Lecture #2: Split Hopkinson Bar Systems by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 2015 1 1 1
Uniaxial Compression Testing Useful experiment to characterize the flow behavior of materials at large strains Hydraulic actuator Fixed cross-head piston loading platen (top) load cell loading platen (bottom Base plate 2 2 2
Uniaxial Compression Testing undeformed deformed F c F F c = applied compression force l = current length L l L 0 = initial length l = current length F c L 0 = initial length Axial strain definitions l 1 (engineering or nominal strain) ln[ 1 ] L eng Axial stress definitions F F eng (engineering or nominal stress) 1 ] A [ eng eng 0 A 3 3 3 eng Only valid for incompressible materials (true or logarithmic strain) 0 0 (true stress) 0 0 (extension) (shortening) (tension) (compression)
Applied force [kn] Applied force F [kn] Engineering stress [MPa] Engineering stress [MPa] True stress [MPa] F c F Uniaxial Compression Testing l F c Force-displacement curve 00-10 -10-7.5-5 -5-2.5 00-50 -50 Engineering stress-strain curve 0 0-0.6-0.6-0.4-0.4-0.2-0.2 0 0-500 -500 True stress-strain curve 0-1 -0.75-0.5-0.25 0-200 -100-150 -1000-1500 -400-600 -800-200 Change in in length l-l l-l 0 0 [mm] -2000 Engineering strain [-] [-] -1000 Logarithmic strain [-] 4 4 4
Uniaxial Compression Testing Numerical Simulation ( also topic of computer lab #2) Source: https://www.youtube.com/watch?feature=player_detailpage&v=ozzwc-sff-w 5 5 5
Uniaxial Compression Testing Experimental challenges Recommended size for metals testing: H=15mm D=10mm Plastic buckling Shear buckling H/D too large Poor alignment anisotropy barreling Friction too high H/D too small optimal Excellent lubrication H/D ~ 1.5 6 6 6
Strain rate in a compression experiment undeformed deformed L u l L l u u 0 0 (extension) (shortening) Nominal strain rate eng L l 1 eng l L u L True strain rate ln[ 1 eng ] eng 1 eng Equiv. strain rate eng 1 eng 7 7 7
Compression of a cylindrical specimen Principle of Quasi-static Equilibrium F in L initial F out Quasi-static equilibrium F in ( t) F ( t) out 8 8 8
Time scale #1 Strain at the end of the experiment: Average strain rate (over time): max av Duration of experiment T max av Note: T is independent of specimen dimensions! Example: 0.15 max av 500 / s T 0.15 500 0.0003s 0.3ms 300s 9 9 9
Time scale #2 Wave propagation speed: c E Specimen length: L L c Wave travel time t L c Note: t does depend on specimen dimensions! Example: c 5000m / L 10mm s t 10 6 2 10 s 510 6 2s 10 10 10
Principle of Quasi-static Equilibrium Long time scale: Duration of experiment T max av Short time scale: Wave travel time t L c L c Condition for quasi-static equilibrium: (when testing an elasto-plastic material) Fin( t) Fout( t) t T 11 11 11
Principle of Quasi-static Equilibrium Condition of quasi-static equilibrium t T L c max av Example for challenging experiments (w/ regards to quasi-static equilibrium) Brittle materials, e.g. ceramics max ~ 0.01 Soft materials, e.g. polymers c ~ 1000m / s Materials of coarse microstructure, e.g. metallic foams L 1mm 12 12 12
Dynamic testing of foams Example for challenging experiments (w/ regards to quasi-static equilibrium) Zhao et al. (1997) 13 13 13
LIMITATION OF UNIVERSAL TESTING MACHINES Vibration issues t 200 2 s t 20 1 s t 2s Condition of validity max Texp max[ t av breaks often down at around 10/s i ] 14 14 14
LIMITATION OF UNIVERSAL TESTING MACHINES Ringing of conventional load cell 15 15 15
SEPARATION OF TIME SCALES Characteristic time scale of testing system t sys versus Duration of experiment T exp max av Universal testing machines t sys T exp SHPB t sys Texp [ s 3 10 10 2 10 1 2 3 4 1 10 10 10 10 16 16 16 1 ]
Wave Equation Derived wave equation for bars under the assumption of uniaxial stress x, u dx dx x position axial displacement Differential equation for axial displacement u[x,t]: 2 2 u c 2 x 2 u 2 t 0 longitudinal elastic wave velocity: the particle velocity: c E v[ x, t] u [ x, t] u t 17 17 17
18 18 18 General Solution General solution of wave equation 0 2 2 2 2 2 t u x u c ] [ ] [ ], [ ct x g ct x f t x u ], [ ], [ ], [ ], [ t x t x c x g c x f c t x u t x v r l displacement: particle velocity [ ct] x f [ ct] x g Leftward traveling rightward traveling ], [ ], [ ], '[ ], [ t x t x x g x f t x u t x r l strain
Elastic Wave Velocities Particle velocity depends on applied loading, while the wave velocity is an intrinsic material property Material Density [g/cm 3 ] Young s modulus [GPa] Wave velocity [m/s] Air 340 Steel 7.8 210 5188 Aluminum 2.7 70 5091 Magnesium 1.7 44 5087 Tungsten 19.3 400 4552 Lead 10.2 14 1171 PMMA 1.2 3 1581 Concrete 3 30 3162 All values are rough estimates and may vary depending on the exact material composition and environmental conditions 19 19 19
Hopkinson Bar Experiment Typical system characteristics: Striker bar length: L s =1000 mm Input bar length: L i = 3000 mm Output bar length: L o = 3000 mm Bar diameter: D= 20 mm Specimen characteristics (for simplified theoretical analysis): Ideal plastic, constant force 20 20 20
rightward leftward total Hopkinson Bar Experiment 21 21 21
rightward leftward total Hopkinson Bar Experiment 22 22 22
rightward leftward total Hopkinson Bar Experiment 23 23 23
rightward leftward total Hopkinson Bar Experiment 24 24 24
rightward leftward total Hopkinson Bar Experiment 25 25 25
rightward leftward total Hopkinson Bar Experiment 26 26 26
rightward leftward total Hopkinson Bar Experiment 27 27 27
rightward leftward total Hopkinson Bar Experiment 28 28 28
rightward leftward total Hopkinson Bar Experiment 29 29 29
rightward leftward total Hopkinson Bar Experiment 30 30 30
rightward leftward total Hopkinson Bar Experiment 31 31 31
rightward leftward total Hopkinson Bar Experiment 32 32 32
rightward leftward total Hopkinson Bar Experiment 33 33 33
Force Strain HOPKINSON BAR TECHNIQUE Measuring force with a slender bar R (, t) x A applied 2( L x ) / A c ( t) A ( x R A measured, t) ( x L A, t) Time x A / c A R Time ( x, t) ( 2 L x ) / A c L (, t) x A F x A L L ( x, t) 34 34 34
Force Force Force HOPKINSON BAR TECHNIQUE Measuring force with a slender bar duration of valid measurement Time Time Time F A x A L x B R ( x, t) ( x, t) B 35 35 35
HOPKINSON BAR TECHNIQUE Direct impact experiment: One force measurement (output bar only) specimen output bar u out striker specimen output bar v 0 out L out Strategy: Experiment ends before leftward traveling wave in output bar reaches strain gage Length of output bar L out T 2c out duration of the experiment 36 36 36
Strain Strain Strain HOPKINSON BAR TECHNIQUE Split Hopkinson Pressure Bar (SHPB) experiment Two force measurements L L L 2L st / c R ( 0, t) R R Time Time Time ( 0, t) ( / 2, t) (, t) L in L in v 0 striker input bar specimen output bar L st L in 37 37 37
Strain HOPKINSON BAR TECHNIQUE Criterion to avoid wave superposition at strain gage position 3L in / 2c L 3Lin / 2c Lin / 2c 2Lst / c L in / 2c Lin 2L st 2L st / c R Time ( / 2, t) L in v 0 striker input bar specimen output bar L st L in 38 38 38
Strain Strain SPLIT HOPKINSON PRESSURE BAR (SHPB) SYSTEM Input force measurement: Kolsky (1949) measured in L calculated ( t) ( t L / 2c) re L in wave transport F in EA re inc Time in R Time inc in ( t) ( t L / 2c) R in F in v 0 striker input bar specimen output bar L in 39 39 39
SPLIT HOPKINSON PRESSURE BAR (SHPB) SYSTEM v 0 striker input bar specimen output bar Calculate forces and velocities at specimen boundaries Input bar/specimen interface: in R in L inc re tra F v Output bar/specimen interface: out R in in F v Specimen specific post-processing 40 40 40 EA re c out out re EA c tra tra inc inc F in F out v in Verify quasi-static equilibrium Coupling with other measurements (e.g. high speed photography) Calculate stress, strain and strain rate v out
DESIGN OF A STEEL SHPB SYSTEM Duration of the experiment: Minimum output bar length: T max av 0.5 1000 500s L out Tc / 2 0.0005s 5000m / s / 2 1. 25m Minimum striker bar length: Minimum input bar length: L st Tc / 2 1. 25m Lin 2Lst 2. 5m v 0 striker input bar specimen output bar The bar diameters need to be chosen in accordance with the forces required to deform the specimen (force associated with incident wave should be much higher than the specimen resistance) 41 41 41
EXAMPLE EXPERIMENT v 0 striker input bar specimen output bar 42 42 42
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 Launching system striker bar L/2 L/2 L/2 L/2 input bar specimen output bar strain gage strain gage 43 43 43
Kolsky bar formulas s l s s re Stress in specimen: EA s( t) tra( t) A Strain rate in specimen: 2c s( t) re( t) l s s tra inc Launching system striker bar L/2 L/2 L/2 L/2 input bar specimen output bar strain gage 44 44 44 strain gage
compressive strain Wave Dispersion Effects compressive strain v [t] recorded by strain gage 1D THEORY EXPERIMENT Pochhammer-Chree oscillations v 2c long rise time time time 45 45 45
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! D ( 1 )D Inertia forces along the radial direction delay the radial expansion upon axial compression x ( 1 )x 46 46 46
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 [ x, t*] c L Theoretical wave speed (1D analysis): c E L x Theoretical wave speed (3D analysis): 1.0 c / L c D Pochhammer-Chree 3D analysis 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 47 47 47 0.5 0. 0.5 1.0 D L
Geometric Wave Dispersion Example: Rightward propagating wave in a steel bar [t] 2000mm [t] 20mm 2 1 L 40mm 1 L 200 2 mm high frequency D L 0.5 1 D L 0. 2 1 c 1 3.1km / s c 4.9 2 km / s 2 1 f 1 78kHz f 2 25kHz high frequency T 642 1 s T 406 2 s 0 0-1 -1-2 -2 2 1 low frequency 2 1 low frequency 0 0-1 -1-2 -2 4 2 superposition 4 2 superposition 0 0-2 -4-2 48 48 48-4 0 10 20 30 40 50 60 70 80
Example Geometric Wave Dispersion v [t] recorded by strain gage Chen & Song (2010) 49 49 49
Modern Hopkinson Bar Systems Launching system striker bar L input bar High speed camera L output bar specimen strain gage 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) 50 50 50
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. 51 51 51
Reading Materials for Lecture #2 George T. Gray, High-Strain-Rate Testing of Materials: The Split-Hopkinson Pressure Bar : http://onlinelibrary.wiley.com/doi/10.1002/0471266965.com023.pub2/abstract M.A. Meyers, Dynamic behavior of Materials (chapter 2): http://onlinelibrary.wiley.com/book/10.1002/9780470172278 W. Chen, B. Song, Split Hopkinson (Kolsky) Bar : http://link.springer.com/book/10.1007%2f978-1-4419-7982-7 52 52 52