dynamic fracture of materials: experimental and …framcos.org/framcos-9/bazant/ozbolt.pdf ·...
TRANSCRIPT
1
Dynamic Fracture of Materials: Experimental and Numerical Studies
J. Ožbolt
H.W. Reinhardt, A. Sharma, E. Sola, B. Irhan and D. Ruta
Institute of Construction Materials, University of Stuttgart, Germany
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Supported by National Science Foundation of Germany (DFG, AOBJ: 612266)
2
Contens
• Introduction • Theoretical framework • Examples (analysis & experiments):
• Brittle & quasibrittle materials (e.g. glass & concrete) • Ductile materials (e.g. steel)
• Split Hopkinson Bar (SHB) • High velocity projectile impact • Summary and Conclusions
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
3
Traffic: 10-6 – 10-4 s
-1
Earthquake: 5x10-3 – 5x10-1 s
-1
Airplane impact: 5x10-2 – 2x100 s
-1
Hard impact: 100 – 5x101 s
-1
Hypervelocity impact: 102 – 106 s
-1
Introduction
1
5
1E-06 1E-05 0.0001 0.001 0.01 0.1 1 10 100 1000
Tension
Compression
10-6 30·10-6 10-4 10-2 1 10 30 102 103
NMC (Draft)
5
4
3
2
1
imp
stat
f
f
1[ ]s
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Concrete R
ela
tive
ap
pa
ren
t str
en
gth
4
Introduction
Macro & meso scale modeling level
(1) Effects covered by constitutive law
• Rate dependent growth of microcracks • Viscosity due to the water content, e.g. concrete • Thermo-mechanical coupling (impact)
(2) Inertia effects – covered by dynamic analysis
• Structural inertia • Hardening & softening of materials • Crack propagation: velocity & branching • ……..
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
5 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Experimental investigations (e.g. tension)
• Direct tensile tests: d/dt < 0.1/s
• Indirect tensile tests
• Split Hopkinson Bar (SHB): 0.1/s < d/dt < 10/s
• Modified Split Hopkinson Bar (MSHB), spalling tests: 10/s < d/dt < 200/s
• Compact tension specimen (CTS)
d/dt < 50/s
6
General framework for modeling of rate dependent response
• Continuum mechanics
• Irreversible thermodynamics
• Constitutive law – temperature dependent microplane model
• Non-linear fracture mechanics
– smeared crack approach – crack band method
• Basic principles of contact mechanics & Thermo-mechanical coupling
• Standard finite elements & fragmentation
Numerical analysis
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
7
Microplane model
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Relaxed kinematic constraint (Ožbolt at al. 2001, 2014)
Rate dependency: Rate process theory (Bažant et al. 2000, Ožbolt et al. 2006)
Large strain generalization: Green-Lagrange strain & Co-rotated Cauchy stress tensor (Bažant et al. 2000)
Thermo-mechanical coupling (Ožbolt at al. 2005, Irhan 2014)
FE integration point
microplane
Materials: brittle, quasi-brittle & ductile
8 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
+V,D
+sV,D
compression tension
+Tr
+sTr
V
D
P(sp,p)+ +
P'(sp',p')- -
D
+
sD
+
A
BC
A'
B' C' 0
ED,0
Concrete
Steel
Cyclic rules
Volumetric & deviatoric
Shear
Deviatoric Volumetric Shear
9
Effect of inertia due to softening (or hardening) (direct tension)
, ,
or
with ;
++
+
s s c cs s c c t t t
t
s s c c s s c ct t s t I
t t t t
A M uA M u A
A
A M u A M u
A A A A
ss s s
s ss s s
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Non-uniform displacement field Consequence: acceleration field
Contribution of material strength
Contribution of inertia due to softening Experimentally measured
10
True vs. apparent tensile strength
-12
-8
-4
0
4
8
12
Load o
r re
action [
kN
]
0 0.5 1 1.5 2 2.5Time [ms]
Load (Apparent strength)
Reaction (True strength)
-50
0
50
100
150
200
250
Load o
r re
action [
kN
]
0 0.01 0.02 0.03 0.04 0.05
Time [ms]
Load (Apparent strength)
Reaction (True strength)
Concrete
Linear
Elastic
a
a
a
a
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Quasi-static
Dynamic
Loading in constant displacement rate
True strength
Apparent strength
11
True vs. apparent tensile strength
Brittle
Linear
Elastic
a
a
a
a
-50
0
50
100
150
200
250
Load o
r re
action [
kN
]
0 0.01 0.02 0.03 0.04 0.05
Time [ms]
Load (Apparent strength)
Reaction (True strength)
-15
-10
-5
0
5
10
15
Load o
r re
action [
kN
]
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Time [ms]
Load (Apparent strength)
Reaction (True strength)
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Concrete
Brittle material (glass)
12
CT concrete specimen
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
symmetry
Loaded
Loaded Loaded
Numerically predicted, loading rate 2.5 m/sec (Ožbolt et al. , 2011)
Fixed (Reaction)
Experimental result, loading rate 3.3 m/sec (Ožbolt et al. , 2013)
Pre-test analysis
13 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Loading rate: 4.30 m/s
Loading rate: 0.035 m/s Experiment (4.30 m/s)
14 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
• DIF for strength follows rate dependent constitutive law
• DIF for reaction: • Up to strain rate ≈ 50/s, linear rise in DIF (controlled by constitutive law) • For strain rate > 50/s, sudden and progressive increase in DIF (controlled by inertia)
Zone of strain rate evaluation
Crack branching
CT specimen – concrete
16 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Brittle (glass)
Crack velocity
Quasi-brittle (concrete)
Loading rate: 4.30 m/s Loading rate: 3.30 m/s
18 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Properties of steel
Uniaxial tension
Shear
Young`s modulus 210 GPa Poisson`s ratio 0.33 Yield stress 350 MPa Strength 600 MPa Ultimate strain 0.25
Constitutive law:
microplane model for steel
19 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Crack velocity & crack branching
Loading rate: 1 m/s Loading rate: 50 m/s Loading rate: 100 m/s
20 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Crack branching: experiment (left, Bousquent et al., 2011) and
numerical prediction (right, loading rate 100 m/s)
21 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Strength (von Mises stress) along the plastification zone
Yield limit
Quasi-static
50 m/s
22 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
90 m/s 75 m/s
1 m/s Static (0.002 m/s) 50 m/s
Size of the plastification zone ~
5 m
m
~3 m
m
~2 m
m
~2 m
m
~2 m
m
grey: stress < 350 MPa = yield stress
23 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
DIF on load & toughness
Apparent fracture toughness as a function of
crack velocity at -40°C (Kanazawa et al.,
1981)
Numerically predicted resistance
24
SHB: experiment vs. numerical analysis
(Schuler et al., 2005)
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
(Irhan et al., 2015)
Vimpact= 4.1 m/s
upb,exp= 2.63 m/s
t= 0.065 ms
25
Vimpact= 4.1 m/s
upb,num = 2.81 m/s
t= 0.070 ms
Failure modes
upb,num
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Vimpact= 4.1 m/sec
Vimpact= 7.6 m/sec
Vimpact= 11.1 m/sec
SHB: experiment vs. numerical analysis
26
Evaluated using
wave equation (elasticity)
Universität Stuttgart
Institut für Werkstoffe im Bauwesen
SHB: experiment vs. numerical analysis
27 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Projectile impact with plain concrete slab (Cargile, 1999)
Slab diameter = 1.52 m Concrete: Ec= 28 GPa; = 0.18 fc= 45.5 MPa; ft= 3.1 MPa; GF= 90 J/m2
30 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Parametric Study: contribution of bulk viscosity
31 Universität Stuttgart
Institut für Werkstoffe im Bauwesen
Parametric Study: influence of deletion criteria
Optimum deletion value (max. principal strain)= 1.0
32
Summary & Conclusions
• The employed 3D FE code based on the microplane model together with simple
finite elements is able to realistically reproduce fracture behavior of different materials at high loading rates.
• Inertia effects are responsible for: (i) rate dependent resistance and toughness, (ii) failure mode and (iii) crack branching.
• There are different kind of inertia effects on different scales: inertia due to micro-cracking, hardening & softening, speed of crack propagation & branching, structural inertia, …
• Consequently, inertia effects should not be a part of the rate sensitive constitutive law, they should come out from dynamic analysis automatically.
• Although steel does not exhibit strain rate sensitivity, due to high non-linear behavior (hardening) its dynamic fracture is strongly influenced by loading rate.
Universität Stuttgart
Institut für Werkstoffe im Bauwesen