Lecture #3: Split Hopkinson Bar Systems (cont

10/5/2015 1 1 Lecture #3 – Fall 2015 1 D. Mohr

151-0735: Dynamic behavior of materials and structures

by Dirk Mohr

ETH Zurich, Department of Mechanical and Process Engineering,

Chair of Computational Modeling of Materials in Manufacturing

Lecture #3:

• Split Hopkinson Bar Systems (cont.) • Introduction to 1D Plasticity

© 2015



Kolsky bar system

Requirements: • Striker, input and output bar made from the same bar stock (i.e. same material, same diameter)

• Length of input and output bars identical • Striker bar length less then half the input bar length • Strain gages positioned at the center of the input and

output bars

striker bar

strain gage

strain gage

specimen input bar output bar

Launching system

L/2 L/2 L/2 L/2



Kolsky bar formulas

striker bar

strain gage

strain gage

specimen input bar output bar

Launching system

L/2 L/2 L/2 L/2

)(ttra

)(tre

)(tinc

)()( tA

EAt tra

s

s

)(2

)( tl

ct re

s

s

• Stress in specimen:

• Strain rate in specimen:

s

sl

s



Wave Dispersion Effects co

mp

ress

ive

str

ain

time

com

pre

ssiv

e s

trai

n

time

“Pochhammer-Chree” oscillations

long rise time

[t] recorded by strain gage

v

c

v

2

1D THEORY EXPERIMENT



Wave Dispersion Effects

Simplified model: axial compression only:

Reality: axial compression &

radial expansion:

In reality, the wave propagation in a bar is a 3D problem and lateral inertia effects come into play due to the Poisson’s effect!

Inertia forces along the radial direction delay the radial expansion upon axial compression

x

D

x )1(

D)1(



Geometric Wave Dispersion • Consider a rightward traveling sinusoidal wave train of wave length L in an

infinite bar of radius a raveling at wave speed

*],[ tx

L

x

D

1.0

cc /L

LD

0.5 0.

0.5

1.0

Lc

• The wave propagation speed depends on wave length!

• The 1D theory only true for very long wave lengths (or very thin bars)

• High frequency waves propagate more slowly than low frequency waves

Ec • Theoretical wave speed (1D analysis):

• Theoretical wave speed (3D analysis):

Pochhammer-Chree 3D analysis



-2

-1

0

1

2

-2

-1

0

1

2

-4

-2

0

2

4

0 10 20 30 40 50 60 70 80

Geometric Wave Dispersion • Example: Rightward propagating wave in a steel bar

mm20

5.01

LD

1.02

LD

skmc /1.31

skmc /9.42

mm401 L

mm2002 L

kHzf 781

kHzf 252

mm2000

sT 6421

sT 4062

-2

-1

0

1

2

-4

-2

0

2

4

-2

-1

0

1

2

[t] [t]

superposition

low frequency

high frequency

superposition

low frequency

high frequency



Geometric Wave Dispersion • In practice:

A[t] B[t]

N

n

nnA tnbtnaa

t1

0 ]sin[]cos[2

][

1. Spectral decomposition (Fourier) of measured strain history at location A

2. Compute travel times tn from A to B as function of frequency n

3. Compute strain history at location B

N

n

nnnnB ttnbttnaa

t1

0 )](sin[)](cos[2

][

(evaluation of wave dispersion relationship)



Geometric Wave Dispersion • Example

[t] recorded by strain gage

v

Chen & Song (2010)



Modern Hopkinson Bar Systems

striker bar

strain gage

specimen

input bar output bar

Launching system

L L

High speed camera

Features: • Striker and input typically made from the same bar stock (i.e. same material, same diameter)

• Small diameter output bar for accurate force measurement • Similar length of all bars • Output bar strain gages positioned near specimen end • Wave propagation modeled with dispersion • Strains are measured directly on specimen surface using Digital

Image Correlation (DIC)



ADVANCED TOPICS related to SHPB technique

• Accurate wave transport taking geometric wave dispersion into

account

• Use of visco-elastic bars (slower wave propagation than in metallic

bars, more sensitive for soft materials)

• Torsion and tension Hopkinson bar systems

• Lateral inertia at the specimen level

• Friction at the bar/specimen interfaces

• Dynamic testing of materials (where quasi-static equilibrium cannot

be achieved)

• Pulse shaping

• Intermediate strain rate testing

• Infrared temperature measurements

• Experiments to characterize brittle fracture

• Multi-axial ductile fracture experiments

• Experiments under lateral confinement

… and many others.



INTRODUCTION TO ONE-DIMENSIONAL RATE-INDEPENDENT PLASTICITY



Uniaxial tension test of a mild steel (Round bar specimen, D0=10mm, L0=100mm)

https://www.youtube.com/watch?feature=player_detailpage&v=D8U4G5kcpcM



Uniaxial tension test of a mild steel (Round bar specimen, D0=10mm, L0=100mm)



Uniaxial tension test of aluminum (Round bar specimen, D0=10mm, L0=100mm)



0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01

Stress-strain response of metals

E E

① Elastic loading

② Elasto-plastic

loading

③ Elastic

unloading

Permanent change in length

Recoverable change in length

= plastic deformation

= elastic deformation



True Stress Definition (1D)

If a homogeneous bar is uniformly stretched or compressed, the true stress is defined as the applied force F divided by the current cross-sectional area A. All material models will be formulated in terms of the true stress.

A

F

Recall that the engineering stress is defined as the applied force divided by the initial cross-sectional area A0. The difference between the true and engineering stresses vanishes when working in the framework of infinitesimal strains (e.g. elastic behavior of metals)



True Stress Definition (1D)

If a homogeneous bar is uniformly stretched or compressed, the true stress is defined as the applied force F divided by the current cross-sectional area A. All material models will be formulated in terms of the true stress.

A

F

Recall that the engineering stress is defined as the applied force divided by the initial cross-sectional area A0. The difference between the true and engineering stresses vanishes when working in the framework of infinitesimal strains (e.g. elastic behavior of metals)



Kinematics (1D)

A body is considered as a closed set of material points. We first consider the case of a simple one-dimensional body (e.g. very thin wire) where all material points lie on a straight line.

The initial configuration B0 describes the position of all materials points before applying any loading. The position coordinate of a material point in the reference configuration is denoted by a capital X.

Coordinate system origin

0 X

undeformed one-dimensional body B0

(blue line)

One material point X ϵ B0



Kinematics (1D)

The current configuration describes the position of all materials points after applying loading. The position coordinate of a material point in the current configuration is denoted by x (lower case letter).


0 X

one-dimensional body in its INITIAL CONFIGURATION

(thick blue line)

Position of material point in initial configuration


0 ],[ tXxx

one-dimensional body in its CURRENT CONFIGURATION

(thick red line)

Position of material point in current configuration



Kinematics (1D)

The displacement u=u[X,t] needs to be added to the initial position coordinate to obtain the position of a material point in the current configuration.

X INITIAL CONFIGURATION

],[ tXxx

CURRENT CONFIGURATION Xxu



Kinematics (1D) The deformation gradient F is mathematically defined by

X

tXxtXF

],[],[

In 1D, it is also called stretch l, l=F[X,t]. It is a first measure of change in length in the neighborhood of a material point.

dXINITIAL CONFIGURATION

CURRENT CONFIGURATION dx

A differential segment of length dX in the initial configuration at the material X, changes its length to dx in the current configuration:

)(dXdx l



Kinematics (1D)

The length of a differential element of a solid body always remains positive (even under compression) which implies

0],[ tXF

CURRENT CONFIGURATION ! 0dx

The deformation gradient is related to the displacement gradient by the relationship

1],[

X

tXuF



Kinematics (1D)

With the stretch definition at hand, the logarithmic strain definition reads

]ln[l

This dimensionless quantity is also often called the true strain.

• If a material has been stretched, we have: 1l 0and hence

• If a material has been compressed, we have: 1l 0and hence

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3

l



Rigid Body Motion

In case of a rigid body motion, a body remains undeformed and the stretch is

1],[ tXl

dXINITIAL CONFIGURATION

CURRENT CONFIGURATION dXdx

and the strain 0],[ tX



Elastic and plastic deformation

INITIAL CONFIGURATION

DEFORMED (CURRENT) CONFIGURATION

0l

STRESS-FREE PERMANTLY DEFORMED

CONFIGURATION

pl

l

0l

ll

0l

lp

p l

p

el

ll

plastic stretch

elastic stretch

total stretch



Elastic and plastic deformation

ep

p

p

l

l

l

l

l

llll

00

In the theory of plasticity, it is assumed that the total stretch l can be multiplicatively decomposed into an elastic stretch le and a plastic stretch lp

In polycrystalline metals, the elastic stretch represents the macroscopic average stretching of the crystal lattices, while the plastic stretch represents the macroscopic deformation associated with permanent changes in the crystal structure (such as dislocation glide)



Elastic and plastic deformation source: http://www.ganoksin.com/borisat/nenam/metal-rolling-n-drawing.htm

Elastic lattice deformation plastic lattice deformation Initial lattice

By definition of the logarithmic strain, the multiplicative decomposition of the stretch implies an additive decomposition of the total strain into elastic and plastic parts:

peepep lllll

]ln[]ln[]ln[]ln[

pe



0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01

Important difference

E E

① Elastic loading

② Elasto-plastic

loading

③ Elastic

unloading

pe

① Elastic loading

② Elastic

unloading

e

ELASTO-PLASTIC NON-LINEAR ELASTIC (e.g. metals, concrete, thermoplastics ) (e.g. rubbers, foams)



Rate-independent perfect plasticity

p

• Simplified rheological model:

The strain is split into an elastic and a plastic part

i.e. the elastic strain is

INITIAL CONFIGURATION

DEFORMED (CURRENT) CONFIGURATION

pe

pe

linear spring frictional device




In our basic theory of elasto-plastic materials, the stress is a function of the elastic strain only,

)( pe EE

with E denoting the Young’s modulus.

kf ][

A scalar-valued function of the stress, the so-called yield function f is introduced

to define a constraint on the admissible stress states (as represented by the frictional device:

0][ kf

• Constitutive equation for stress:

• Yield function




According to the definition of the domain of admissible stresses,

0][ kf

is called yield surface.

0][ kf

0][ kf The boundary

the absolute value of the stress cannot be greater than the current flow stress k. The elastic domain is defined by

kk 0

Elastic domain

Yield surface

Admissible stress states




The length of the frictional element does not change when the applied stress is lower than the flow stress, i.e.

0][ kf 0p if

k


0p




When the yield condition

k0p if


p

k

is satisfied, the length of the frictional element can change:

0][ kf

k0p if




The evolution of the plastic strain is prescribed by the flow rule

defines the magnitude of the plastic strain rate.

][sign p

0f0 if

• Flow rule

0

0f0 if

0f0 if

0fand

0fand

(elastic unloading)

(elasto-plastic loading)

(elastic loading)

① Elastic loading

③ Elastic

unloading

② Elasto-plastic loading

k

• Loading/unloading conditions

The sign of the applied stress gives the direction of plastic flow, while the non-negative scalar multiplier



i. Constitutive equation for stress

)( pE

ii. Yield function kf ][

iii. Flow rule ][sign p

iv. Loading/unloading conditions

0f0 if

0f0 if

0f0 if

0fand

0fand

Rate-independent perfect plasticity - Summary

Material model parameters: (1) Young’s modulus E, and (2) flow stress k.



Rate-independent perfect plasticity - Application

p

Ek /

Ek /

time

time

time

k

Total strain

Plastic strain

Stress

k

①

①

②

③

③

④

④

⑤

① ② ③ ④ ⑤



Rate-independent isotropic hardening plasticity

The magnitude of the stress increases due to strain hardening when the material is deformed in the elasto-plastic range. For isotropic hardening materials, it is described through an evolution equation for the flow stress k.

E E

① Elastic loading

② Elasto-plastic

loading

③ Elastic

unloading

④ Elastic

re-loading

⑤ Elasto-plastic

loading




to measure the amount of plastic flow (slip). This measure is often called equivalent plastic strain. Unlike the plastic strain, the magnitude of the equivalent plastic strain can only increase!

][ pkk

Firstly, we introduce a scalar valued non-negative function

dtp

It is then assumed that the flow stress is a monotonically increasing smooth differentiable function of the equivalent plastic strain

This equation describes the isotropic hardening law.




Frequently used parametric forms of the function are the Swift and Voce laws:

n

pS Ak )( 0

][ pkk

]exp[10 pV Qkk

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

SV kkk )1(

Swift Voce Swift-Voce

Qkkd

dk

p

0 ,0

Hardening saturation

pp

p

k k k



0

50

100

150

200

250

300

350

400

0 0.05 0.1 0.15 0.2


In engineering practice, the isotropic hardening function is often represented by a piece-wise linear function

][ p

][ MPak

PEEQ k

0.000 199.1

0.020 246.3

0.050 283.9

0.100 321.0

0.200 365.6



i. Constitutive equation for stress

)( pE

ii. Yield function ][],[ pp kf

iii. Flow rule ][sign p

iv. Loading/unloading conditions

0f0 if

0f0 if

0f0 if

0fand

0fand

Isotropic hardening plasticity - Summary

v. Isotropic hardening law

][ pkk with dtp

Documents

Lecture #3: Split Hopkinson Bar Systems (cont