation calculations with experiment



Agadir, July 08-12, 2013 73

R.B. Pcherski, Modelling of inelastic


FEM calculations of notched specimens.

C f t ti f l l ti ith i t


74/134

Confrontation of calculations with experiment



Agadir, July 08-12, 2013 74



FEM calculations of notched specimens.


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Agadir, July 08-12, 2013 75



FEM calculations of notched specimens.Confrontation calculations with experiment


76/134

Confrontation calculations with experiment


Agadir, July 08-12, 2013 76






FEM calculations of notched specimens.Confrontation of calculations with experiment


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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


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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



FEM calculations of deformation of Inconel.Confrontation of experiment with calculations


79/134

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 79





80/134



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





81/134



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




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The idea of shear-compression test


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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



Analytical study of the shear-compression test


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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



Analytical study of the shear-compression test


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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



State of stress and strain for w


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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



Analytical results


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



Experimental investigations in the lab ofthe Division of Applied Plasticity IPPT


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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



AlMg 5%SiC Composite


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Agadir, July 08-12, 2013 89



Experimental results


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Agadir, July 08-12, 2013 90



Numerical simulation of the shecompression test


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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



Approximation of material characteristic


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eqeqeq

C

DCBA exp1

19.3018

21.264

02.2124.235

D

C

MPaBMPaA

Agadir, July 08-12, 2013 92



Results of numerical simulation versus

experiment


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experiment

Agadir, July 08-12, 2013 93



Paraboloidal criterion of Burzyski


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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



Identification of the strength differential factorkby means of FEM simulation of compression test


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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



Aluminum alloy PA6


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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



Experimental results obtained in the lab of the


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Division of Applied Plasticity, IPPT

Agadir, July 08-12, 2013 97



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



Experiment versus analytical results


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Agadir, July 08-12, 2013 99



Analysis of the effect of Lode angle


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2

3

3

2

271

eq

J

jkikij SSSJ313

kkp3

1

eqq

Agadir, July 08-12, 2013 100



The influence of the slit angle


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= 45


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Literature


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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




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1. Inntroduction


2. Phenomenological description of inelastic flow viscoplasticity theory accounting for micro-shear

banding


Part II. Phenomenological description

of micro-shear banding

Outline


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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.



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


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.


Physical motivation


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.


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



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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]).


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.


Multiscale hierarchy of shear bands inpolycrystalline metals


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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]


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)


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


Multiscale hierarchy of shear bands inpolycrystalline metals


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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



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



Literature


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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)



Multiscale hierarchy of shear bands

Schematic illustration


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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]



Some relations of the model


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118

System of active micro-shear bandsas surface of strong discontinuity


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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)


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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


Contribution function of shear banding

identification with use of the channel-die test


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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


Account for the change of deformation path (Lode angle )K. Kowalczyk Gajewska, R.B. Pcherski (2005)


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( ( ))1

oMS

ff a b

epp

pp

D

Ddet

2

36)cos(

))cos(1()(


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]


Simple compression

(experimental data: Anand et al.)


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124

Simple shear

(experimental data: Anand et al.)

numerical simulations: K Kowalczyk Gajewska&R B Pcherski [2009]


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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


Balance of plastic deformation power in RVE


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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


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12

0

2 2 2

0

1

1 (1 )(1

( ) , 0


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Viscoplasticity model accounting

for SDE

G


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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

=


Viscoplasticity model accounting

for micro-damage

G


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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=



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3 1 3 0x10vp s

SBf

Dynamic compression




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Concluding remarks

1. The novelty of presented approach consists in derivation of energy-based


134/134

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

Ryszard B. Pecherski, Evolution of elastic range accounting for strength differential effect and...

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