lecture #3: split hopkinson bar systems (cont
TRANSCRIPT
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
10/5/2015 2 2 Lecture #3 – Fall 2015 2 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 3 3 Lecture #3 – Fall 2015 3 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 4 4 Lecture #3 – Fall 2015 4 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 5 5 Lecture #3 – Fall 2015 5 D. Mohr
151-0735: Dynamic behavior of materials and structures
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(
10/5/2015 6 6 Lecture #3 – Fall 2015 6 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 7 7 Lecture #3 – Fall 2015 7 D. Mohr
151-0735: Dynamic behavior of materials and structures
-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
10/5/2015 8 8 Lecture #3 – Fall 2015 8 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 9 9 Lecture #3 – Fall 2015 9 D. Mohr
151-0735: Dynamic behavior of materials and structures
Geometric Wave Dispersion • Example
[t] recorded by strain gage
v
Chen & Song (2010)
10/5/2015 10 10 Lecture #3 – Fall 2015 10 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 11 11 Lecture #3 – Fall 2015 11 D. Mohr
151-0735: Dynamic behavior of materials and structures
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.
10/5/2015 12 12 Lecture #3 – Fall 2015 12 D. Mohr
151-0735: Dynamic behavior of materials and structures
INTRODUCTION TO ONE-DIMENSIONAL RATE-INDEPENDENT PLASTICITY
10/5/2015 13 13 Lecture #3 – Fall 2015 13 D. Mohr
151-0735: Dynamic behavior of materials and structures
Uniaxial tension test of a mild steel (Round bar specimen, D0=10mm, L0=100mm)
https://www.youtube.com/watch?feature=player_detailpage&v=D8U4G5kcpcM
10/5/2015 14 14 Lecture #3 – Fall 2015 14 D. Mohr
151-0735: Dynamic behavior of materials and structures
Uniaxial tension test of a mild steel (Round bar specimen, D0=10mm, L0=100mm)
10/5/2015 15 15 Lecture #3 – Fall 2015 15 D. Mohr
151-0735: Dynamic behavior of materials and structures
Uniaxial tension test of aluminum (Round bar specimen, D0=10mm, L0=100mm)
10/5/2015 16 16 Lecture #3 – Fall 2015 16 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 17 17 Lecture #3 – Fall 2015 17 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 18 18 Lecture #3 – Fall 2015 18 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 19 19 Lecture #3 – Fall 2015 19 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 20 20 Lecture #3 – Fall 2015 20 D. Mohr
151-0735: Dynamic behavior of materials and structures
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).
Coordinate system origin
0 X
one-dimensional body in its INITIAL CONFIGURATION
(thick blue line)
Position of material point in initial configuration
Coordinate system origin
0 ],[ tXxx
one-dimensional body in its CURRENT CONFIGURATION
(thick red line)
Position of material point in current configuration
10/5/2015 21 21 Lecture #3 – Fall 2015 21 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 22 22 Lecture #3 – Fall 2015 22 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 23 23 Lecture #3 – Fall 2015 23 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 24 24 Lecture #3 – Fall 2015 24 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 25 25 Lecture #3 – Fall 2015 25 D. Mohr
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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
10/5/2015 26 26 Lecture #3 – Fall 2015 26 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 27 27 Lecture #3 – Fall 2015 27 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 28 28 Lecture #3 – Fall 2015 28 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 29 29 Lecture #3 – Fall 2015 29 D. Mohr
151-0735: Dynamic behavior of materials and structures
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)
10/5/2015 30 30 Lecture #3 – Fall 2015 30 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 31 31 Lecture #3 – Fall 2015 31 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
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
10/5/2015 32 32 Lecture #3 – Fall 2015 32 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
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
10/5/2015 33 33 Lecture #3 – Fall 2015 33 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
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
linear spring frictional device
0p
10/5/2015 34 34 Lecture #3 – Fall 2015 34 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
When the yield condition
k0p if
linear spring frictional device
p
k
is satisfied, the length of the frictional element can change:
0][ kf
k0p if
10/5/2015 35 35 Lecture #3 – Fall 2015 35 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
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
10/5/2015 36 36 Lecture #3 – Fall 2015 36 D. Mohr
151-0735: Dynamic behavior of materials and structures
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.
10/5/2015 37 37 Lecture #3 – Fall 2015 37 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 38 38 Lecture #3 – Fall 2015 38 D. Mohr
151-0735: Dynamic behavior of materials and structures
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
10/5/2015 39 39 Lecture #3 – Fall 2015 39 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent isotropic hardening plasticity
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.
10/5/2015 40 40 Lecture #3 – Fall 2015 40 D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent isotropic hardening plasticity
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
10/5/2015 41 41 Lecture #3 – Fall 2015 41 D. Mohr
151-0735: Dynamic behavior of materials and structures
0
50
100
150
200
250
300
350
400
0 0.05 0.1 0.15 0.2
Rate-independent isotropic hardening plasticity
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
10/5/2015 42 42 Lecture #3 – Fall 2015 42 D. Mohr
151-0735: Dynamic behavior of materials and structures
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