ELASTIC WAVES and particulate materials J. Carlos Santamarina - - PowerPoint PPT Presentation

ELASTIC WAVES

and particulate materials

• J. Carlos Santamarina

Georgia Institute of Technology

Aussois 2012

References: Santamarina, J.C., in collaboration with Klein, K. and Fam, M. (2001). Soils and Waves, J. Wiley and Sons, Chichester, UK, 488 pages. Lee, J. S. and Santamarina, J. C. (2005a). "Bender Elements." ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 131, No. 9, pp. 1063-1070. Lee, J. S. and Santamarina, J. C. (2005b). "P-wave Reflection Imaging." ASTM Geotechnical Testing Journal,

• Vol. 28, pp. 197-206.

Wang, Y. H., Santamarina, J. C., and Cascante, G. (2003). "Counter EMF effects in Resonant Column Testing." ASTM Geotechnical Testing Journal, Vol. 26, No. 4, pp. 410-420. Cascante, G., Santamarina, J. C., and Yassir, N. (1998). "Flexural Excitation in a Standard Torsional-Resonant Column Device." Canadian Geotechnical Journal, Vol. 35, No. 3, pp. 478-490. Some pdfs (these and related papers) available at http://pmrl.ce.gatech.edu under "Publications"

ELASTIC WAVES

Let's assume… infinite homogeneous isotropic single-phase linear elastic continuum

dy y

yx

dx x

x

dx dz dy

dz z

zx

z y x t u

zx yx x 2 x 2

Mechanics - 1: Equilibrium

Mechanics - 2: Constitutive Equations

z z

E

z z

z x x x

E

x x z

y z y

E

y y z

y x z z

E 1

Mechanics - 2: Constitutive Equations

E G 2 ( 1 )

x y

z z

E ( 1 ) 4 M B G ( 1 ) ( 1 2 ) 3

• vol

E B 3 ( 1 2 )

z z

Mechanics - 3: Compatibility

spring beam

u u

in the continuum

x u x

x

x u y u

y x xy

z y x t u

zx yx x 2 x 2

Equilibrium Constitutive

2 x 2 2 x 2 2 x 2 z 2 y 2 2 x 2 2 x 2

z u y u x u G z x u y x u x u G M t u

Wave Equation Compatibility

x u x

x

x u y u

y x xy

z y vol x

G 2 M

xy xy

G

Wave Equation

Two Propagation Modes

x ux x uy

Longitudinal Transverse

Compression P-wave

Wavelength

P

(Bolton) (Bolt)

2 2 x x 2 2

u M u t x

Direction of Propagation

Shear S-wave

Wavelength

S

Direction of Propagation

(Bolton) (Bolt)

2 2 y y 2 2

u u G t x

Solution of the Wave Equation

( i t x )

u Ae

P

M V

2 2 x x 2 2

u M u t x

2 2 y y 2 2

u u G t x

S

G V

Spectrum

1 Hz 10 Hz 100 Hz 1 kHz 10 kHz 100 kHz 1 MHz sound (human) earthquakes consumer electronics medical imaging field testing (P) resonant column S Lab P-waves bender element S quasi- static

So far… infinite homogeneous isotropic single-phase linear elastic continuum

P & S

Finite:

Geometry dispersion Other propagation modes Reflection & refraction

Stiffness: M Stiffness: E

P

M V

P

E V

S

G V

infinite medium rod

S

G V

Longitudinal Transverse

torsional

P

M V

P

E V

infinite medium rod Geometry Dispersion

low f or high λ/d high f or low λ/d

R

V V

S-Wave: NO geometry dispersion

S

G V

Finite:

Reflection …. resonance

Reflection:

Add weight W at time t=0

W

At time t>0

u(t) L

u t L t t V E

u V E

W A

At time t=H/ V

u(t) H t

At time t>H/ V

u(t) H t

W A W 2 A

At time t=2H/ V

u(t) H t

W 2 A

At time t>2H/ V

u(t) H t

W A W 2 A

At time t=3H/ V

u(t) H t

W A

At time t>3H/ V

u(t) H t

W A

At time t=4H/ V

u(t) H t

Resonance

u(t) H t T

4H T V

TV 4H

Non-Elastic: Lossy

Lossy

u(t) H t

LW u t EA

t

• u t

u e

Lossy

u(t) x x ( i t x )

u t A e e

2 2 x x 2 2

u M u t x

* * P

M V i

Lossy

* * P

M M' iM" M' V 1 i tan i

P

M' V

P

tan 2V

Lossy

tan f V V 2 D 2 Q 1

d

1 2 D f tan S V

Total Attenuation

2 1

r r 1 1 2 2 1

A r e T A r

Material Attenuation Dispersion

b a b

V V 1.5D V

for

b a

10

Kramers–Kronig

Multiple reflections Interference Diffraction

Audi

Interference - Directivity

Diffraction Healing

Gradual Heterogeneity Non-linear: Shock

Vertically heterogeneous Cross-anisotropic Linear Elastic Homogeneous Isotropic Linear Elastic

HK Kim

Ray Bending: Fermat

Correlated Random heterogeneous Isotropic Linear Elastic

HK Kim

Homogeneous Isotropic Linear Elastic

Random Heterogeneity

Shock waves

Wave Phenomena: Complexity Richness

Infinite, homogeneous, isotropic, single-phase, linear elastic, continuum P, S Finite medium R – L – Rod - Tube Interfaces Reflection - Transmission, Refraction - Mode conversion Gradually Heterogeneous Curved rays (Fermat) Anomalies Diffraction - Scattering (Huygens) Anisotropic Quasi-propagation - Splitting Multiphase (poroelastic) Slow-P (Biot) Visco-elastic Attenuation & dispersion - Relaxation Non-linear Shock waves - Non-Lin. coupling Discrete Dispersion - Low-pass filtering

Measurement

Laboratory Testing

f Quasi static Wave propagation Standing wave fres

>> cell ~ cell << cell

E or G hyteresis E or G D VP or VS α

LDT

Quasi-static

• r
• r
• r
• r

B A G O

ABO area 4 loop inside area D slope G

• r

E

STANDING WAVE: Step response Resonant Column

Step Response

u(t) t

2 D

t

t f 2 cos e A ) t ( u

r t

t

Wavelength? Fixed-Fixed or Free-Free =2H Fixed-Free =4H

r

f T V

H

Resonant Column

Resonant column Driving head

1 2 3 4 4 3 2 1 Normalized Frequency [ ] Phase angle [rad] 1 2 3 4 5 10 Normalized Frequency [ ] Amplitude H [ ]

u/ n

D = 70% D = 20% D = 10% D = 5%

u/ n

Hu k

D = 70% D = 20% D = 10% D = 5%

u

(a) (b)

Induced Counter emf

• Equivalent circuit
• Induced counter emf

Vemf L R Vemf R L Vemf R L

Signal Generator

Vemf L R Coil set A Coil set B Voltmeter Vemf

543 . , V

. emf

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.50 1.00 1.50 2.00

Angular velocity [rads/sec] Vemf [V]

emf effects

WAVE PROPAGATION

S: bender elements P: standard piezo-elements

Bender element types

EXTERNAL PLATES INTERNAL PLATE PIEZO CERAMIC PLATES EXTERNAL PLATES INTERNAL PLATE PIEZO CERAMIC PLATES INTERNAL PLATE SOLDING SITE

Devices and materials

• 1. Soldering iron and accessories
• 2. Soldering flux
• 3. Coaxial cable
• 4. Epoxy
• 5. Multimeter
• 6. Silver conductive paint
• 7. Heat-shrink tubing
• 8. Nylon flat point socket screw
• 9. Polyurethane

Preparation

Remove outer shield from one end of coaxial cable. Separate the inner core from the copper mesh. Remove the end of inner core shield. If making a parallel BE, divide the copper mesh into two branches. Coat the ends of the cable and the BE with soldering flux.

Preparation

If making a series type BE, solder the core to one external plate and the copper mesh to the other one.

Preparation

If making a parallel BE, solder the core of the cable to the BE internal plate. Caution: The core/soldering metal should not touch the external plates. Solder the two copper branches to the external plates.

Check connections

Check the circuits with a multimeter. The core-to-shield resistance must be infinite (open circuit).

Coating

Water-proof the BE by coating the BE and the exposed portion of the cables with low viscosity

• polyurethane. Be sure to coat all BE faces, including the edges. Allow the polyurethane to dry

with the BE in the upright position. A second coat may be applied if needed.

An electric shield is needed to prevent cross talk phenomena (critical in wet soils – Parallel bender elements are “self-grounded”). Spread a layer of silver conductive paint over the surfaces of the coated bender element. The conductive paint must contact the shield in the coaxial cable, i.e., ground.

Electric shield

Cable reinforcement

Reinforce the connections using heat-shrink tubing. Shrink the tube using a hair dryer. May use more than one shrink-tube layers.

Housing in nylon socket screw: 1-drill

Take a nylon socket screw and make a hole through its center with a drill Nylon screw Drilled Nylon screw

Slide the BE into the hole inside the nylon screw. Fill in the air gap between the BE assembly and the screw with epoxy.

Housing in nylon socket screw: 2-fix

Done !

The BE assembly is ready for use once the epoxy has cured. The threaded nylon screw housing can be conveniently installed in any geotechnical cell, and easily replaced in case of malfunction.

Cross-talk

0.5 1 1.5 2 2.5 3 1 1 Time [ms] Output [Normalized]

Series-to-Series (without shielding)

0.5 1 1.5 2 2.5 3 1 1 Time [ms] Output [Normalized]

Series-to-Series (with shielding)

0.5 1 1.5 2 1 1 Time [ms] Output [Normalized]

Series-to-Parallel (without shielding)

0.5 1 1.5 2 1 1 Time [ms] Output [Normalized]

Parallel-to-Parallel (without shielding)

Directivity

Transverse

S-wave

In-plane

In-plane directivity

10 20 30 40 50 60 70 90 Crosshole tomographic configuration

30 60 300 330 0.15 0.1 0.05

Directivity

10 20 30 40 50 60 70 90 Base-to-borehole tomographic configuration

30 60 300 330 0.15 0.1 0.05

Transverse directivity: Side lobe P-wave (specimen size)

20 40 60 80 100 20 40 60 80 100 Tip-to-tip distance [mm] Cell radius [mm] =0.45 =0.30 =0.15 =0.00 P-Wave H R S-Wave

S-wave P-wave P-wave

Input and output - Convolution

Square, fr 4kHz Impulse Sine: f = 40kHz Sine: f = 12kHz Sine: f = 4kHz Sine: f = 1kHz Sine: f = 0.5kHz

1 2 3 Time [ms]

Resonant frequency

Experimental study Analytical formulation In Air In Soil

be

E L t f 12 2 875 . 1

2 2 2 1 2 2 3 4

) 1 ( 2 875 . 1 2 1 L b btL L V L EI f

sl be sl s

Operating Frequency - Comparison



Experimental Results



Analytical Results Controlling parameter: Short cantilever length Bender element Long cantilever length Soil properties

L b

Vs=500m/s Vs=160m/s Vs=50m/s In Air 1 10 100 2 4 6 8 10 12 Cantilever Length [mm] Frequency [kHz] 1 10 100 2 4 6 8 10 12 Cantilever Length [mm] Frequency [kHz]

First arrival?

A B C D

Source Receiver

A: First deflection B: First inflection C: Zero after first inflection D: Second inflection

Multiple Reflection

Soil

L

BE BE 2nd event 1st event 1st event 2nd event 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1 1 Time [microsec] Output [Normalized]

Goals:



High R-boundaries



No P-wave from side walls



No uncertainty in length



No uncertainty in time

Experimental Study - Results

A B C D

Source Receiver

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [ms] Cross Spectral Density

Time difference b/w 1st and 2nd event

Near Field: Signal matching

Mathematical Solution Cruse and Rizzo (1968) Stokoe and Sanchez-Salinero (1987) Procedure: Signal Matching For given values L and μ 1: Measure the signal Sm 2: Estimate fr and Vs 3: Compute predicted signal Sp= f(Vs, fr) 4: Change fr and Vs until Sp~ Sm

0.1 0.2 0.3 0.4 0.5 1.5 1 0.5 0.5 1 1.5 Arrival Time [ms] S-motion

Dotted line : Measured Solid line: Signal Matching

100 200 300 400 500 600 700 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Time [microsec] Meausred Signal

Analytical Approach

Measured signals Predicted signals

’ increases ’ decreases

100 200 300 400 500 600 700 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Time [microsec] Analytical Signal

P-Transducer

Ultrasound Transducer

Matching layer Piezoelectric material Backing block Insulator Probe case Electric cable

Damping dependent Heavy Intermediate Light

Ultrasound Transducer



Transducer (A3441):



GE Panametrics



Immersion type To avoid z mismatch with water.



High frequency (fr 500kHz)



Goals:

 Assess homogeneity

Layer detection



Position objectives (e.g., Transducers)

Directivity

0.00 1.00 2.00 3.00 4.00 5.00

• 40 -30 -20 -10

10 20 30 40

0.00 1.00 2.00 3.00 4.00 5.00

• 40 -30 -20 -10

10 20 30 40 Transducer

0.00 1.00 2.00 3.00 4.00 5.00

• 40 -30 -20 -10

10 20 30 40

25.4mm 25.4mm 53.3mm

Source Receiver

Fixed axial distance

Directivity

Fixed center-to-center distance (=25mm)

30 60 90 120 150 180 1 0.5

Source Receiver

Wave Parameters

P

4 B G M 3 V

S

G V

x 2 1

A e A

Velocity and Attenuation

Mechanical Waves attenuation S-waves P-waves

Mindlin contact: Inherently non-elastic

(Fretting damage after 10000 cycles - steel)

(Johnson, 1961)

= 60 = 90 = 20 = 30

0.4 mm

N Po Po=0.4N

Photoelasticity and Thermal IR Imaging

Photoelasticity and Thermal IR Imaging

Thermo-mechanical coupling

IR image Photoelastic image

Atomic Force Microscopy (AFM)

• Surface topography
• Surface properties
• Forces at nano-

scale

• Atomic-scale

experiments

Tip radius: 20 nm Stiffness :0.58 N/m

Environmental chamber (A) and Isolation box Laser beam Photodector

1

nN

Immersed in water nN nm

1

nN

Results of AFM Test

• Force curve

Dry, ambient, saturation A

A

B

B

C

C

D

D

nN Approach Retraction nm nN 500 500 nm 60 100 Relative humidity (%)

(1) (2) (3) (4) (5) (9) (6) (7) (8)

5 10 15 20 25 1 Average: 50

• Pull-out force

Summary

Gravelly Soils D = 0.008 – 0.018 Sand Air-dry D = 0.002 – 0.01 Saturated D = 0.003 – 0.021 Clayey soils D = 0.01 – 0.052 Residual soils D = 0.009 – 0.054 Peat (wg 200%) D 0.025

The effect of frequency

3 2 1 0.01 0.1 1 10 100

Loading Frequency, f, Hz Dmin /Dmin 1Hz Gmax /Gmax 1Hz

(Stokoe et al.1999)

Gmax Gmax 1Hz Dmin Dmin 1Hz

Mechanical Waves attenuation S-waves P-waves

1: Effective Stress

’ increases ’ decreases

’=1.4 ’=10.1 ’=27.4 ’=62.1 ’=131.5 ’=270.3 ’=409.1 ’=603.9 ’=798.8 ’=1062.5 ’=798.8 ’=603.9 ’=409.1 ’=270.3 ’=131.5 ’=62.1 ’=27.4 ’=10.1 ’=1.4 [sec]

a y x S

P 2 ' ' = V

= 0.36 - /700 0.10 0.20 0.30 0. 100 200

• factor [m/s]

exponent

Very soft clays Sands OC clays Cemented soils

kPa ' log s m V log

1: Effective Stress

200 400 600 800 1000 1200 0.0 0.2 0.4 0.6 0.8 1.0 Degree of saturation S Shear wave velocity [m/s]

2: Suction - Unsaturated Soils

a r m S

P suction S V

3: Cementation

’ increases ’ decreases sand 4% cement

[sec] ’=18.7 ’=36.0 ’=70.7 ’=140.1 ’=278.9

’=417.7

’=612.5 ’=807.4 ’=1071.1 ’=807.4 ’=612.5 ’=417.7 ’=278.9 ’=140.1 ’=70.7 ’=36.0 ’=18.7

kPa ' log s m V log Cementation controlled Stress controlled

’ increases ’ decreases

[sec] ’=18.7 ’=36.0 ’=70.7 ’=140.1 ’=278.9

’=417.7

’=612.5 ’=807.4 ’=1071.1 ’=807.4 ’=612.5 ’=417.7 ’=278.9 ’=140.1 ’=70.7 ’=36.0 ’=18.7

loose sand – 2% cement

kPa ' log s m V log

3: Cementation - Loading

T.Y. Yun

Uncemented

550 600 650 700 Vs [m/s] 100 150 200 250 Vs [m/s] 100 200 300 400 500 Confining Pressure [kPa]

kPa ' log s m V log

3: Cementation - Unloading

• A. Fernandez

3:  Sampling effects

• V. Rinaldi

0.0 0.5 1.0 1.5

Measurement error

Sandy Soils

(a) 0.0 0.5 1.0 1.5 200 400 600 800 1000 Vf [m/s]

Measurement error

Clayey Soils

(b)

Vlab / Vfield Vfield

Mechanical Waves attenuation S-waves P-waves

Bulk Stiffness

1 r r fl w a

S 1 S B B B

fl r a r w

1 S S

1 sus g fl

1 n n B B B

sus g fl

1 n n

soil sus g fl

1 n n

Fluid Mixture Suspension Soil (fluid + skeleton)

P

4 B G M 3 V

soil sus sk

B B B

from G= Vs

2

• K. Ishihara

Saturation

1 sk sk w a g P g w

4 S 1 S 1 n B G n 3 B B B V 1 n nS

Velocity and Impedance (S=100%)

VP -vs- n Impedance -vs- n

0.2 0.4 0.6 0.8 1 1 2 3 4

porosity n Z soil / Z fluid

n n)G (1 n) α(1 n 1 V V

s fluid P soil P

n) α(1 n n n)G (1 z z

s fluid P mix P 0.2 0.4 0.6 0.8 1 0.75 1 1.25 1.5 1.75 2

porosity n VP soil / VP fluid

=Bf/Bg

=0.01 =0.05 =0.1

=Bf/Bg

=0.01 =0.05 =0.1

Mechanical Waves closing….

Summary: P- and S-waves

Waves Small-strain phenomena May be used to monitor large-strain processes Vs Skeletal stiffness: G  Geo-mechanical design Effective stress, suction, cementation Sampling: pronounced effect  measure in situ ! Simple lab & field devices and methods VP Fluid & skeletal stiffness: B & G Proximity to full saturation VP &Vs: Dry  skeletal Poisson's ratio Saturated  porosity Spatial variability

VP in water 1482 VP in saturated soils 1450-1900 VP in unsaturated soils <100-800 VP in lightly cemented soils 400-1000 VS in saturated soils <50-400 VS in unsaturated clayey soils <100-500 VS in lightly cemented soils 250-700 VP in air 343

Summary

Porosity (S=100%)

2 2 2

4 1 2 1 2 2

g fl fl g g sk P S sk g fl

B V V n

VP and VS

see Foti & Lancelotta

1 V V 1 V V 2 1

2 S P 2 S P

Poisson's ratio (~dry)

Venice (M. Jamiolkowski)

Some Applications

P-waves

Data Fusion

http://sunsite.tus.ac.jp/multimed/pics/animals/bat.jpg http://www.moorhen.demon.co.uk

Navigational Homing in

• Mat. Eval. 1999

Massive data Processing Information

Paracoccus denitrificans Nitrate broth F110 + 3%Kaolin

0.1 0.2 0.3 0.4 Time (ms)

1 day

P-monitoring: Bio-gas

• V. Rebata

Elapsed Time (log scale in minute) Time ( sec)

20 40 60 80 100

Laboratory: Sedimentation

Clay Water

Settlement (mm)

Distance [mm] Time [ -sec] Bottom Interface

Anomaly Detection

Anomaly

28.5mm 60.5mm 85mm Anomaly

P-wave scanning – Before Liquefaction

Distance [mm] Time [ -sec]

Distance [mm] Time [ -sec]

After Liquefaction (~2 hr)

Boiling Water film

Distance [mm] Time [ -sec]

After Liquefaction (2 days)

Before Katrina

Before Katrina

NSF - D. Fratta

After Katrina

NSF - D. Fratta

Biloxi D’Iverville

I-110 Bridge

Pile 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Bathymetry: 200 kHz Sub bottom profiling: 20 kHz

NSF - D. Fratta

Massive data Display Information

500m 100m

• J. Cartwright - www.3DLab.org.uk

New Phenomena: Polygonal Faults

S-waves

Post Event Time [s] Post Event Time [s] Shea Wave Velocity [m/s]

S-monitoring: Liquefaction

S-monitoring: Excavation & Retaining Walls

Wall displacement / H Velocity [m/s] 80 140 200 0% 1% Force [kN] 10 20 30 40 Wall displacement / H 0% 1%

Boulanger

A1 A2 A8

B1 B2 B6 C1 C2 C8

Vs (m/s)

35 50 65 80 95 110 >125

Pixel Parametric (RLSS) (L-norms)

Fernandez, Lee

Imaging the mean stress

Before Loading With Loading Difference

a y x S

P 2 ' ' = V

Around tunnels

Vs (m/s)

35 50 65 80 95 110 >125

Pixel Parametric

Around tunnels: velocity tomograms

Field: Surface Waves (non-invasive)

• G. Rix

x

Active Pasive Sensor Arrays

0.1 0.2 0.3 5 10 15 20 25 30 Time (sec) Depth (m) Sledgehammer Seismic S

Penetration-based Field: Penetration-based (invasive)

S-CPTU

• P. Mayne

Measured Signals

JS Lee

S-fork

5 10 15 20 25 30 Depth (m)

0 0.2 0.4 Time (msec)

>100 100-125 125-150 150-175 175-200 200-225 >200 Velocity(m/s) NFT BRT TDT

100 m/s 200 m/s 150 m/s

0.0 m +2.8 m +7.2 m

Under dams