PLASTICITY

Problems related with asymmetry of elastic

range: polymer & metallic materials

Ryszard B. Pęcherski

Institute of Fundamental Technological Research,

Polish Academy of Sciences, Warsaw

&

AGH University of Science and Technology, Krakow

February 9, 2012

1. Introduction

2. Physical motivation

3. Experimental foundations

4. The Burzyński hypothesis of material effort

accounting for asymmetry of elastic range

4. Experimental specification of asymmetry

of elastic range

5. Conclusions

Introduction

Asymmetry of elastic range for isotropic materials

is related with:

- pressure sensitivity of the limit state,

- influence of the Lode angle on the limit state,

- dependence of the limit state on both:

pressure and Lode angle.

limit state – limit of linear elasticity or yield limit.

Lode angle – related with the third invariant of

stress deviator.

Explanation of the Lode angle concept

From: Y. Bai, T. Wierzbicki, A new model of

metal plasticity and fracture with pressure and

Lode angle, Int. J. Plasticity, 24, 1071-1096, 2008

Physical motivation

Two experimental observations about

metals by P.W. Bridgman [1947]:

no influence of hydrostatic pressure on yielding,

incompressibility for plastic straining,

became the basic tenets of classical metal plasticity.

Percy Williams

Bridgman

(1882 – 1961)

1946 Nobel Prize

Bridgman, P.W., 1947,"The Effect of Hydrostatic Pressure on the Fracture of Brittle

Substances," Journal of Applied Physics, Vol. 18, p. 246.

Physical motivation

P.W. Bridgman: Studies in Large Plastic Flow

and Fracture with Special Emphasis on the

Effects of Hydrostatic Pressure, [1952], p. 64:

By the time the last series of measurements was being

made under the arsenal contract, however, skill in

making the measurements had so increased, and

probably also the homogeneity of the material of the

specimens had also increased because of care in

preparation, that it was possible to establish a definite

effect of pressure on the strain hardening curve.

Experimental foundations

W.A. Spitzig et al. [1975],

O. Richmond, W.A. Spitzig [1980],

W.A. Spitzig, O. Richmond [1984].

4330 steel

2 1

31

1Y

I c aI

c ap

a c

Drucker-Prager

yield condition

C. D. Wilson:

A Critical Reexamination

of Classical Metal

Plasticity, J. Appl. Mech.,

2002, 69, 63-68

Experimental foundations triaxial state of stress in notched

specimen:

results of

tension test

alluminium alloy: 2024-T351 Al

Yield criteria:

result of calculations with use

of J2 theory

Experimental foundations

From: Y. Bai, T. Wierzbicki [2008]

New plasticity model is required

Material effort

Material effort is the state of material point of

continuous body produced by the increase of internal

forces (stresses) and resulting in the change of the

strength of chemical bonds in the RVE of condensed

matter under investigation.

A measure of material effort is

required to estimate the distance

of considered state of stress from

the postulated surface

of limit states. Y. Bai, T. Wierzbicki [2008]

Energy-based hypotheses of material effort

Energy as a multilevel scalar quantity can be assumed

as the universal measure of the change of the

strength of chemical bonds – material effort.

E. Beltrami [1885] – the density of total elastic energy.

M.T. Huber [1904] - the density of elastic energy of

distortion.

J.C. Maxwell [1936] - the density of elastic energy of

distortion (in private letter 1856).

Density of elastic energy

2 2 2

1 2 1 3 2 3

1

12f

f v

2

1 2 3

1 2

6v

E

shear modulus , Young modulus , Poisson ratioG E

volumetric

change

distortion

Distribution of elastic energy density for isotropic solids

derived by Stokes [1855] and Helmholtz [1907]:

Elastic energy density for moderate strains

2

2 2 2

1 2 3 1 2 3

ε - infinitesimal strain

2

22 2 2

1 2 3 1 2 3

Lame constant

,2

ln , 1,2,3

1 1 2

i i

e e e e e e

e i

E

L. Anand, On H. Hencky’s approximate strain-energy function for moderate

deformations, Journal of Applied Mechanics (trans. ASME), 46, 1979, 78-82.

W. BURZYŃSKI: Study on Material Effort Hypotheses, Lwów,

1928 (in Polish) ; English translation: Engineering Transactions,

vol. 57, No. 3-4, 185-215, 2009.

2

1 2

2

3

2 2

2

, 0, 0;

0, 0,

,

6

;

0

,

,

,3 3

39 3 0

3

,

I II III

I II III

I I

C T

eq f Y

I III

Y

T

Y

C

Y

S S

C TC T C TY YY Y Y Y

S

C TY Y

eq

S

vf

G

Energy based hypothesis of material effort

pp

p p

p K

f

v

density of elastic energy of distortion

density of elastic energy of volume change

I II III

W. T. BURZYŃSKI (1900-1970)

„Study on Material Effort Hypotheses”,

Lwów 1928 – PhD thesis (in Polish).

„Ueber die Anstrengungshypothesen”,

Schweizerische Bauzeitung, 94, 259-162,1929.

changevolumeofenergyelasticofdensity

distortionofenergyelasticofdensity

v

f

2 2

2 2

0 0

' '23

39 3 0

3

0 1

13

,

, :

e m m

f m v cr

m

T C T CC T T CY Y Y YY Y Y Y

etr

σ σ σ

2 2

0 0

0

3

T C

Y Y

Huber Mises Hencky

condition

Huber-Mises-Hencky cylinder

Burzynski-Drucker-Prager cone

Burzynski-Torreparaboloid

ellipse

m

e

YC

T

Y3

C

Y3

YT

0 3 T C

Y Y 0 3 T C

Y Y

2

03 T C

Y Y

Criteria resulting from Burzyński’s hypothesis

SDE

SDE – Strength Differential Effect C

Y

T

Y

k

G.Vadillo, J. Fernandez-Saez, R.B. Pęcherski, Some application

of Burzyński yield condition in metal plasticity, Material and

Design, 2011

Experimental specification

Coincidence of the experimental results with the modified yield locus

for metallic materials (P.S. Theocaris [1988]).

1.3C

Y

T

Y

k

Limit surface for steel and cupper – historical data

according to Burzynski criterion

Teresa Frąś with use of Mathcad 15

3 s r ck k k

Experimental specification

The yield locus for various polymers revealing the SDE for

(P.S. Theocaris [1988]).

1.3C

Y

T

Y

k

Limit surface for polymers – historical data

according to Burzynski criterion

3 s r ck k k

Teresa Frąś with use of Mathcad 15

Limit surface for grey cast iron – historical data

according to Burzynski criterion

Teresa Frąś with use of Mathcad 15

3 s r ck k k

Limit surface for Al2O3 foam

23 s r ck k k

Teresa Frąś with use of Mathcad 15

Extension of Burzyński hypothesis accounting

for Lode angle effect

Extension of Burzyński hypothesis accounting

for Lode angle effect

The idea of shear-compression test

D. Rittel, S. Lee, G. Ravichandran, Exp. Mech. [2002]

Originally the idea of the

applications of SCS

was used for obtainning

the stress-strain

characteristics of metallic

materials with symmetric

elastic range under

quasi-static and dynamic

conditions.

3 ,S C C T

Y Y Y Huber-Mises condition

Analytical study of the shear-compression test

(M.Vural, A. Molinari, N. Bhattacharya, Analysis of slot orientation

in shear-compression specimen (SCS), Exp. Mech., 2010)

Boundary conditiongs:

displacements

tractions

0, 0 for 0 y xu u y

sin( ) cos( ) for x x y w

h

x y

P

A

B

D

C

*

w

cos( ) sin( ) 0 for xy yy y w

0 on ABCDze t

' ' ' '0 on BCCB and ADDAxe t

22

22

cossin4

cos

2

3

3

4:

3

2

hyyxyeq

εε

22 cossin4

cos

wyy

22 cossin4

sin2

wxy

yyzz

tD

Pyyxyeq

2222 cossin4cos

2

34

4

3:

2

3SS

Analytical study of the shear-compression test

x y

w

h

x y

P

A

B

D

C

*

State of stress and strain for w << t

000

00

00

xy

xy

ε

ε

ε

000

0

0

p

p

xy

xy

σ

23 3: 2 3

2 2eq xy xy ' '

σ σ

w

t

210

exp p

eeq qk kP

D t

1

1eq

k h

22

1 cossin4cos2

3k

222

cossin4

cos

2

3

k

Analytical results

h

x y

P

A

B

D

C

*

Experimental investigations in the lab of

the Division of Applied Plasticity, IPPT

shear

compression

specimen

(SCS)

dimensions: L= 20.0 mm D = 7.0 mm w = 2.0 mm t = 1.0 mm = 45 h = 1.42w

w

h

t

L

D

AlMg 5%SiC Composite

Experimental results

Load – displacement curve

Numerical simulation of the shear-compression test

Assumptions:

element: C3D8

friction: 0.0001

vertical displacement: 1.0 mm

number of elements: 11540

number of nodes: 13871

ABAQUS Standard

Approximation of material characteristic

eqeqeq

C

DCBA exp1

19.3018

21.264

02.21

24.235

D

C

MPaB

MPaA

Results of numerical simulation versus experiment

Paraboloidal criterion of Burzyński

T

Y

C

Yk

C

YT

Y

S

- yield limit in compression

- yield limit in tension

- yield limit in shear

1σ :3

1m

3:

2eq ' '

σ σ

2 2 21

3 1 9 1 4 02

C

m m eq Yk k k

3 C TY YSfor paraboloid of revolution

Solution of elasto-plasticity problem

0

0,,

C

Y

qepmp

m

q

e

p

C

Yem

tt

t

e

q

t

e

t

t

e

t

t

e

t

e

q

CG

CKG

CKGG

GKCK

' '

222

2

'

21

'

12

2

11

9

33621

σσ

σ11σI11ε

σ '

Stiffness matrix:

Newton- Raphson

q

t

ee

p

t

mm

G

K

3

G.Vadillo, J. Fernandez-Saez, R.B.

Pęcherski, Some application of Burzyński

yield ocndition in metal plasticity, Material

and Design, 2011

The UMAT was programmed

and implemented in Abaqus

by Marcin Nowak, IPPT

G.Vadillo, J. Fernandez-Saez, R.B.

Pęcherski, Some application of Burzyński

yield ocndition in metal plasticity, Material

and Design, 2011

Identification of the strength differential factor k

by means of FEM simulation of compression test

1

3

S C

Yk

235.24MPaC

Y

For AlMg 5%SiC:

15.1k

126.64MPaS

C

Y

T

Y

k

The preliminary results

Skład chemiczny:

Si 0.2 -0.8

Fe 3.5 - 0.7

Cu 0.4 - 4.5

Mn 0.4 -1.0

Mg < 1.0

Cr < 0.1

Zn < 0.25

Ti + Zr Al < 0.25

Np

eqeq BA

6.0

70

220

N

MPaB

MPaA

Aluminum alloy – PA6

Experimental results obtained in the lab of the

Division of Applied Plasticity, IPPT

o

p

eqeqgD

Pkk

21 exp

hk

p

eq

1

1

22

1 cossin4cos2

3k

222

cossin4

cos

2

3

k

o

p

eqeqgD

P

2.0185.0

h

p

eq

Analytical results

Rittel et al.

(2002)

Vural et al.

(2010)

Effect of Lode angle

2

3

3

2

271

eq

J

jkikij SSSJ3

13

kkp 3

1

eqq

The influence of the slit angle θ

= 45

Literature

W. BURZYŃSKI: Study on Material Effort Hypotheses, Lwów, 1928

(in Polish) ; English translation: Engineering Transactions, vol. 57,

No. 3-4, 185-215, 2009.

D.Rittel, S. Lee, G. Ravichandran, A Shear-Compression Specimen

for Large Strain Testing, Experimental Mechanics, 2002

M.Vural, A. Molinari, N. Bhattacharya, Analysis of Slot Orientation

in Shear-Compression Specimen (SCS), Experimental Mechanics,

2010

G.Vadillo, J. Fernandez-Saez, R.B. Pęcherski, Some application of

Burzyński yield ocndition in metal plasticity, Material and Design,

2011

Literature

R.B. Pęcherski, P. Szeptyński, M. Nowak, An extension of Burzyński

hypothesis of material effort accounting for the third invariant of stress

Tensor, Archives of Metallurgy and Materials, 56, 503-508, 2011.

M. Nowak, Ostrowska-Maciejewska, R. Pęcherski, P. Szeptyński, Yield

criterion accounting for the third invariant of stress tensor deviator.,

Part. I. Proposition of the yield criterion based on the concept of influence

functions, Engineering Transactions, 59, No. 4, 2011.

Thank you for your attention

MARIA SKŁODOWSKA CURIE (1867-1934)

born in Warsaw, Poland