Experimental Rock Deformation

(651-4050-00 G)

Spring Semester; 2012

Laboratory note 3

Uniaxial Deformation of Cylindrical Rock Sample

Uniaxial Deformation of Cylindrical Rock Sample

AIM

Quantitative measurement and analysis of experimental rock deformation at room pressure and temperature.

Obtain the values of: Young Modulus (E), Yield strength and Ultimate strength for some rock samples.

Eventually find the Poissons ratio.

HOW

1) Apply a uniaxial load (increasing with time) on the specimen. 2) Record the load applied and longitudinal shortening across the sample 3) To obtain Poisson ratio: measure the increase of diameter during the test.

Figure 1. Stress/strain relation for a sandstone sample 1 inch diameter 60 mm length during an unconfined stress test. *Engineering definition for metals and plastics.

Stre

ss -

(P

a)

Strain - (dl/L)

Linear elastic part, E=slope

Machine adjustment

Linear elastic limit

Yield strength

Ultimate or failure strength

*Elastic curve +0.2 - 2% of

A)

B)

C,D,E)

Figure 2. A) Sample at the beginning of the test. B) Sample at the end of the elastic part. C) Starting of failure. D, E) Failure.

What tools we need

1) A sensor to measure the force applied on the sample => Load cell (strain gages bridge). 2) A sensor to measure the shortening across the sample => Strain sensor (LVDT). 3) A sensor to measure the barreling of the sample => Strain gages glued on the sample

attached perpendicularly to the axis of the cylindrical sample.

Preliminary phase

A sensor is generally an instrument which produces or controls electric signal in function of a physical dimension applied on it. The calibration is the function which relates the output (electric signal) with the input (physic dimension).

Calibration of the load cell

Following the Calibration script calibrate the load cell called Francesca. The standard sensor to employ is a dynamometric ring like in fig.3. The dynamometric ring has to be placed in series with the load cell between the jaws of the hydraulic press (fig.3). Than with the hydraulic press apply some steps of force to the column and register the values of force read on the dynamometric ring and the voltage from the sensor.

Figure 3. Hydraulic press, load cell and dynamometric ring or load cell of calibration.

A) B) C) D) E)

Calibration of the displacement sensor

Following the Calibration script calibrate the displacement sensor (LVDT). The standard sensor to use is a micrometric head like in fig.4 which pushes on the LVDT (fig.4).

Figure 4. Micrometer head and LVDT

In order to measure the Poisson ratio a strain gage can be placed normal to the axis of application of the force on the specimen (see fig.2).

Strain gage

A Strain Gage is used to measure strain. It is a small (10x5x0.1 (thick) mm) plastic layer in which an electric circuit is printed. The electric circuit is a resistance. The main law of the resistances says:

=

Where is the resistivity of the material used to build the resistance, A is the area of the section of the resistance and L is the length of the resistance. When the plastic foils is stretched the resistance wire is stretched as well. The deformation of the wire implies a change in length (L) and area (A) of the resistance wire and, as consequence, a change in resistance of the strain gage.

Figure 5. Strain gage. The red arrow shows the stretching

The strain gages are applied to the materials (metals, rocks or plastics) which are tested. They give an indication of deformation of the specimen in according with the positions of the strain gages and the direction of application of the stress.

A non-stressed strain gage exhibit the nominal resistance(Rg) which could be, for instance, 120 Ohms (check this value on the strain gage label). The change of resistance (Rg) in function of strain () is expressed by the gage factor (GF):

GF = Rg Rg

eq.1

SPECIFICATIONS Foil Thickness: 5 m Carrier Material: Polyimide Carrier Thickness: 50 m Connections: Solder pads (constantan gages); solder dots (karma) Nominal Resistance: 350 Ohms Resistance Tolerance: 0.5% Gage Factor: 2.0 nominal (actual value printed on package) Gage Factor Tolerance: 1.0% Thermal Properties:

Reference Temp.: 23C (73F) Service Temp.: Static: -30 to 250C (-22 to 482F) Dynamic: -30 to 300C (-22 to 572F) Temperature Comp: (zero) Carbon Steel (ferritic): +11 ppm/C Stainless Steel (austenitic): +17 ppm/C Aluminum: +23 ppm/C Compensated Temp.: -5 to 120C (23 to 248F) Tolerance of Temp Comp.: 1 ppm/C

(0.5 ppm/F) Gage Factor Temp Coefficient: Constantan: 0.0090%/C (+0.0050%/F) Karma (SS comp): 0.0103%/C (-0.0057%/F) Mechanical Properties: Maximum Strain: 3% or 30,000 S Hysteresis: Negligible Fatigue (@ 1,500 S): >10,000,000 cycles Smallest Bending Radius: 3 mm

Strain Gages are always inserted in a resistance circuit (Bridge). The most common bridge is the Wheastone Bridge:

Figure 6. Strain gage (Rg) in a Wheastone bridge.

1 = 2 = 4 = 1 = ( + ) 2 = (2) = ( 1) 2

= 12 + eq.2 Example (under stress)

R1=R2=R4=R=120 Ohms

GF=2

Rg=120 Ohms (not under stress)

Vin=10 Volts

Vout=0.01 Volts

Rg*=Strain gage resistance under stress

= ?

(from eq.2) = 2

Rg*=120.48 Ohms

Rg=0.48 Ohms

(from eq. 1) = Rg RgGF =0.002 = 0.2%

Vin Vout

R2 R1

R4 Rg

(Strain gage)

I1 I2

LVDT

Linear Variable Differential Transducer or LVDTs are common sensors used to measure displacement. A primary coil is supplied with an alternate current (AC). The magnetic field generates by the primary is concatenated with a secondary circuit which is formed by 2 coils. The concatenation between primary and secondary is modulated by a ferromagnetic road (movable magnetic core fig. 7) which is the item that measures the displacement.

The advantage of this sensor is that there is not friction between mechanical parts (e.g. the brushes in a linear potentiometer) because the magnetic core is not touching the coils but it is just connected with the movable part. LVDT with full scale of 1 mm can measure nanometric (10e-9 m) displacements.

Figure 7. LVDT scheme.

Test

Materials:

1) Uniaxial press (JENNY max load 200kN) 2) Load cell (full scale 200kN) 3) LVDT (full scale 5 mm) 4) Strain gauges (120 Ohm) 5) Analog to Digital converter (A/D) + Power supplier 6) PC + Matlab script 7) Excel or Matlab or Octave* to elaborate the data 8) Cylindrical Rock samples (20 < diameter [mm] < 28)

*http://www.gnu.org/software/octave/

PC A/D

Power supplier

Load cell

LVDT

Strain gages

Force

Sample

Example of result (without error evaluation)

References

John Dunnicliff, Geotechnical Instrumentation for monitoring field performance, Wiley Interscience - 1993

http://www.omega.com/literature/transactions/volume3/strain.html