detecting and charactorizing flaws in concrete using impedance

USING IMPEDANCE SPECTROSCOPY TO DETECT FLAWS IN CONCRETE

A Thesis Submitted to the Faculty of Purdue University by Mark Niemuth In Partial Fulfillment of the Requirements for the Degree of Master of Science December 2004

ii

Dedicated with love to my family.

iii

ACKNOWLEDGEMENTS

I would like to thank my advisors, Professor Weiss, Professor Olek, and Professor Haddock, for all their dedication and time. I would also like to acknowledge the support and help of all those in the materials department. I would also like to express my appreciation to my family and fiancée who have given love and support allowing me to complete this thesis. I would also like to thank God for giving me strength and wisdom throughout this entire process.

iv

TABLE OF CONTENTS

Page LIST OF TABLES............................................................................................................ vii LIST OF FIGURES ........................................................................................................... ix ABSTRACT..................................................................................................................... xiv CHAPTER 1: INTRODUCTION ....................................................................................... 1 1.1 Background............................................................................................................... 1 1.2 Objectives and Scope................................................................................................ 1 1.3 Organization of Contents .......................................................................................... 2 CHAPTER 2: A REVIEW OF LITERATURE RELATING TO ELECTRICAL IMPEDANCE SPECTROSCPY AND DAMAGE IN CONCRETE ................................. 5 2.1 Introduction............................................................................................................... 5 2.2 A Review of Impedance Spectroscopy in Concrete and Cement ............................. 5 2.3 Applications of Tomography Using Non Invasive Measurements......................... 13 2.4 Typical Types of Field Damage.............................................................................. 13 2.5 Summary................................................................................................................. 15 CHAPTER 3: RELATING FINITE ELEMENT MODELS OF ELECTRICAL RESISTANCE WITH PHYSICAL PROPERTIES, SPECIMEN GEOMTRY, AND CRACK GEOMETRY...................................................................................................... 17 3.1 Introduction............................................................................................................. 17 3.2 Research Approach ................................................................................................. 18

v Page 3.2.1 Analytical Modeling ........................................................................................ 19 3.2.1.1 Defining the Finite Element Model Specimen Geometry and Properties. 21 3.2.1.2 Influence of Resistivity and Current on Geometry Factor........................ 21 3.2.2 Typical Experimental Measurements............................................................... 23 3.2.3 Experimental Procedure for Impedance Spectroscopy Measurement ............. 24 3.2.4 Typical Specimen Geometry............................................................................ 26 3.2.5 Mixture Proportions, Specimen Preparation and Curing Conditions .............. 28 3.3 Determination of Resistivity................................................................................... 29 3.3.6 Method 1 for Determination of Resistivity ...................................................... 30 3.3.7 Method 2 for Determination of Resistivity ...................................................... 32 3.4 Comparison of Model and Experiment................................................................... 34 3.4.8 Unflawed Cylinder Specimen .......................................................................... 35 3.4.9 Flawed Cylinder............................................................................................... 40 3.4.10 Flawed Slab Specimen ................................................................................... 45 3.5 Summary and Conclusions ..................................................................................... 50 CHAPTER 4: USING IMPEDANCE SPECTROSCOPY TO DETECT AND CHARACTORIZE FLAWS IN CYLINDRICAL CONCRETE SPECIMENS............... 51 4.1 Introduction............................................................................................................. 51 4.2 Analytical Geometry Functions .............................................................................. 52 4.3 Unflawed Specimen................................................................................................ 53 4.4 Flawed Specimen.................................................................................................... 54 4.4.1 Influence of the Position of Cut (θ) ................................................................. 55 4.4.2 Influence of the Diameter (D) of Flawed Specimen ........................................ 56

vi Page 4.4.3 Influence of the Width of Cut (w).................................................................... 60 4.4.4 Influence of Depth of Cut (a)........................................................................... 65 4.4.5 Influence of Conductivity of Cut ..................................................................... 70 4.5 Conclusions............................................................................................................. 73 CHAPTER 5: USING IMPEDANCE SPECTROSCOPY TO DETECT AND CHARACTORIZE FLAWS IN SLAB CONCRETE SPECIMENS ............................... 75 5.1 Introduction............................................................................................................. 75 5.2 Analytical Geometry Functions .............................................................................. 76 5.3 Experimental Details............................................................................................... 78 5.4 Comparison of Model and Experiment................................................................... 79 5.5 Unflawed Specimen – Slab Depth Determination.................................................. 89 5.6 Flawed Specimen.................................................................................................... 95 5.6.1 Influence of the Position of Cut (Lcut).............................................................. 95 5.6.2 Influence of Cut Width (w).............................................................................. 97 5.6.3 Influence of Cut Depth (a) ............................................................................... 99 5.6.4 Influence of the Conductivity of Cut, σcut ...................................................... 103 5.7 Conclusion ............................................................................................................ 106 CHAPTER 6: SUMMARY AND CONCLUSIONS...................................................... 109 6.1 Summary............................................................................................................... 109 6.2 Findings and Conclusions..................................................................................... 110 REFERENCES ............................................................................................................... 114

vii

LIST OF TABLES

Table

Page

Table 3.2.1 Influence of resistivity on the geometry factor (k) ........................................ 22 Table 3.2.2 Influence of current load on the geometry factor (k)..................................... 22 Table 3.2.3 Mixture Proportions....................................................................................... 28 Table 3.3.1 Bulk resistance of small cylinder cement specimens..................................... 32 Table 3.3.2 Method 2 resistivity determination data. ....................................................... 33 Table 3.4.1 Cylinder setup 16 electrode configuration alignment numbers ..................... 36 Table 3.4.2 Prism specimen electrode alignment numbers............................................... 37 Table 3.4.3 Geometry factor k for 4" diameter and 4" tall cylinder specimen ................. 38 Table 3.4.4 Geometry factors for 3” x 3” x 4” prism specimen ...................................... 40 Table 3.4.5 Parameters of cylinder with varied cut depth. ............................................... 42 Table 3.4.6 Experimental data for 7/8” x 1/8” cut along the height of 4” diameter cylinder specimen. .......................................................................................................................... 42 Table 3.4.7 Modeled geometry factors (k) of cylinder with varied cut depth. ................. 43 Table 3.4.8 Parameters for comparison of modeled and experimental for a slab specimen ........................................................................................................................................... 45 Table 3.4.9 Measured data for 3" deep slab specimen...................................................... 46 Table 3.4.10 Parameters of Modeled 3” deep slab specimen ........................................... 48 Table 3.4.11 Modeled 3" deep slab specimen, calculated data......................................... 49

viii Table

Page

Table 4.3.1: Parameters of cylinder with varied diameter without cut............................. 53 Table 4.4.1: Parameters of cylinder with varied diameter with cut. ................................. 56 Table 4.4.2 Cylinder with varied diameter, parameters of k vs. θ log fit. ........................ 58 Table 4.4.3 Parameters of cylinder with varied cut width. ............................................... 61 Table 4.4.4 Parameters of cylinder with varied cut depth. ............................................... 66 Table 4.4.5 Parameters of cylinder with varied conductivity. .......................................... 70 Table 5.4.1 Influence of slab depth on bulk resistance..................................................... 84 Table 5.4.2 Data for conversion between 3-d and 2-d...................................................... 85 Table 5.4.3 Data for slab specimen, B of varied Dslab (in) for bulk resistances 3-d/2-d bulk resistance vs. L (in). .......................................................................................................... 87 Table 5.5.1 Parameters of slab for varied slab depth........................................................ 90 Table 5.6.1 Parameters of slab for varied position of cut. ................................................ 96 Table 5.6.2 Parameters of slab for varied cut width ......................................................... 97 Table 5.6.3 Parameters of slab for varied cut depth. ........................................................ 99 Table 5.6.4 Parameters of slab for varied cut conductivity. ........................................... 103

ix

LIST OF FIGURES

Figure

Page

Figure 1.3.1 An overview of the approach used in this thesis for assessing whether IS can be used to detect flaws in concrete elements (a) elements used in phase I to compare finite element analysis with experimental results (b) set-up for phase II, which focuses on cylinder specimen, (c) set-up for phase III, which focuses on the slab specimen. ............. 3 Figure 2.2.1 Example of a typical Bode plot. ..................................................................... 7 Figure 2.2.2 Example of a Nyquist Plot.............................................................................. 8 Figure 2.2.3 Ideal bulk arc of Nyquist plot......................................................................... 9 Figure 3.2.1 Cylinder with a cut perpendicular to the surface.......................................... 19 Figure 3.2.2 Two dimensional view of finite element model. Grey scale levels of the figure on the right represent different levels of electric field vector sum nodal solution. The lighter the levels indicate higher concentration of electric field analogous to stress concentration..................................................................................................................... 21 Figure 3.2.3 Impedance/gain-phase analyzer.................................................................... 24 Figure 3.2.4 Electrodes with sponges and spacers............................................................ 25 Figure 3.2.5 Prism specimen 3" x 3" x 4" with 20 electrodes. The electrodes are 1/4" x 1/4" x 5" each covering 1/4" x 4” area of the specimen. The electrodes are spaced 1/4” apart with a 3/8” gap from a corner to the first electrode. ................................................ 26 Figure 3.2.6 Cylinder specimen 4" diameter x 4" tall with 16 1/4" x 1/4" x 5" tall electrodes spaced evenly around the specimen................................................................. 27

x Figure 3.2.7 Slab specimen 4” x 3” x 18” with ¼” x ¼” x 5” electrodes with a contact area of ¼” x 4” along the height of the specimen. Sponges are between the specimen and electrodes. Only one of the sets of electrode alignments are shown. ............................... 27 Figure 3.3.1 Small cylinder with geometry factor k of 631,000 1/km with screws for electrodes. ......................................................................................................................... 30 Figure 3.3.2 Prism specimen 3” x 3” x 4” with ¼” x 3” x 4” electrodes across entire surfaces with sponge between specimen and electrodes................................................... 33 Figure 3.4.1 Cylinder setup 16 electrode configuration with electrode numbers (numbers next to electrodes) and electrode alignment numbers (numbers between electrodes)...... 36 Figure 3.4.2 Top view of prism specimen with electrode numbers (numbers next to electrodes) and electrode alignment numbers (numbers between electrodes).................. 37 Figure 3.4.3 Comparison of modeled geometry factors verses measured geometry factors on cylinder specimen and prism specimen. ...................................................................... 39 Figure 3.4.4 Two dimensional view of unflawed cylinder with definitions of parameters. ........................................................................................................................................... 41 Figure 3.4.5 Modeled and measured ratios of cut/uncut bulk resistances with 7/8” wide cut. The angle from cut as shown in Figure 3.4.4............................................................. 44 Figure 3.4.6 A comparison between the modeled and measured ratios of cut/uncut bulk resistances with 7/8” wide cut........................................................................................... 44 Figure 3.4.7 Modeled 3" slab deep specimen ................................................................... 47 Figure 3.4.8 Parameters of modeled 3” deep slab specimen ............................................ 47 Figure 3.4.9 Comparison of experimental and modeled slab specimen. The linear fit line represents a perfect model of the experimental results. The 2.75” and 2.25” cuts have a relatively large offset from the line. The 2.75” cut depth deviated from the 1:1 ratio because around the tip of the cut there was a relatively large concentration of current (analogous to stress concentration) for the same element size, similar to using a coarse element in FEA. ................................................................................................................ 49

xi Figure 4.1.1 Phase II of research ...................................................................................... 51 Figure 4.3.1 Unflawed cylinder specimen of k (1/in) vs. D (in)....................................... 54 Figure 4.4.1 Two-dimensional view of flawed cylinder with definitions of parameters.. 55 Figure 4.4.2 Cylinder specimen with varied D (in) for k (1/in) vs. θ (degrees). .............. 57 Figure 4.4.3 Cylinder specimen with varied θ (degrees) for k (1/in) vs. D (in) ............... 57 Figure 4.4.4 Cylindrical specimen of intercepts and slopes of varied D (in) for k (1/in) vs. θ (degrees)......................................................................................................................... 58 Figure 4.4.5 Prediction of k vs. diameter of specimen for varied angle from cut. ........... 59 Figure 4.4.6 Prediction of k vs. angle from cut for varied diameter of specimens........... 60 Figure 4.4.7 Cylinder specimen with varied w/D for k (1/in) vs. θ (degrees). ................. 62 Figure 4.4.8 Cylinder specimen with varied θ (degrees) for k (1/in) vs. w/D. ................. 62 Figure 4.4.9 Cylinder specimen, intercept and slopes of varied θ for k (1/in) vs. w/D .... 63 Figure 4.4.10 Definitions of parameter for cylinder specimen......................................... 64 Figure 4.4.11 Cylinder with varied w/D for k (1/in) vs. a/D with electrode directly over cut...................................................................................................................................... 64 Figure 4.4.12 Cylinder specimen with varied a/D for k (1/in) vs. w/D with electrode centered over cut. .............................................................................................................. 65 Figure 4.4.13 Cylinder specimen with varied a/D for k (1/in) vs. θ (degrees). Linear fits for θ greater than 50o......................................................................................................... 66 Figure 4.4.14 Cylinder specimen with varied θ (degrees) for k (1/in) vs. a/D. ................ 67 Figure 4.4.15 Cylinder specimen with intercept (θ = 0) and slope of varied a/D for k (1/in) Vs θ (degrees). .................................................................................................................. 68 Figure 4.4.16 Prediction of cut depth for cylindrical specimen with varied cut depth/diameter for geometry factor (1/in) vs. the angle from the cut to the electrodes (o) (k vs. θ). ............................................................................................................................ 69

xii Figure 4.4.17 Predictions of cut depth for cylindrical specimen with varied angle from cut the to electrode (θ) (o) for geometry factor (1/in) vs. cut depth/diameter (k vs. a/D)...... 69 Figure 4.4.18 Cylinder specimen with varied σ (S/in) for k (1/in) vs. θ (degrees)........... 71 Figure 4.4.19 Cylinder specimen with varied θ (degrees) for k (1/in) vs. σ (S/in) less than specimen conductivity. ..................................................................................................... 71 Figure 4.4.20 Cylinder specimen with varied θ (degrees) for k (1/in) vs. σ (S/in) greater than specimen conductivity............................................................................................... 72 Figure 5.1.1 Illustration of the Specimen Geometry Investigated in Phase III of research ........................................................................................................................................... 76 Figure 5.2.1 Two dimensional view of slab specimen with definitions of parameters. ... 77 Figure 5.4.1 (a) A graphical illustrated (b) FEM representation 3-d of a slab specimen of semi-infinite width, length, and depth. ............................................................................. 80 Figure 5.4.2 Three dimensional view of slab specimen of infinite depth and length and finite width. ....................................................................................................................... 81 Figure 5.4.3 Two dimensional view of slab specimen of infinite depth and length with specimen width equal to electrode length......................................................................... 82 Figure 5.4.4 Two dimensional view of slab with infinite width and length and finite depth. ........................................................................................................................................... 82 Figure 5.4.5 Bulk resistance (Ohms) vs. distance from cut to closest electrode (in)........ 83 Figure 5.4.6 Slab specimen with varied D (in) for bulk resistances 3-d/2-d vs. L (in). ... 86 Figure 5.4.7 Slab specimen, intercept and slope of varied L (in) for bulk resistances 3d/2-d vs. D (in).................................................................................................................. 86 Figure 5.4.8 Slab specimen, B of varied Dslab (in) for bulk resistances 3-d/2-d bulk resistance vs. L (in). .......................................................................................................... 88 Figure 5.5.1 Slab specimen with varied distance between electrodes, L, (in) for m vs. Dslab less than 10”.............................................................................................................. 90

xiii Figure 5.5.2 Slab specimen with varied slab depth, Dslab, (in) for m vs. L (in). ............... 91 Figure 5.5.3 Slab specimen M and B of fits of varied slab depth, Dslab, (in) for m vs. L (in) where L is greater than 2 inches. ...................................................................................... 92 Figure 5.5.4 Predictions for Slab specimen with varied distance between electrodes (in) for m vs. Dslab (in). ............................................................................................................ 93 Figure 5.5.5 Predictions for slab specimen with varied slab depth (in) for m vs. L (in). . 93 Figure 5.5.6 Slab specimen with varied slab depths (in) for geometry factor m vs. length between electrodes normalized by the slab depth (in) ...................................................... 94 Figure 5.6.1 Slab specimen for m vs. distance from cut (in). ........................................... 96 Figure 5.6.2 Slab specimen with varied length between electrodes (in) for m vs. w (in). 98 Figure 5.6.3 Slab specimen with varied w (in) for m vs. L (in)........................................ 98 Figure 5.6.4 Slab specimen with varied Lcut (in) for m vs. a (in). .................................. 100 Figure 5.6.5 Slab specimen with varied a (in) for m vs. Lcut (in). .................................. 100 Figure 5.6.6 Slab specimen, M and B of varied a (in) for m vs. Lcut greater than 1 ½”. 101 Figure 5.6.7 Prediction of slab specimen with varied Lcut (in) for m vs. a (in). ............. 102 Figure 5.6.8 Predictions of slab specimen with varied a (in) for m vs. Lcut (in)............. 102 Figure 5.6.9 Slab specimen with varied Lcut (in) for m vs. σcut (S/in). ........................... 103 Figure 5.6.10 Slab specimen with varied conductivity of cut (S/in) for m vs. Lcut (in).. 104 Figure 5.6.11 Slab specimen with varied Lcut (in) for m vs. normalized σcut with bilinear log fits. ............................................................................................................................ 105 Figure 5.6.12 Specimen conductivity prediction from varied cut conductivity data...... 106 Figure 5.7.1 Prediction for the position of cut. ............................................................... 107

xiv

ABSTRACT

Niemuth, Mark. M.S., Purdue University, December 2004. Using Impedance Spectroscopy to Detect Flaws in Concrete. Major Professors: Jason Weiss and Jan Olek.

Electrical measurements in concrete and cement research have been used for a wide variety of applications, to determine a wide variety of properties. This research broadens the applications of electrical measurements to now be able to determine flaw characteristics in concrete. One important byproduct of use and abuse of concrete is flaw development. The major motivation is to determine if electrical measurements can be used in a nondestructive manor to evaluate flaws in the concrete. The feasibility of using electrical measurements is determined in specific setups using impendence spectroscopy and finite element analysis. Once it was determined that electrical measurements are related to the flaws the specific characteristics of the flaw (geometry and conductivity) were varied to produce a quantitative relationship between electrical measurements and the flaw characteristics. Electrical measurements were determined to be insensitive to cut width. By varying the conductivity of the cut the specimen conductivity can be predicted. The quantitative relationship can be used in conjunction with simple electrical measurements to predict cut depth. Although this research looked at two specific cases the methodology can be reproduced for other specific setups of interest.

1

CHAPTER 1: INTRODUCTION

1.1 Background It has been the said that you can count on concrete to do two things: harden and crack. This research investigated the development of a testing technique to detect the later of these two “absolutes” and was aimed at determining the general state of the geometry of flaws that may exist in concrete. Simply stated, this research investigated the influence a flaw has on the electrical bulk resistance of a concrete specimen. This research investigates the use of external electrodes to enable exceptional flexibility in assessing the performance of in situ concrete.

1.2 Objectives and Scope

The main objective of this research was to investigate whether electrical properties can be used to detect flaw characteristics in cementitious systems. Specifically, the flaw characteristics that are of interest are the width and depth of the flaw. An array of external electrodes was used to create an electrical “image” of the geometry of a crack in a specimen. The hypothesis of this research is that the bulk resistance can be used to locate and quantify flaw characteristics. Furthermore, after establishing the validity of this hypothesis mathematical relationships were developed to establish a quantitative relationship between the bulk resistance and the geometry of the flaw characteristics. In addition to assessing the geometry of the flaws, the specimen geometry (i.e., slab thickness and cylinder diameter) was also assessed. The results of the finite element

2 analysis (FEA) were compared with physical measurements and then used to link the bulk resistance measurements with the specimen geometry and flaw geometry.

1.3 Organization of Contents

This research was divided into three phases (I, II, and III). An overview of the first phase of this research (Chapter 3) is shown in Figure 1.3.1, which focused on comparing the results of FEA with experimental measurements. The FEA was compared to physical measurements for unflawed specimens, flawed cylinder specimens, and flawed slab specimens. The reasonable correlation of the FEA with experimental results enabled it to be used in phases II and III. Phase II (Chapter 4) relates the electrical bulk resistance to the geometry characteristics of a flaw in a cylindrical specimen. The specific nature of the experimental program consists of a cylindrical specimen with diametrically opposed electrodes around the circumference, thereby creating an array of electrodes around the entire specimen, as shown in Figure 1.3.1 (b). The influence of specimen diameter on the bulk resistance was measured. The influence of the position of the electrodes relative to the flaw was measured from for every different setup. The flaw is defined as a cut perpendicular to the surface. In addition, several other features were varied including the cut conductivity, cut depth, cut width, and position of the electrodes. Equations were developed to relate the bulk resistance as a function of the position of the electrodes relative to the cut to the specimen diameter, cut depth, cut width and cut conductivity. Phase III (Chapter 5) extends this work to the more general case where there is only single sided access to a slab (Figure 1.3.1 (c)). The FEA was preformed using a pseudo two-dimensional environment, which is valid if the electrodes are the same size as the width of the specimen (i.e., there are no effects from the ends of the electrodes). In many instances it would be impossible to have an electrode that is the same size as the slab. As a result, a correction is necessary to include the effects for the ends of the electrodes. FEA was performed for an unflawed slab to determine the link between bulk resistance

3 and the distance between electrodes and thickness or depth of the slab. Using the pseudo two-dimensional setup, a flaw was introduced, characterized as cut perpendicular to the surface, with a defined width, depth, and conductivity. The cut width, cut depth, and cut conductivity were each independently varied along with distance from electrodes to the flaw to quantify the effects of each of these parameters on the electrical impedance. Mathematical relationships were then developed between each of these parameters and the bulk resistance.

(a) Figure 1.3.1 An overview of the approach used in this thesis for assessing whether IS can be used to detect flaws in concrete elements (a) elements used in phase I to compare finite element analysis with experimental results (b) set-up for phase II, which focuses on cylinder specimen, (c) set-up for phase III, which focuses on the slab specimen.

4 PHASE Cylinder specimen with an external array of electrodes

Cut Electrod Cylinde Specime Idealization

Finite Element Analysis

(b)

Slab Specimen

Cut Electrode

(c)

Test Specimen Geoemtry

5

CHAPTER 2: A REVIEW OF LITERATURE RELATING TO ELECTRICAL IMPEDANCE SPECTROSCPY AND DAMAGE IN CONCRETE

2.1 Introduction

This chapter provides a general overview of what impendence spectroscopy is and how the cement and concrete community has used electrical impedance spectroscopy. In addition, it discusses how electrical measurements have been used in other disciplines. Finally, this chapter describes some typical types of damage in cement and concrete.

2.2 A Review of Impedance Spectroscopy in Concrete and Cement

Electrical impedance spectroscopy (EIS) can be used to detect the electrical resistance of a specimen (i.e., the bulk resistance, Rb). The bulk resistance, which is geometry dependent, can be related to the resistivity (ρ), which is geometry independent material property (k) as shown in equation (2.2.1). Rb = ρ⋅ k

(2.2.1)

where k is a geometry factor, which is a function of specimen geometry and electrode geometry and electrode alignment. In electrical impedance spectroscopy (EIS) an alternating current is applied to the specimen and the resulting voltage drop (V) is measured. Electrical impedance (Z) is determined using equation (2.2.2), which is a more complete form of Ohm’s law in which V, I, and Z can be complex variables (real and imaginary components).

6 V = I⋅Z

(2.2.2)

where Z is the impedance (measure of resistance), I is the alternating current (AC), and V is the voltage. The output of EIS are impedances which can be shown in a Bode plot as seen in Figure 2.2.1 The total impedance (|Z|) can be subdivided into a real component (resistance) and imaginary component (capacitance and inductance) defined in (2.2.3) Z = Z ''2 + Z '2

(2.2.3)

where Z’’ is the imaginary impedance and Z’ is the real impedance. The phase angle (θ) is defined in (2.2.4) as:

⎛ Z '' ⎞ θ = tan −1 ⎜ ⎟ ⎝ Z' ⎠

(2.2.4)

The Bode plot shows the parameters measured by the gain-phase analyzer over a wide range of frequencies. Figure 2.2.1 (a) shows the total impedance, which is a combination of imaginary and real impedance. Figure 2.2.1 (b) shows the real impedance, which is the resistive component of the system. Figure 2.2.1 (c) shows the imaginary impedance, which is a capacitive component of the system. The phase angle shown in Figure 2.2.1 (d) indicates how far the total impedance is out of phase. The phase angle minimum falls on the imaginary impedance minimum.

7 800 Imaginary Impedance (Ohms)

Total Impedance (Ohms)

800

600

400

200

0 1.E+02

1.E+04 1.E+06 Frequency (Hz)

1.E+08

600 400 200 0 -200 1.E+02

(a)

1.E+08

(c)

800

1.8 1.6 Phase Angle (degrees)

Real Impedance (Ohms)

1.E+04 1.E+06 Frequency (Hz)

600

400

200

1.4 1.2 1 0.8 0.6 0.4 0.2

0 1.E+02

1.E+04 1.E+06 Frequency (Hz)

1.E+08

0 1.E+02

(b)

1.E+04

1.E+06

1.E+08

Frequency (Hz)

(d)

Figure 2.2.1 Example of a typical Bode plot.

The complex electrical impedance (|Z|) has both a resistive and capacity component (McCarter, 1996a) while the true bulk resistance of a material does not have a capacitive component. The capacitive component can be seen in the bulk arc of the Nyquist plot. An example of a typical Nyquist plot is shown in Figure 2.2.2.

8 -350

Imaginary Impedance (Ohms)

-300

Bulk arc

Electrode arc

-250

-200

-150

-100

-50

0 250

Bulk resistance

300

350

400

450

500

550

600

Real Impedance (Ohms)

Figure 2.2.2 Example of a Nyquist Plot The Nyquist plot is made over a wide range of frequencies since large errors in interpreting the bulk resistance may occur if one single frequency is used (Shane et al., 1997). The bulk resistance is the real impedance, which occurs at the intersection of the bulk arc and the electrode arc (where the imaginary impedance is a minimum), which is equivalent to DC resistance. On the Nyquist plot, the electrode arc (the resistance due to the electrode) seen in Figure 2.2.2 occurs at low frequencies (generally kilohertz and lower) while the bulk arc (resistance due to specimen interference) occurs at high frequencies (generally kilohertz to megahertz). The bulk resistance is a measure of the purely resistive component (i.e., no capacitive component). An idealized view of the bulk portion impedance plot can be seen in Figure 2.2.3. The total impedance or amplitude (|Z|) and phase angle (θ) can be seen in this figure as well. This bulk arc can be represented as a combination resistors (R1 and R2) and capacitor (C). The frequency at the peak point (w) is defined as the reciprocal of the product of R2*C.

9

-Imaginary Impedance

Bulk Arc R1

C

Electrode Arc

R2 w = 1/(R2 * C) |Z| R1

0

R2 Real Impedance

Figure 2.2.3 Ideal bulk arc of Nyquist plot

When comparing the example of the Nyquist plot from the experiment (Figure 2.2.2) and the ideal bulk arc (Figure 2.2.3) an obvious difference can be noticed. In the actual experiment data the Nyquist plot does not enable the bulk arc to be projected back to intercept with the real impedance axis. This occurs due to equipment limitations at high frequencies needed to acquire a complete plot of the bulk arc. According to the ideal bulk arc, the bulk resistance is simply R1 + R2. This bulk resistance is equivalent to the direct current (DC) resistance (Coverdale et al., 1995). DC measurements used in cement and concrete systems are not typically used since the resistivity that occurs due to direct current may change due to a polarizing effect (Mindess et al., 2003). AC is also used to minimize electrochemical processes in the electrode/concrete interface (McCarter et al., 1990). From the phase angle and total impedance data, the real and imaginary impedances can be calculated. The frequency (w) at the peak of the bulk arc as seen in Figure 2.2.3

10 can be used to estimate the dielectric constant of the material, which is governed by the microstructure of the material (Coverdale et al., 1994). The frequency at which the bulk resistance occurs decreases with an increase in the drying of the concrete specimen (McCarter, 1990; Schieβl et al., 2000; Weiss et al., 1999). In the ideal case, where the electrode and the specimen have the same cross sectional areas (A) and the electrodes are parallel to each other, equation (2.2.5) holds because there is an uniform electric field. This allows for a direct measurement of the resistivity (inverse of conductivity) of the specimen. For specimens with a high conductivity (σ) and/or a small distance between electrodes (L) relative to the cross sectional area of the electrodes (A) a uniform electric field can be assumed. This enables the assumptions that external electrodes apply uniform electric field to the specimen. This case corresponds to a uniform electric field applied by the electrodes (Wagnesson, 1986). σ=

L A⋅Rb

(2.2.5)

where σ is the conductivity. Yang (2004b) conducted research using uniform electric fields in concrete specimens subjected to freezing and thawing damage to correlate conductivity with the elastic modulus. In that work, a prismatic specimen was used with a cross sectional area that was equal to the cross sectional area of the electrodes, which resulted in an effective conductivity measurement. It was observed that as the damage increased the effective conductivity increased. This increase in conductivity was due to connectivity of water filled cracks that increased as the damage progressed. The amount of damage was quantified by measuring the dynamic elastic modulus. A bilinear relationship between bulk resistance and elastic modulus was then established. For the general case of measuring the bulk resistance of concrete using external electrodes, equation (2.2.5) may not be applicable, because the electrode configuration may not result in a uniform electrical field. To quantify the nonuniformity of the electric field, a geometry factor (k) can be introduced as illustrated in equation (2.2.6) (Christensen et al., 1994).

11 σ=

k Rb

(2.2.6)

This geometry factor (k) accounts for distance between electrodes (L), the cross sectional area of the electrodes (A), the cross sectional area of the specimen, specimen size, and electrode alignments (nonparallel alignments). This geometry factor (k) can be calculated using equation (2.2.7). For electrodes that are spaced relatively close or cross sectional areas of the electrodes and specimen are equal the constant m is considered to be equal to one, which simplifies to (2.2.6) to (2.2.5) (Hummel 1998).

k=

L A⋅m

(2.2.7)

For setups using external electrodes with cross sectional areas smaller then the cross sectional areas of the specimen equations (2.2.6) and (2.2.7) are valid, because m can account for the nonparallel electric fields that will result. The constant m is only constant for specific electrode alignments and specimen geometries. Varying either of these parameters can change m, thus it can be considered as a geometry factor. The conductivity of the sample is dominated by the conductivity of the capillary porosity. The conductivity is a function of amount of evaporable water in the paste (φcap), which is a function of the concentration of mobile ions in the pore fluid, the connectivity or tortuosity of the pore network (β), and the conductivity of pore solution (σo) (Christensen et al., 1994; Coverdale et al., 1995).

σ = ϕcap ⋅β ⋅ σo

(2.2.8)

The degree of saturation has been linked to the resistance ratio, which is the ratio of the bulk resistance after a certain time of drying to the bulk resistance in the initial undried state (McCarter et al., 1996b; Rajabipour et al., 2004b). The increase in resistivity during the first 50 hours is due to the change in tortuosity in the pore structure (as well as the changing conductivity of the pore solution) during that time.

12 A procedure developed by Neithalath et al. (2004) can be used to determine the pore connectivity by saturating the pore network with an electrolyte of known conductivity, measuring the effective conductivity, and then saturating the pore network with another electrolyte of different conductivity. In drying concrete, conductivity has been shown to be highly dependent on the time of drying, relative humidity, and depth from the drying surface. Drying also decreases the frequency at which the bulk resistance occurs (Schieβl et al., 2000). Rajabipour et al. (2004a) have investigated the effects of drying on local conductivity as a function of depth of from drying surface in concrete systems. These conductivity profiles were then translated into a humidity profile and moisture diffusion coefficient, which can be used for moisture loss and drying shrinkage profile simulations (Rajabipour, 2004a). A corollary to this work is a technique to predict a moisture profile in cementious system from a single electrical measurement at any time. This technique was based on impedance measurements and finite element analysis (Rajabipour et al., 2004b). Two point spectra (measurements using only two electrodes) are useful for bulk resistance measurements, but measurement of dielectric constant and arc depression angle require multipoint techniques employing more than two electrodes as in the two point spectra method. The multipoint techniques have the same two electrodes as two point technique, but the current or voltage is applied from different electrodes. Problems may occur in reproducibility for dielectric constant and arc depression when using single point techniques. The ease and reproducibility of the two point technique makes it advantageous to multipoint techniques. In two point techniques the AC amplitude is kept low to minimize and inductive effects. To obtain precise bulk electrode arc values, multipoint techniques need to be employed (Ford et al., 1995). IS has been used to measure many different things; moisture profiling of drying concrete (Schieβl et al., 2000), determining depth of penetration of water, estimating the molar concentration of sodium chloride within the pore water (McCarter et al., 1996b; Rajabipour, 2003; Rajabipour et al., 2004a; Rajabipour et al., 2004b), ionic diffusivity

13 and water permeability (Christensen et al., 1994), determining the existence and amount of pulverized fly ash (McCarter et al., 1996b)

2.3 Applications of Tomography Using Non Invasive Measurements

Electrical impendence techniques have proven valuable in other fields of research by working off the premise that differences in conductivities can be measured, which is what this research is based on. Electrical impedance tomography (EIT) has been used in medical applications, which uses electrical measurements to reconstruct and display approximate pictures of electrical conductivity and permittivity in the body (Cheney et al., 1999). An AC current is applied to various pairs of electrodes in an array of electrodes. The other electrodes in the array measure the resultant voltages (Brown et al., 1985; Brown et al., 1987). EIT has also been used in environmental engineering applications to look at the structural and hydraulic nature of porous media and fluids contained in such media (Kemna et al., 2000). EIT is useful for producing high resolution images of hydrogeological structures and assessment of dynamic processes in the subsurface environment such as groundwater flow by injecting tracers measured from electrodes in bore holes (Binley et al., a2002; Binley et al., 2002b).

2.4 Typical Types of Field Damage

Damage in concrete can lead to many long term performance related problems. Damage in concrete, specifically cracking, can lead to reductions in freeze thaw resistance, corrosion resistance, and ultimate strength. Damage can occur due to freezing and thawing in concrete. The damaged is characterized by gradual formation of microcracking. Measured crack densities correlate fairly well with the measured durability factors (i.e., resonant frequency) (Jacobson et al.,

14 1996; Jacobson et al., 1995a). Acoustic emission events caused by freezing and thawing took place during freezing and thawing period and AE can be used to determine the amount of frost damage in mortar (Shimada et al., 1991). Restricting water transport through the cover zone (first few millimeters) brings on mechanical damage due to freezing and thawing (Litvan, 1992). Freezing temperature is dependent on size of the pore neck, along with presence of alkalies. Water freezing will not take place if the amount of water is less then amount needed to form two layers of ice (Litvan, 1970). When freezing does occur, severe dilation causes internal tensile stresses, which leads to cracking. The hydraulic pressure from water increasing its volume 9% upon freezing was thought to be the reason for the damage, but it seems that the damage is mainly due to water in the concrete moving towards the freezing sights due to either osmotic pressure or a difference in the vapor pressure. As the water freezes, the hydraulic pressure from the excess water near the freezing sight causes cracking in the capillary pores severely damaging the cement paste. (Mindess et al., 2003) Damage can also occur because static fatigue and surface scaling, which is usually associated with high w/c ratios (Zhou et al., 1994). The degree of saturation is the most important parameter in determining the damage due to freezing and thawing. The porosity of the concrete also influences the effects of freezing and thawing (McCarter et al., 1996b). Air entrainment is an effective and reliable mean of protecting the concrete from freeze thaw damage. The free space provided by the air bubbles inside concrete allows a place for excess water to migrate thus relieving the hydraulic pressure. Air entrainment is not effective if the specimen is saturated and the air bubbles are filled with water. Such concrete has a greater susceptibility to freezing because there is more water in the concrete (Mindess et al., 2003). Some aggregates with very fine pores, which have a relatively high porosity and low permeability, are susceptible to cracking during freezing and thawing because of hydraulic pressure. This occurs when the aggregates become saturated and then freezes. This phenomena is labeled as D-cracking and occurs most frequently near the surface. D-

15 cracking occurs when water is forced from aggregates with high absorption into the surrounding paste causing hydraulic pressure at the paste-aggregate interface. This hydraulic pressure forms cracks in the concrete (Mindess et al., 2003). Jacobson showed that cracks followed around most aggregate particles (1995a; Samaha et al., 1992) Zhou et al. (1994) performed freeze thaw testing on mortar specimens that had flexural preloading up to 50% of the flexure strength. They showed a correlation existed between the preloading level and with the chances that there is earlier failure. Cracking in concrete increases permeability, allowing more moisture into the concrete. Water permeability tests showed that cracks induced by tensile loading that are less than 50 microns wide have little impact on permeability. The permeability increases rapidly in the 50 to 200 micron range, becoming a steady increase after 200 microns (Wang et al., 1997). Work by Samaha et al. also showed that compressive loading increased liquid transport. Loading increased the transport by up to 20%. The loading started to affect the transport when 75% of the maximum load had been reached (1992). Cracking and flaws in concrete influence corrosion. Cracking allows the initiation of corrosion to happen sooner and increases the rate at which it occurs (Yoon et al, 2000). Liquid chlorides can be transported faster and with less impedance to reinforcement and lead to corrosion when there are flaws present in the protective coating (Locoge et al., 1992). The importance of flaws and damage translates to the importance of evaluation of those flaws and damage in concrete, which lead to effective methods for prevention and repair of concrete structures.

2.5 Summary

The interrelated nature of freeze/thaw durability, absorption, and cracking means that any single one of those properties can play a significant role in the durability and life time of concrete. The ability to find and describe cracking and flaws in concrete allows for

16 predictions of the service lifetime of the concrete and corrective measures to be applied to extend the life of the concrete, which leads ultimately to safer more efficient structures.

17

CHAPTER 3: RELATING FINITE ELEMENT MODELS OF ELECTRICAL RESISTANCE WITH PHYSICAL PROPERTIES, SPECIMEN GEOMTRY, AND CRACK GEOMETRY

3.1 Introduction Cracking in concrete structures can lead to various durability problems that can significantly reduce the service live of the structure. The ability to characterize these flaws with greater ease and accuracy has the potential to enable a more complete and accurate evaluation of structures in-situ. While several techniques have been proposed to assess cracking, alternating current impedance spectroscopy (abbreviated in this document as ACIS) is one proposed approach that shows promise in these areas. While impedance spectroscopy (IS) has been used in concrete over the last two decades, the majority of the reported work has used either embedded electrodes or electrodes configurations that enable a parallel electrical field (i.e., the electrode cross sectional area was the same as the specimen cross sectional area). This chapter describes results of an investigation to determine whether local variations in conductivity caused, for example, by flaws or cuts in a cylinder can be measured and described in concrete using IS methods with externally applied electrodes. Toward this end electrical measurements were performed on unflawed concrete elements and compared with results from a finite element model (FEM). This model was then modified to consider specimens with cracks. Geometry functions were developed to relate the cut geometry to changes in the bulk resistance measurements.

18 3.2 Research Approach

To determine whether a single crack (or cut1) can be detected in a concrete specimen using electrical resistance a systematic approach was taken. This approach consisted of applying the IS techniques to simple specimen geometries and crack geometries and then moving to more complex specimen geometries. Figure 1.3.1 shows the research approach for all three phases of this study: Phase I (Figure 1.3.1 (a)), Phase (Figure 1.3.1 (b)), and Phase (Figure 1.3.1 (c)). Phase I was primarily concerned with conflating the finite element model predictions of electrical resistance to the physical bulk resistance measurements of the specimen with specific electrode alignments. The first specimen geometry investigated consisted of a cylindrical specimen with a cut perpendicular to the outer circumference that runs along the length of the specimen as illustrated in Figure 3.2.1. Using this geometry, Phase II was performed by considering two electrodes that were placed at diametrically opposed positions. An array of measurements were then taken from diametrically opposed electrode alignments by rotating the electrode pair systematical around the circumference of the specimen. These measurements were used to determine if the cut can be characterized using the array of electrodes. In Phase III a second geometry was considered consisting of a slab specimen with a cut perpendicular to the top surface with electrodes placed on the top surface. This geometry was used to determine whether ACIS can be used to determine properties of a flaw that originates from the same surface as the electrodes, which is frequently the case in the field when access is limited to one side of the structure.

1

Note the word cut will be used to describe a defect that is placed in the specimen with a measurable width (typically X mm x Y mm x Z mm) and straight orthogonal walls while a crack will used to describe a defect in a specimen with tortuous walls and a generally smaller width that may vary from point to point.

19

Cylinder

Cut perpendicular to the surface

Figure 3.2.1 Cylinder with a cut perpendicular to the surface.

To fully assess whether the use of ACIS may permit flaw detection, an analysis was performed using two approaches. The first approach was to physically take measurements of the bulk resistance using an electrical impedance gain phase analyzer. The second approach consisted of using finite element modeling (FEM). The finite element (FE) simulations were advantageous because they allowed for many different specimen geometries and electrode alignments to be analyzed quickly. In addition the FE simulations reduce sources of error from laboratory measurements. The advantage of taking the actual lab measurements is that the data can be directly correlated to field applications. Some disadvantages of physical measurements are that the concrete and cement specimens are highly susceptible to environmental conditions (i.e., moisture content change). In addition, there is inherent variability from the setup, operator, equipment, and aggregate (which has a low conductivity), which can cause errors if there is a high concentration in certain areas.

3.2.1 Analytical Modeling ANSYS University Advanced 7.1 (ANSYS 2002) software was used to create a finite element model for the specimens investigated in this study using an approach similar to

20 that described by Rajabipour et al. (2004). This FEM considered the specimens, electrodes, and applied current loads. The load in this case was electrical current, in amps. The elements that were used were two dimensional, linear, plane three or four-node elements depending on the geometry. This permitted a temperature and voltage to be defined at each node. The temperature was considered to be constant throughout modeling and was assumed to be 23°C while taking measurements. While modeling was performed in three dimensions, the solutions were adjusted to consider variations in two dimensions while the third dimension was one unit thick. A current source with a constant amplitude was used to apply the electrical load (I) to the electrode elements. As this electrical load was applied, a voltage drop occurred across the electrodes and specimen elements, which was measured. The application of ohms law enabled the voltage drop (V) and the defined current to be used to calculate the bulk resistance (Rb) as seen in (3.2.1)2. Rb =

V I

(3.2.1)

where I is the current in amps. There is an analogous relationship between stress-strain response and the electrical response of a specimen, where effective elastic moduli (E) can be thought to be analogous to effective resistivity (ρ); voltage drop (V) analogous to the strain (ε); electric current (I) analogous to the stress (σ) (Torquato, 2002; Weiss, 2003). This relationship is shown in equation (3.2.2).

⎛1 I⎞ ⎛ σ⎞ ⎜ ∝ ⎟ ∝ ⎜E ∝ ⎟ ε⎠ ⎝ρ V⎠ ⎝

2

(3.2.2)

Bulk resistance replaces the impedance (Z), because the bulk resistance is the purely resistive component impedance (Z).

21 3.2.1.1 Defining the Finite Element Model Specimen Geometry and Properties

The specimen was idealized using three different types of elements, which included concrete elements, electrode elements, and cut elements. An example of these elements can be seen in Figure 3.2.2. The specimen, electrode, and cut geometries were defined assuming perfect conductivity between different elements of different conductivity. Each element was characterized by resistivity.

Specimen Cut

Electrode

Figure 3.2.2 Two dimensional view of finite element model. Grey scale levels of the figure on the right represent different levels of electric field vector sum nodal solution. The lighter the levels indicate higher concentration of electric field analogous to stress concentration.

3.2.1.2 Influence of Resistivity and Current on Geometry Factor

In order to assess the influence of the magnitude of the resistivity and current on the results obtained from finite element analysis (FEA), a series of simulations were

22 performed in order to determine how these properties influence the measured geometry factor (k) defined in equation (3.2.3). k=

Rb ρ

(3.2.3)

where Rb is bulk resistance and ρ is resistivity. For the modeling case the resistivity was simply a input parameter and was kept constant at 1000 ohm•in. The current load was also kept constant at 1 amp. An unflawed cylindrical specimen geometry was used (as seen in Figure 3.2.2 except without the cut). The geometry factor is not a function of resistivity is as seen in Table 3.2.1 where the resistivity is varied significantly without significant change in k. Also, varying current does not change the geometry factor (k) significantly in the modeling case as seen in Table 3.2.2.

Table 3.2.1 Influence of resistivity on the geometry factor (k) Resistivity (ohm*in) 0.01 0.1 1 10 100 1000 10000

Geometry factor k (1/in) 2.2437 2.2435 2.2434 2.2434 2.2434 2.2434 2.2435

Table 3.2.2 Influence of current load on the geometry factor (k) Current (Amp) 0.001 0.01 0.1 1 10 100 1000

Geometry factor (1/in) 2.2434 2.2434 2.2434 2.2434 2.2434 2.2434 2.2434

23 3.2.2 Typical Experimental Measurements

The electrical impedance gain-phase analyzer that was used to measure the impedance (Z) was a Solartron SI 1260 Impedance/Gain-Phase Analyzer. The imaginary impednace (Z’’) and real impedance (Z’) were then calculated by ZPlot for Windows (Johnson 2000) and ZView for Windows (Johnson 2000) and related using equations (3.2.4), (3.2.5), and (3.2.6).

Z = Z '+ j ⋅ Z ''

(3.2.4)

Z = Z '2 + Z ''2

(3.2.5)

⎛ Z '' 180o ⎞ θ = tan −1 ⎜ − ⋅ ⎟ ⎝ Z' π ⎠

(3.2.6)

where j is an imaginary number defined as

−1 ,θ is the phase angle, and |Z| is the total

impedance A Nyquist plot, imaginary impedance versus real impedance, can be plotted from the impedance/gain-phase analyzer output. Similarly Bode plots can be constructed which constructed which consist of total impedance (|Z|) and the phase angle (θ) plotted as a function of frequency. Examples of typical Nyquist plot and Bode plots of a shown in Figure 2.2.2 and Figure 2.2.1. The setup for that data was a 4” diameter by 4” tall concrete cylinder with ¼” by ¼” by 4” tall electrodes (i.e., ¼” x 4” area was in contact with the cylinder) aligned axially and diametrically opposed. Bulk resistance is the parameter that is of interest in this study. Bulk resistance can be obtained from a Nyquist, as seen in Figure 2.2.2, which is the real impedance at the minimum imaginary impedance where capacitance effects have been minimized. This bulk resistance is equivalent to DC resistance.

24 3.2.3 Experimental Procedure for Impedance Spectroscopy Measurement

The gain phase analyzer, seen in Figure 3.2.3, was attached to the specimen using external electrodes, which were placed on the surface of the specimen with wet sponges in-between the electrodes and specimens. The sponges were used to ensure uniform contact between the electrodes and specimen. The electrodes consisted of ¼” (0.64 cm) square stainless steel rods that were cut approximately 1 inch longer then the specimen for easy attachment to the impedance/gain-phase analyzer, which translates to a cross sectional area of ¼” x height of the specimen. The impedance/gain-phase analyzer worked by applying a potential over a range of frequencies and measuring the electrical impedance. The typical range was 10 MHz to 100 Hz while taking 10 measurements per decade.

Figure 3.2.3 Impedance/gain-phase analyzer

Because drying changes the resistivity of the specimen, the effect of drying was minimized by carefully controlling the contact with the air (Christensen et al., 1994;

25 Coverdale et al., 1995; McCarter et al., 1996b; Schieβl et al., 2000). Measurements were not taken before 12 minutes of exposure to the atmosphere to allow the surface moisture to disappear and to minimize error from surface moisture loss. The overall drying during testing was minimized with acrylic spacers between the electrodes on the prism specimen and covering the top and bottom of the specimen with acrylic plates as seen in Figure 3.2.5. The acrylic spacers were epoxied to the electrodes so that when the electrodes and spacers were placed on the side of the specimen there would be ~1/16 gap between the electrode and specimen. This gap was filled with a wet sponge to insure uniform contact between the electrode and specimen. The sponges, electrodes, and spacers can be seen in Figure 3.2.4. The electrode/spacer/sponge assembly was then clamped to the specimen. The cylinder setup was then covered in plastic wrap to further prevent drying during testing.

electrodes

sponges spacers

electrode, spacers, and sponge

Figure 3.2.4 Electrodes with sponges and spacers.

The electrode contact area (¼” width x the height of the specimen) was assumed to be constant for the modeling and measurements. For the cylinder specimen the electrode

26 configuration that was predominantly used was diametrically opposed alignments; the shortest path between electrodes through the center of the cylinder. Electrodes were always parallel relative to each other and the flaw if applicable.

3.2.4 Typical Specimen Geometry

Specific geometries used for measurements are shown in Figure 3.2.5, Figure 3.2.6, and Figure 3.2.7.

Figure 3.2.5 Prism specimen 3" x 3" x 4" with 20 electrodes. The electrodes are 1/4" x 1/4" x 5" each covering 1/4" x 4” area of the specimen. The electrodes are spaced 1/4” apart with a 3/8” gap from a corner to the first electrode.

27

Figure 3.2.6 Cylinder specimen 4" diameter x 4" tall with 16 1/4" x 1/4" x 5" tall electrodes spaced evenly around the specimen.

Figure 3.2.7 Slab specimen 4” x 3” x 18” with ¼” x ¼” x 5” electrodes with a contact area of ¼” x 4” along the height of the specimen. Sponges are between the specimen and electrodes. Only one of the sets of electrode alignments are shown.

28 3.2.5 Mixture Proportions, Specimen Preparation and Curing Conditions

The mixture proportions for the specimens used in this research are shown in Table 3.2.3. The cement was Type I portland cement manufactured by Lonestar, Inc. The course aggregate used was limestone with a maximum size of ¾” (19mm). The fine aggregate was a river sand. The water reducer used in this investigation was WRDA-82 made by Grace Construction Products. The air entrainer was Daravair 1400 made by Grace Construction Products.

Table 3.2.3 Mixture Proportions Mixture #

1

2

3

Mixture description

Paste

Concrete without air

Concrete with air

Coarse Aggregate (kg/m3)

N/A

821

821

Fine Aggregate (kg/m3)

N/A

1003

1003

w/c

0.35

0.43

0.43

Water Reducer (kg/m3)

N/A

0.435

0.435

Air entrainer (kg/m3)

N/A

N/A

0.035

The specimens were cast according to ASTM C-192. The coarse and fine aggregate were premixed with 1/3 of the water for 1 minute before the cement, water, water reducer, and air entrainer were added. That mixture was then mixed for 3 minutes, stopped for 3 minutes, and then mixed for 2 more minutes. The molds were filled half way, rodded, filled the rest of the way, rodded, and then vibrated on a vibrating table. The surface was finished with a smooth steel trowel. The specimens were then covered with wet burlap and plastic sheets. The specimens were demolded after 24 hours and placed in a 98 ± 2% relative humidity chamber until the time of testing. The cylinders and prism specimens from mixtures 1 and 2 were stored in a 98 ± 2% relative humidity chamber for the first week

29 after being cast after which they were submerged in a 2 M solution of limewater at room temperature until the electrical measurements were taken. The concrete beam made from mixture 3 was kept in a 98 ± 2% relative humidity chamber for the entire period before electrical measurements were taken.

3.3 Determination of Resistivity

The resistivity (ρ) for each system of external electrodes was determined using two different methods. The first method (i.e., method 1) used equation (3.2.7), while the second method (i.e., method 2) used equation (3.2.8). In both of these expressions the resistivity is given as ρ. ρ=

Rb k

(3.2.7)

ρ=

Rb ⋅ A L

(3.2.8)

where Rb is the measured bulk resistance, A is the area of the electrodes in contact with the specimen and the cross sectional area of the specimen, and L is the length of the specimens or the length between electrodes, and k is a geometry factor. The first method (i.e., method 1) consisted of using a known resistivity of liquid (see section 3.3.1) to find the geometry factor of a particular setup, which was used to find the resistivity of the cement paste, while the second method (i.e., method 2) consisted of using equation (3.2.8) directly to find the resistivity. Of the two methods discussed the second method was considered to be more appropriate and accurate for this research. The assumption of perfect contact between electrode and specimen in method 1 for the small cylinder specimen could lead to error. Method 2 is more consistent with the rest of the measurements described in this thesis where a sponge is placed between the specimen and the electrodes.

30 3.3.6 Method 1 for Determination of Resistivity

Since the exact relationships are not known that relate the measured bulk resistance of the concrete system with the resistivity, the geometry factor k needed to be determined experimentally. It is known that boundaries of the specimen geometry influence the geometry factor k, but the exact effect needs to be determined experimentally. To determine the geometry factor for each electrode configuration the resistivity of the specimen must be known prior to testing. To determine the geometry factor for paste specimens (mixture design 1) the specimens were cast into a prism mold (Figure 3.2.5), a cylindrical mold (Figure 3.2.6), and a small cylinder mold (Figure 3.3.1). This small cylinder had one electrode cast into each end.

Specimen

4" 1/2" Diameter

Figure 3.3.1 Small cylinder with geometry factor k of 631,000 1/km with screws for electrodes.

The geometry factor was previously determined for the small cylinder (ksmall cylinder = 631000 km-1) by Rajabipour et al. (2003) by preparing a mold of the same size as the small cylinder specimen and filling it with a liquid of a known resistivity (ρknown). Electrodes were inserted into the mold of the liquid into the same position as they would be if an actual specimen was being cast, and the bulk resistance was measured across the electrodes (Rb-measured-1). The resistivity of the cement paste sample (ρcement) was then determined by measuring bulk resistance of cement paste (Rb-measured-2) and using the

31 small cylinder with the geometry correction factor ksmall

cylinder.

The resistivity of the

cement paste sample (ρcement), which is assumed to be the same for the all the cement paste specimens cast, was used to find geometry factors for each electrode alignment (kAlignment N) by measuring the bulk resistance (Rb-measred-3), which, along with the bulk resistance for alignment N (Rb-measured-4), was used to find the resistivity of the concrete specimens (ρconcrete). The process is shown in (3.2.9) by which ksmall cylinder is solved for in (a), which allows ρcement to solved for in (b), which allows kAlignement N to be solved for in (c), which allows ρconcrete to be solved for in (d). R b − measured−1 = k small cylinder ρ known

(a)

R b − measured− 2 = ρ cement k small cylinder

(b)

R ⇒ b − measured−3 = k Alignment N ρ cement

(c)

R b − measured− 4 = ρ concrete k Alignment N

(d)

⇒

⇒

(3.2.9)

Four small cylinders were cast, one (MSC3) was not used because the saturation condition had been altered. Table 3.3.1 shows the bulk resistance measured for each small cylinder cement specimen. The difference in the bulk resistance trial #429 is due to the reduced time being exposed to unsaturated conditions. Equation (3.2.10) and (3.2.11) illustrate the calculations that were used for the determination of the cement paste resistivity, ρcement, and the cement paste conductivity, σcement. ρcement =

σ cement =

R b − measured 10,753 Ω = = 17.041 Ω ⋅ m = 670.9 Ω ⋅ in k small cylinder 631,000 km -1

1 ρ cement

=

1 = 0.058681 S/m 170.41 Ω ⋅ m

(3.2.10)

(3.2.11)

32

Table 3.3.1 Bulk resistance of small cylinder cement specimens. Trial

Specimen

Bulk resistance

Resistivity

(#)

(#)

(Ohms)

(Ohm*in)

423 424 426 428 429 430 431 432 434 423 424 426 428 430 431 432 434 Average

MSC1 MSC2 MSC2 MSC1 MSC4 MSC1 MSC4 MSC2 MSC4 MSC1 MSC2 MSC2 MSC1 MSC1 MSC4 MSC2 MSC4

10687 10731 10827 10734 10428 10803 10633 10826 10780 10687 10731 10827 10734 10803 10633 10826 10780 10753

666.80 669.54 675.53 669.73 650.64 674.03 663.43 675.47 672.60 666.80 669.54 675.53 669.73 674.03 663.43 675.47 672.60 670.9

3.3.7 Method 2 for Determination of Resistivity

The second method uses equation (3.2.8), which assumes that the electrode cross sectional area is equal to the specimen cross sectional area as shown in Figure 3.3.2 and therefore the electric field will remain parallel and constant across the specimen. The cross sectional area (A) and the distance between electrodes (L) are easily measured and the bulk resistance is measured thereby enabling the resistivity to be solved for (resistivity is the inverse of conductivity) using (3.2.8).

33

Figure 3.3.2 Prism specimen 3” x 3” x 4” with ¼” x 3” x 4” electrodes across entire surfaces with sponge between specimen and electrodes.

The average bulk resistance measured across a 3” x 3” x 4” cement prism made using mixture 1 was 259 Ω with a cross sectional area of 12 in2, and a distance between electrodes of 3”. Using equation (3.2.8) the bulk resistance, cross sectional area, and length between electrodes are used to determine resistivity as seen in Table 3.3.2.

Table 3.3.2 Method 2 resistivity determination data. Bulk resistance Cross sectional area Length between electrodes Resistivity Trial # (Ohms) (in^2) (in) (Ohm*in) 1 253.87 12 3 1015 2 254.21 12 3 1017 3 264.01 12 3 1056 4 264.20 12 3 1057 Average 259.07 12 3 1036

34 The amount of drying in method 2 was consistent with actual measurements, because the specimen geometries are similar (i.e., surface area to volume ratio). Method 1, because of the larger surface area to volume, had a much greater sensitivity to drying compared to method 2 (i.e., lower surface area to volume ratio). Method 1 had a lower resistivity value (671 ohm*in), because the measurements were taken immediately (10-15 seconds) after removing the specimens from the limewater bath, therefore not much drying could take place, which affects the effective resistivity of the specimen in contrast to method 2 measurements, which were taken until at least 12 minutes after removing the specimen from the limewater bath, which is consistent with experimental procedure used for the experimental measurements compared with the FEM.

3.4 Comparison of Model and Experiment

The geometry factor k is defined by (3.2.12) for unflawed specimens with electrode alignments seen in Figure 3.2.5 and Figure 3.2.6. k=

Rb ρ

(3.2.12)

where ρ is the resistivity of the specimen and Rb is the bulk resistance measured between electrodes. The geometry factor m is defined in (3.2.13). m=

ρ⋅L Rb ⋅ A

(3.2.13)

where L is the length between electrodes, A is the cross sectional are of the electrodes, Rb and ρ are the same as in (3.2.12). Since the prism specimen is a more specific case the geometry factor m was used instead of geometry factor k, allowing for more easily measured variables (i.e., length between electrodes and cross sectional area of electrodes) to be accounted for so that the effects of the specimen boundary conditions and flaws will be highlighted in the data

35 analysis. The geometry factor k lumps all of those variables together, which is fine for the cylinder setup because the distance between the electrodes and the cross sectional area of the electrodes are kept constant for all the diametrically opposed alignments on specimens of constant diameter. The units of k are in-1 and m is unitless. The geometry factors m and k are inversely proportional. Consistent sign convention was the reason for not defining them proportionally. The geometry factors from FEM were normalized by the height of the experimental specimens to take into account the height of the specimen. The resistivity of the specimens was determined by Method 2 outlined above.

3.4.8 Unflawed Cylinder Specimen

Two specimen geometries were used to compare the experiments and the finite element models. These specimen geometries consisted of a cylindrical specimen and prism specimen prepared using cement paste (i.e., mixture design 1). In the case of the cylindrical specimen a 4” diameter specimen was used with electrodes alignments seen in Figure 3.4.1 and Figure 3.2.6 and where the electrode alignment numbers are defined in Table 3.4.1. The experimental specimens were 4” tall while the modeled specimen was 1” tall. The prism specimen with the various electrode alignments can be seen in Figure 3.2.5 and Figure 3.4.2 and where the electrode alignment numbers are defined in Table 3.4.2. The alignment numbers in Table 3.4.1 and Table 3.4.2 correspond to paths of same length and geometry. The tables with the alignment numbers hold only for unflawed specimens.

36

4" Diameter 7 8 9 10 6 11 22.5° 5 12 4 6 8 10 3 13 13 14 2 1 16 15 Figure 3.4.1 Cylinder setup 16 electrode configuration with electrode numbers (numbers next to electrodes) and electrode alignment numbers (numbers between electrodes).

Table 3.4.1 Cylinder setup 16 electrode configuration alignment numbers Electrode # 1 2 3 4 5 6 7 8 9 10 11 12 Alignment numbers 1 1 2 3 4 5 6 7 8 7 6 5 2 1 1 2 3 4 5 6 7 8 7 6 3 2 1 1 2 3 4 5 6 7 8 7 4 3 2 1 1 2 3 4 5 6 7 8 5 4 3 2 1 1 2 3 4 5 6 7 6 5 4 3 2 1 1 2 3 4 5 6 7 6 5 4 3 2 1 1 2 3 4 5 8 7 6 5 4 3 2 1 1 2 3 4 9 8 7 6 5 4 3 2 1 1 2 3 10 7 8 7 6 5 4 3 2 1 1 2 11 6 7 8 7 6 5 4 3 2 1 1 12 5 6 7 8 7 6 5 4 3 2 1 13 4 5 6 7 8 7 6 5 4 3 2 1 14 3 4 5 6 7 8 7 6 5 4 3 2 15 2 3 4 5 6 7 8 7 6 5 4 3 16 1 2 3 4 5 6 7 8 7 6 5 4

13 14 15 16 4 5 6 7 8 7 6 5 4 3 2 1

3 4 5 6 7 8 7 6 5 4 3 2 1

2 3 4 5 6 7 8 7 6 5 4 3 2 1

1 2 1 3 2 1

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

37

1 2 3 4 5 20 19 18 17 16

1 2 3 4 5 6 7 8 9 10

Electrode Sponge Specimen

6 70.25" 8 9

14 13 12 11

3.00"

10

0.25"

15 14 13 12 11

Figure 3.4.2 Top view of prism specimen with electrode numbers (numbers next to electrodes) and electrode alignment numbers (numbers between electrodes).

Table 3.4.2 Prism specimen electrode alignment numbers Electrode 20 19 18 17 16 15 14 13 12 11 10 Number 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

19 18 17 16 5 14 13 12 11 10 9 8 7 6 5 4 3 2 1

19 18 38 38 37 38 17 18 4 3 24 23 33 43 32 33 12 13 9 8 29 28 28 48 27 28 7 8 14 13 24 33 23 43 22 23 2 3

17 16 37 17 38 18 19 19 2 1 22 2 23 3 24 4 14 5 7 6 27 7 28 8 27 9 9 10 12 9 32 8 33 7 24 6 4 5

5 4 3 2 1 19 18 17 16 5 14 13 12 11 10 9 8 7 6

14 24 23 22 2 19 38 37 17 4 24 33 32 12 9 29 28 27 7

9

8

Alignment Number 13 12 11 10 9 8 33 32 12 9 29 28 43 33 13 8 28 48 23 24 14 7 27 28 3 4 5 6 7 8 18 17 16 5 14 13 38 37 17 4 24 33 38 18 3 23 43 38 19 2 22 23 18 19 1 2 3 3 2 1 19 18 23 22 2 19 38 43 23 3 18 38 33 24 4 17 18 38 13 14 5 16 17 18 8 7 6 5 4 3 28 27 7 14 5 23 48 28 8 13 33 5 28 29 9 12 32 33 8 9 10 11 12 13

7

6

5

4

3

2

1

7 27 28 27 9 12 32 33 24 4 17 37 38

6 7 8 9 10 11 12 13 14 5 16 17 18 19

5 14 13 12 9 10 9 8 7 6 5 4 3 2 1

4 24 33 32 8 9 29 28 27 7 14 5 23 22 2 19

3 23 43 33 7 8 28 48 28 8 13 33 5 23 3 18 38

2 22 23 24 6 7 27 28 29 9 12 32 33 5 4 17 37 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 5 16 17 18 19

19 2 22 23 5 14

1 2 3 4 5

19 18 38 17 37 38 16 17 18 19

38 The geometry factor (k) was measured and modeled for the prism and cylinder specimens and calculated using (3.2.12). The modeled geometry factors were normalized by the height of the experimental specimens. The modeled and measured geometry factors (k) for the prism specimen and the cylinder specimen are shown in Table 3.4.3 and Table 3.4.4. The comparison between the modeled and the experimental geometry factors (k) can be seen in Figure 3.4.3. It should be noted that there is relatively constant offset from the ideal case, 1:1 ratio of measured to modeled, which is the best- fit line. The line at 90% of 1:1 ratio is also shown in the figure to show that the modeled geometry factors are proportional to the experimental geometry factor. The offset in the prism case is likely due to resistivity increasing from the time of measurement of resistivity to the time when bulk resistances where measured for different electrode alignments. The offset in either case could also have been due to effects of leads not accounted for in the model.

Table 3.4.3 Geometry factor k for 4" diameter and 4" tall cylinder specimen Alignment Model geometry factor k Measured geometry factor k #

(1/in)

(1/in)

1

0.3793

0.4659

2

0.4905

0.5797

3

0.5506

0.6603

4

0.5892

0.6882

5

0.6151

0.7049

6

0.6319

0.7286

7

0.6415

0.7353

8

0.6445

0.7390

39

Measured geometry factror k (1/in)

1.0

0.8

0.6

0.4

3" x 3" x 4" Prism specimen 4" diameter x 4" tall cylinder 1:1 Ratio 90% of 1:1 Ratio

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Modeled geometry factor k (1/in)

Figure 3.4.3 Comparison of modeled geometry factors verses measured geometry factors on cylinder specimen and prism specimen.

40 Table 3.4.4 Geometry factors for 3” x 3” x 4” prism specimen

Alignment # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16

Modeled geometry factor k (1/in) 0.31855 0.40995 0.49090 0.56103 0.63768 0.65248 0.61715 0.61738 0.64053 0.69293 0.69213 0.63738 0.61060 0.63768 0.61870

Measured Geometry factor k (1/in) 0.36649 0.35992 0.44484 0.45853 0.54495 0.50170 0.46743 0.45578 0.49054 0.54075 0.53673 0.49623 0.46814 0.45427 0.43667

Alignment # 17 18 19 22 23 24 27 28 29 32 33 37 38 43 48

Modeled Geometry factor k (1/in) 0.53670 0.45023 0.31353 0.43085 0.47985 0.53568 0.57768 0.57273 0.59123 0.57938 0.54713 0.44118 0.30880 0.50513 0.56085

Measured Geometry factor k (1/in) 0.39548 0.31711 0.34713 0.32302 0.35073 0.40286 0.40708 0.40899 0.44205 0.45314 0.41376 0.37112 0.28175 0.38896 0.38632

3.4.9 Flawed Cylinder

A cylinder made using mixture design 2 had a cut introduced perpendicular to the surface as seen in Figure 3.2.1 and Figure 3.4.4. It should be noted that only 8 electrodes were used (every other electrode, i.e. # 1, 3, 5, …, 15) for electrode alignments defined in Table 3.4.1. The experimental specimens were 4” tall and 4” in diameter. The modeled specimen was 1” tall and 4” in diameter. The cut was made using a diamond tipped saw and had dimensions of 1/8” wide by 7/8” deep by the height of the specimen. The angle from the cut, cut depth, and cut width are defined in Figure 3.4.4. The angle is defined from the position of the reference point to the closest edge of the closest electrode as shown in Figure 3.4.4.

41

Specimen Cut Cut width, w

Cut depth, a Angle from cut

Electrode

Figure 3.4.4 Two dimensional view of unflawed cylinder with definitions of parameters.

The bulk resistance measurements were taken at various angles away from the cut for diametrically opposed electrode alignments (electrode alignments 8 in Figure 3.4.1 and Table 3.4.1). The parameters for the modeled data are shown in Table 3.4.5. The modeled data is shown in Table 3.4.7 and the experimental data is shown in Table 3.4.6. The modeled data and parameters are the same as in chapter 4. The modeled data is in terms of geometry factor k (1/in) and can be converted to ratio of cut bulk resistance/uncut bulk resistance by (3.2.14). Normalized cut bulk resistance =

ρ ⋅ k cut k Cut Bulk Resistance = = cut Uncut Bulk Resistance ρ ⋅ k uncut k uncut

(3.2.14)

where ρ is the resistivity of the specimen, kcut is the geometry factor for that specific cut depth, and kuncut is the geometry factor k for an uncut specimen.

42 Table 3.4.5 Parameters of cylinder with varied cut depth. Resistivity of specimen

ohm*in

1000

Current

Amp

1

Electrode area

in^2

0.25*height

Diameter of specimen

inch

4

Cut Depth/Diameter of specimen

-

0 to 0.94

Cut Width/Diameter of specimen

-

0.03125

Angle from cut

degrees

Variable

Conductivity of cut

S/in

0

Table 3.4.6 Experimental data for 7/8” x 1/8” cut along the height of 4” diameter cylinder specimen.

6

Uncut Bulk Resistance (Ohms) 1781.3

7/8" Cut Bulk Resistance (Ohms) 2141.7

Ratio 7/8” Cut Bulk Resistance/Uncut Bulk Resistance 1.2023

28

1743.7

1977.0

1.1338

49

1837.0

2005.8

1.0919

67

1877.8

1977.8

1.0533

67

1776.7

1822.5

1.0258

49

1952.8

1935.4

0.9911

28

1710.7

1955.7

1.1432

6

1772.6

2117.2

1.1944

Angle from cut (degrees)

43 Table 3.4.7 Modeled geometry factors (k) of cylinder with varied cut depth. Angle from cut (o)

0.874 3.374 5.874 17.124 28.374 39.624 50.874 62.124 73.37484.624

a/D

Geometry factor k (1/in)

0.00

0.3843 0.3840 0.3863 0.3870 0.3870 0.3870 0.3870 0.3870 0.38700.3870

0.06

0.3479 0.3614 0.3714 0.3826 0.3849 0.3857 0.3860 0.3862 0.38630.3863

0.13

0.3249 0.3385 0.3516 0.3727 0.3794 0.3821 0.3833 0.3841 0.38430.3844

0.25

0.2987 0.3117 0.3230 0.3482 0.3607 0.3674 0.3715 0.3737 0.37480.3752

0.38

0.2816 0.2928 0.3027 0.3251 0.3373 0.3449 0.3499 0.3532 0.35490.3555

0.50

0.2676 0.2773 0.2857 0.3034 0.3123 0.3174 0.3207 0.3227 0.32380.3242

0.63

0.2539 0.2620 0.2687 0.2804 0.2842 0.2849 0.2847 0.2841 0.28370.2835

0.75

0.2380 0.2436 0.2480 0.2512 0.2486 0.2447 0.2415 0.2392 0.23770.2373

0.88

0.2130 0.2141 0.2143 0.2056 0.1983 0.1928 0.1893 0.1870 0.18570.1853

0.94

0.1870 0.1830 0.1796 0.1671 0.1605 0.1565 0.1540 0.1523 0.15140.1511

The cut bulk resistance is normalized by the uncut bulk resistance to define the normalized cut bulk resistance in equation (3.2.14). Log fits were applied to the experimental and modeled data and shown in Figure 3.4.5. The experimental data is an average of the values collected at the same angle from the cut. As can be seen when the individual data points are averaged and log fit is applied a reasonable correlation was obtained between the modeled and experimental results. A direct comparison is shown in Figure 3.4.6.

44 1.5

Normalized cut bulk resistance

Model predictions of 7/8" cut Measured for 7/8" cut

1.4

Model predictions for 1 1/2" cut Model predictions for 1/2" cut

1.3

Fit of measurements for 7/8" cut 1.2 1.1 1.0 0

30 60 Angle from cut (Degrees)

90

Figure 3.4.5 Modeled and measured ratios of cut/uncut bulk resistances with 7/8” wide cut. The angle from cut as shown in Figure 3.4.4.

Normalized modeled 7/8" cut bulk resistance

1.4

7/8" cut 1:1 Ratio 90% of 1:1 ratio

1.3 1.2 1.1 1 1

1.1

1.2

1.3

1.4

Normalized measured 7/8" cut bulk resistance

Figure 3.4.6 A comparison between the modeled and measured ratios of cut/uncut bulk resistances with 7/8” wide cut.

45 The correlation between the experimental measurements and the FEA was shown in Figure 3.4.6 to be relatively close to a 1:1 relationship for flawed cylinder specimens.

3.4.10 Flawed Slab Specimen

This section looks at specific flaws in cylinder and slab geometries. Bulk resistance measurements will be compared with the FEA. Using the assumption that the influence of the specific types of flaws used in this section on the bulk resistance operate in similar manners as other flaws the finite element model can be expanded to flaw geometries not specifically measured using IS techniques. Experimental measurements were performed on a 3” deep specimen made from mixture design 3 with both electrodes on the same surface as the flaw as seen in Figure 3.2.7. The flaw was perpendicular from characterized as a cut. The parameters for the set up are defined in Table 3.4.8. The measured data is shown in Table 3.4.9.

Table 3.4.8 Parameters for comparison of modeled and experimental for a slab specimen Modeled resistivity of specimen

ohm*in

1000

Experimental electrode area

in^2

1

Modeled electrode area

in

1

Depth of specimen

in

3

Cut Depth

in

0 to 2.875

Cut Width

in

0.125

Length between electrodes

in

0.25 to 11.25

46 Table 3.4.9 Measured data for 3" deep slab specimen. Cut depth (in)

0

0.25

0.75

1.25

1.75

2.25

2.75

Distance between electrodes (in)

Bulk resistance (Ohms)

1.0

2607.3 3049.7 4079.0 4791.1 5370.4 6042.0 7304.2

2.0

3223.8 3605.7 4047.7 4551.1 5034.9 5632.3 6779.5

3.0

3715.0 4008.4 4234.9 4648.9 5626.8 5574.6 6529.8

4.0

4358.2 4504.5 4733.3 4992.2 5366.9 5883.9 6907.1

5.1

4868.9 5050.0 5138.6 5469.5 5743.1 6242.7 7051.6

7.5

6176.8 6155.7 6251.4 6435.5 6625.9 7031.3 7927.6

8.5

6514.9 6481.6 6566.3 6792.4 6967.1 7320.2 8425.7

9.5

6960.5 6921.1 7032.2 7224.2 7443.2 7906.2 8979.9

10.5

7489.4 7398.5 7566.7 7769.5 7959.1 8411.4 9388.0

11.5

8111.0 8035.7 8130.4 8360.9 8481.4 9024.7 9826.4

The curves shown in Figure 3.4.7 were generated as a best fit to a polynomial equation of the form shown in (3.2.15).

Modeled Cut Bulk Resistance = A ⋅ a3 + B ⋅ a 2 + C ⋅ a + D Modeled Uncut Bulk Resistance

(3.2.15)

where a is the depth of the cut. The values of coefficients A, B, C, and D are shown in Table 3.4.10 and plotted in Figure 3.4.8 where power fits were applied and shown in (3.2.16), (3.2.17), (3.2.18), and (3.2.19). A = 0.2656 ⋅ L−0.3958

(3.2.16)

− B = 0.7681⋅ L−0.4468

(3.2.17)

C = 1.0917 ⋅ L−0.6899

(3.2.18)

D = 0.9235 ⋅ L0.0276

(3.2.19)

47 where L is the length between electrodes. Equations (3.2.15) through (3.2.19) were then used to create to quantitative relationship between the ratio of cut to uncut bulk resistance as a function of length between electrodes and cut depth.

Cut Bulk Resistance/Uncut Bulk Resistance

8 Distance between electrodes (in)

7

0.25 2 6.5

6 5

0.75 3 9

1.25 5 11

4 3 2 1 0 0

1 2 Cut depth (in)

3

Parameters A, -B, C, and D

Figure 3.4.7 Modeled 3" slab deep specimen

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

C

0

2

4

D

6

-B

8

A

10

Distance between electrodes (in)

Figure 3.4.8 Parameters of modeled 3” deep slab specimen

12

48 Table 3.4.10 Parameters of Modeled 3” deep slab specimen Distance between electrodes (in)

A

B

C

D

0.25

0.5723

-2.1809 3.5415 0.9315

0.75

0.3006

-0.9543 1.4988 0.9070

1.25

0.2330

-0.6572 0.9264 0.9299

2

0.1972

-0.5184 0.6131 0.9486

3

0.1763

-0.4583 0.4647 0.9595

5

0.1478

-0.3904 0.3510 0.9689

6.5

0.1361

-0.3484 0.3044 0.9729

9

0.1090

-0.2917 0.2508 0.9777

11

0.0958

-0.2567 0.2198 0.9804

The cut depths and lengths between electrodes that were used in the experimental measurements in Table 3.4.9 were used determine the

Cut Bulk Resistance shown in Uncut Bulk Resistance

Table 3.4.11, which are the vertical axis values in Figure 3.4.9. A comparison between the measured and modeled values of normalized cut bulk resistance can be seen in Figure 3.4.9. The measured bulk resistances for each distance between electrodes from Table 3.4.9 were normalized by the bulk resistances of the uncut specimen. The normalized values of the modeled data are shown on the horizontal axis of Figure 3.4.9. When looking at Figure 3.4.9 it can easily be seen that the modeled data begins to break down when the cut depth nears the specimen depth (3”). This can be explained by current concentrations at the tip of the cut in the model case which lead to larger bulk resistances. One possible solution to this problem is further refinement of the meshing around the tip of the cut.

49 Table 3.4.11 Modeled 3" deep slab specimen, calculated data. Cut depth (in) Distance between electrodes (in) 1 2 3 4 5.1 7.5 8.5 9.5 10.5 11.5

0.25

0.75

1.25

1.75

2.25

2.75

Cut Bulk Resistance/Uncut Bulk Resistance 1.153 1.078 1.053 1.041 1.034 1.027 1.025 1.025 1.024 1.024

1.422 1.217 1.144 1.106 1.082 1.055 1.049 1.044 1.040 1.037

1.607 1.301 1.193 1.138 1.102 1.062 1.053 1.045 1.040 1.035

1.905 1.482 1.329 1.250 1.198 1.137 1.122 1.111 1.101 1.094

2.517 1.911 1.681 1.558 1.474 1.370 1.343 1.321 1.303 1.288

3.641 2.739 2.379 2.178 2.035 1.851 1.801 1.759 1.725 1.695

Normalized Experimental Cut Bulk Resistance

4

3 Cut depth (in)

2

1 1

2

3

4

0.25 0.75 1.25 1.75 2.25 2.75 90% of 1:1 Ratio 1:1 Ratio

Normalized Modeled Cut Bulk Resistance

Figure 3.4.9 Comparison of experimental and modeled slab specimen. The linear fit line represents a perfect model of the experimental results. The 2.75” and 2.25” cuts have a relatively large offset from the line. The 2.75” cut depth deviated from the 1:1 ratio because around the tip of the cut there was a relatively large concentration of current (analogous to stress concentration) for the same element size, similar to using a coarse element in FEA.

50 3.5 Summary and Conclusions

This chapter included an outline of the research approach, initial application of use impedance spectroscopy to determine resistivity of specimens, an introduction to the finite element model that was used to impedance spectroscopy, and a comparison of that model with experimental data. The electrical measurements were shown to vary with the presence of a flaw. A limitation of the FEM approach exists when the cut depth approaches a length that is similar to the size of the specimen (e.g., specimen diameter or specimen depth). The breakdown of the FEA approach occurs, presumably due to the current concentrations that form near the tip of the cut similar to what occurs in fracture mechanics. The close relationship between modeling and measurements (within 90% of 1:1) allows the assertion that the FEM can be used to simulate the impedance spectroscopy setup. The correlation of the FEM with experimental results allows the next phase to be implemented, Phase II (Chapter 4), which will begin assessing the exact nature of the flaw properties that can be determined by use of impedance spectroscopy. The cylinder geometry was chosen because an array of external electrodes can be placed all the way around the specimen and the effect of specimen dimension can be removed as all the measurements will be taken over constant length, as such the geometry of the cut will be highlighted in those studies.

51

CHAPTER 4: USING IMPEDANCE SPECTROSCOPY TO DETECT AND CHARACTORIZE FLAWS IN CYLINDRICAL CONCRETE SPECIMENS

4.1 Introduction

The objective of this chapter is to determine whether electrical resistance measurements can provide a useful method for determining whether flaws exist in concrete specimens and if these flaws exist, can electrical resistance measurements be used to determine characteristics of the flaw (i.e., depth and width). The cylindrical specimen geometry measurements were taken using finite element analysis (FEA), as seen in Figure 4.1.1. An array of external electrodes was to be placed around the specimen, to enable diametric measurements, which provide a complete “view” of the flaw.

Cut Electrode Cylinder Specimen

Figure 4.1.1 Phase II of research

52 The cylindrical geometry was chosen for this study because the geometry factor (k) could be kept constant for infinite number of electrode alignments. The diametrically opposed electrode alignment was chosen to minimize the effects of the path length and boundary effects (Rajabipour et al., 2004b). By using the cylinder specimen with diametrically opposed electrode alignments, one variable, the influence of specimen geometry (i.e., boundary conditions), is kept constant, which greatly simplifies the data analysis process. The majority of the results presented in this section were determined from FEA any data obtained from physical experiments is noted as such. Use of the FEM is advantageous because it allows for many different specimen geometries and electrode alignments to be analyzed quickly. Sources of error from taking measurements in the lab are reduced with the use of FEA thereby providing more precise results and repeatability. The following sections determine relationships between cut position, cut width, cut depth, cut conductivity, specimen conductivity, and specimen diameter to bulk resistance measured between diametrically opposed electrodes at various distances away from the cut.

4.2 Analytical Geometry Functions

The geometry factor (k) is defined in equation (4.2.1) and was previously discussed in chapter 3. The factors that contribute to k include specimen geometry, placement of electrodes, and area of contact between electrodes so k will need to be determined by modeling. The resistivity of each element is defined and an electrical load is applied as a current (I). The units of the geometry factor (k) are given in terms of in-1. The model results in the computation of a voltage drop (V) which can be measured. k=

Rb V = ρ I ⋅ρ

(4.2.1)

53 4.3 Unflawed Specimen

Bulk resistance (Rb) was measured on FEM specimens using an array of electrodes that were positioned axially around the perimeter of the unflawed cylindrical specimen. The bulk resistance was measured across a diametrically opposed pair of electrodes. There is then a corresponding k that is calculated from equation (4.2.1). The influence of the diameter of the specimen on the geometry factor (k) was measured. The diameter of the cylinder specimen was varied from 0.5” to 100” with parameters as seen in Table 4.3.1. The modeled data is shown in Figure 4.3.1 The data shows a logarithmic relationship between the geometry factor (k) and the diameter of a unflawed specimen seen in and equation (4.3.1). k D − unflawed = 0.5706 ⋅ Ln ( D ) + 1.7102

(4.3.1)

It should be noted that there could be divergence from equation (4.3.1) that could occur for small specimen diameters (i.e., below 0.5 in specimen diameter) because the width of the electrode remains the same, which causes a dramatic curling of the electrode around the specimen.

Table 4.3.1: Parameters of cylinder with varied diameter without cut. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Diameter of specimen

in

0.5 to 100

Cut Depth/Diameter of specimen

-

N/A

Cut Width/Diameter of specimen

-

N/A

Angle from cut

degrees

N/A

Conductivity of cut

S/in

N/A

54

Geometry factor k (1/in)

5 4 3 2 1 0 0.1

1

10

100

Diameter of specimen (in)

Figure 4.3.1 Unflawed cylinder specimen of k (1/in) vs. D (in)

4.4 Flawed Specimen

The objective of the work presented in this section is to determine if flaws in concrete can be detected using electrical resistance measurements with external electrodes placed in a circular array around the specimen. In addition to the standard unflawed cylinder specimen shown in the previous section, a cylinder with a flaw was tested. The flaw was characterized as a cut, starting perpendicular to the outer circumference and running axially along the cylinder as shown in Figure 4.1.1. The flaw was defined as a cut perpendicular to the surface cylinder with a specified width (w), depth (a), angle from cut to electrode (θ) as shown in Figure 4.4.1. The conductivity (σ) or resistivity (ρ) was also defined for each element. The diameter of the cylinder (D) was also defined. Electrodes were spaced around the cylindrical specimen running parallel to the main axis of the cylinder as seen in Figure 4.4.1. The angle from the cut to the electrode was varied along with the other parameters to determine the other properties of the cut.

55

Specimen Cut Cut width, w

Cut depth, a Angle from cut

Electrode

Figure 4.4.1 Two-dimensional view of flawed cylinder with definitions of parameters.

4.4.1 Influence of the Position of Cut (θ)

FEA was conducted to determine the influence of the position of the cut (relative to the position of the electrodes) on the geometry factor (k). The position of the cut was defined in terms of the angle from the cut on the outer circumference of cylinder to the nearest electrode. The angle was measured from the nearest perpendicular edge of the cut on the surface, to the edge of the nearest electrode as seen in Figure 4.4.1. While investigating of the influence of other variables (i.e., D, a, w, and ρ) the position of cut was varied in each case. Therefore the issue of the influence of position of the cut is not explicitly addressed in this section but in upcoming section. In general, as the electrode nears the cut the bulk resistance or the geometry factor (k) increases.

56 4.4.2 Influence of the Diameter (D) of Flawed Specimen

This section investigates at the influence of the diameter on the geometry factor (k). Table 4.4.1 shows the variables and constants used. The data is shown in Figure 4.4.2 and Figure 4.4.3. Cut depth/diameter and width/diameter ratios remain constant even though the diameter changes so the width and depth of the cut is different for each different diameter.

Table 4.4.1: Parameters of cylinder with varied diameter with cut. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25*height

Diameter of specimen

in

0.5 to 100

Cut Depth/Diameter of specimen

-

0.375

Cut Width/Diameter of specimen

-

0.015625

Angle from cut, θ

degrees

Variable

Conductivity of cut

S/in

0

57

Geometry factor k (1/in)

7 Diameter (in)

6

0.50 20.00

1.00 40.00

3.00 100.00

5 4 3 2 1 0 0

10

20

30 40 50 60 Angle from Cut (Degrees)

70

80

90

Figure 4.4.2 Cylinder specimen with varied D (in) for k (1/in) vs. θ (degrees).

Geometry factor k (1/in)

7 6 5 4 Angle from cut (o) 0.9 3.4 5.9 17.1 28.4 39.6 62.2 84.7

3 2 1 0 0

20

40 60 Diameter (in)

80

100

Figure 4.4.3 Cylinder specimen with varied θ (degrees) for k (1/in) vs. D (in)

A regression function was applied using a logarithmic equation to the results of each different specimen diameter in Figure 4.4.2 in the form of equation (4.4.1). The average

58 R2 value obtained was 0.97. The coefficients M and B used in equation (4.4.1) were obtained by a logarithmic fit and are shown in Figure 4.4.4 and Table 4.4.2. The regression fits for M and B are combined with equation (4.4.1), where kD is the geometry factor that is due to the diameter of the specimen to produce equation (4.4.2). The regression fits of M and B result in R2 values of 0.99. k = M ⋅ Ln ( θ ) + B

(4.4.1)

⎧⎪( −0.0566 ⋅ Ln ( D ) − 0.081) ⋅ Ln ( θ ) + 0.7508 ⋅ Ln ( D ) + 2.2969 D < 20" kD = ⎨ D > 20" ⎪⎩ ( −0.0002 ⋅ D − 0.2446 ) ⋅ Ln ( θ ) + 0.7508 ⋅ Ln ( D ) + 2.2969

(4.4.2)

Table 4.4.2 Cylinder with varied diameter, parameters of k vs. θ log fit.

7 6 5 4 3 2 1 0

M -0.0397 -0.0809 -0.1472 -0.2487 -0.2497 -0.2607

B 1.6869 2.2954 3.2467 4.6412 4.9469 5.7451 0.00

B

M

-0.05 -0.10 -0.15 -0.20 -0.25

0

20

40

60

80

M of k Vs. Angle from cut

B of k Vs. Angle from cut

Diameter (in) 0.5 1 3 20 40 100

-0.30 100

Diameter of specimen (in)

Figure 4.4.4 Cylindrical specimen of intercepts and slopes of varied D (in) for k (1/in) vs. θ (degrees).

59 Equation (4.4.2) can be reduced to (4.4.3) with < 0.1 % error by removing the

( −0.0002 ⋅ D )

term in the D > 20” portion of the equation.

⎧⎪( −0.0566 ⋅ Ln ( D ) − 0.081) ⋅ Ln ( θ ) + 0.7508 ⋅ Ln ( D ) + 2.2969 D < 20" k D −1 = ⎨ (4.4.3) D > 20" −0.2446 ⋅ Ln ( θ ) + 0.7508 ⋅ Ln ( D ) + 2.2969 ⎪⎩ Using equation (4.4.3) the prediction for common cylinder specimens are shown in Figure 4.4.5 and Figure 4.4.6 which demonstrate the influence of the diameter on the bulk resistance.

5.0

Geometry factor k (1/in)

4.5 4.0 3.5 3.0 2.5 2.0

o

Angle from cut ( )

1.5

0.5 10 60

1.0 0.5

3 30 85

0.0 0

2

4 6 8 10 Diameter of specimen (in)

12

14

Figure 4.4.5 Prediction of k vs. diameter of specimen for varied angle from cut.

60 5.0

Geometry factor k (1/in)

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Diameter of specimen (in)

0.5

4

6

8

12

0.0 0

10

20

30

40

50

60

70

80

90

o

Angle from cut ( )

Figure 4.4.6 Prediction of k vs. angle from cut for varied diameter of specimens.

4.4.3 Influence of the Width of Cut (w)

This section investigates the influence of the cut width on the geometry factor (k). The first approach to determine the influence of the cut width, was to vary the position of the electrodes around the specimen with respect to the cut. The second approach was to place the electrode directly over the cut and vary the surface area of the electrode that is in contact with the specimen by varying the cut width. Both techniques were unsuccessful in determining a quantitative relationship between bulk resistance and cut width for smaller cut widths (less then ½”) that could be practically used to predict cut width. The width of cut is defined as shown in Figure 4.4.1. Table 4.4.3 shows the variables and constants used in this investigation. The data that was obtained from the FEA analysis are shown in Figure 4.4.7 and Figure 4.4.8. A regression function using a linear

61 equation in the form of (4.4.4) was applied to each of the data sets in Figure 4.4.8. The parameters of the linear fits (M and B) were then graphed versus θ in Figure 4.4.9 where polynomial fits were applied to determine the geometry factor due to the width, kw/D, as seen in (4.4.5). k = M⋅

w +B D

(4.4.4)

k w/D = ( -0.000003 ⋅ θ3 + 0.0006 ⋅ θ 2 - 0.0363 ⋅ θ + 2.2323) ⋅

w + 0.000003 ⋅ θ3 D

+0.0005 ⋅ θ 2 - 0.03 ⋅ θ + 3.5287

Table 4.4.3 Parameters of cylinder with varied cut width. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25*height

Diameter of specimen

in

4

Cut Depth/Diameter of specimen

-

0.375

Cut Width/Diameter of specimen

-

0.00025 to 0.0625

Angle from cut

degrees

Variable

Conductivity of cut

S/in

0

(4.4.5)

62

Geometry factor k (1/in)

4 3 Cut Width/Diameter

2

0 1/400 1/64 1/16

1 0 0

10

20

30

40

50

60

70

80

90

Angle from cut (degrees)

Figure 4.4.7 Cylinder specimen with varied w/D for k (1/in) vs. θ (degrees).

Geometry factor (1/in)

4 3 2

Angle from Cut (Degrees)

1

0.874

3.374

5.874

17.124

28.374

39.624

62.124

84.624

0 0

0.01

0.02

0.03

0.04

0.05

0.06

Cut Width/Diameter

Figure 4.4.8 Cylinder specimen with varied θ (degrees) for k (1/in) vs. w/D.

0.07

63

M and B

4 3 2 1 M

B

0 0

10

20

30

40

50

60

70

80

90

Angle from cut (Degrees)

Figure 4.4.9 Cylinder specimen, intercept and slopes of varied θ for k (1/in) vs. w/D

The second approach to predict the width of the cut was implemented by placing the electrode directly over the flaw as seen in Figure 4.4.10. The flaw was assumed to be nonconducting and had a smaller width than the electrode. The electrode was centered on the flaw. The working hypothesis was that as the width of crack increases the cross sectional area near the electrode, as well as in the electrode path, to conduct the current decreases. This increases the measured bulk resistance along with decreasing the effective cross sectional area of the electrode, which also increases the measured bulk resistance. The data for varied cut depths and widths are shown in Figure 4.4.11 and Figure 4.4.12, respectively. As can be seen the sensitivity of geometry factor (k) to the cut width is not very high in either the approaches for determining the cut width as a function of bulk resistance,. In fact, by the time that the geometry factor (k) is high enough to help with a prediction about the cut width, visual inspection would be a much more applicable method.

64

Specimen Cut Cut width, w

Electrode

Cut depth, a

Figure 4.4.10 Definitions of parameter for cylinder specimen

4.0 Geometry factor k (1/in)

3.5 3.0 2.5 2.0

Cut width/Diameter 0 1/400 1/64 1/32 1/20

1.5 1.0 0.5 0.0 0

0.2

0.4

0.6

0.8

1

Cut depth/Diameter

Figure 4.4.11 Cylinder with varied w/D for k (1/in) vs. a/D with electrode directly over cut.

65

Geometry factor k (1/in)

4 3

Depth of cut/Diameter

2

0 0.375 0.95

1

0.0025 0.5 0.9975

0.05 0.625 1

0.125 0.75

0.25 0.875

0 0

0.01

0.02 0.03 Cut width/Diameter

0.04

0.05

Figure 4.4.12 Cylinder specimen with varied a/D for k (1/in) vs. w/D with electrode centered over cut.

4.4.4 Influence of Depth of Cut (a)

The depth of the cut (a) was varied, along with the position of electrodes relative to the cut. The cut depth is defined in Figure 4.4.1. Table 4.4.4 shows the parameters of the measurements. The data are shown in Figure 4.4.13 and Figure 4.4.14.

66 Table 4.4.4 Parameters of cylinder with varied cut depth. Resistivity of specimen

ohm*in

1000

Current

Amp

1

Electrode area

in^2

0.25*height

Diameter of specimen

In

4

Cut Depth/Diameter of specimen

-

0 to 0.94

Cut Width/Diameter of specimen

-

0.03125

Angle from cut

degrees

Variable

Conductivity of cut

S/in

0

7

Geometry factor k (i/in)

6 5 4 3 2

Cut Depth/Diameter

1

0.00

0.06

0.13

0.25

0.38

0.50

0.63

0.75

0.88

0.94

10

20

30

40

0 0

50

60

70

80

90

Angle from cut (degrees)

Figure 4.4.13 Cylinder specimen with varied a/D for k (1/in) vs. θ (degrees). Linear fits for θ greater than 50o.

67 8 Geometry factor k (i/in)

7 6 5

Angle from cut (o) 0.9 3.4 17.1 28.4 50.9 62.1 84.6

5.9 39.6 73.4

4 3 2 1 0 0.00

0.20

0.40

0.60

0.80

1.00

Cut Depth/Diameter

Figure 4.4.14 Cylinder specimen with varied θ (degrees) for k (1/in) vs. a/D.

Regression functions were applied in the form of linear fits in the form of equation (4.4.6) to data shown in Figure 4.4.13 for θ > 50o. The parameters (M and B) from the linear fits are shown in Figure 4.4.15. Combining equation (4.4.6) and the linear fits of Figure 4.4.15 produced equation (4.4.7), where ka/D is the geometry factor expressed as a function of the cut depth and relative position from the cut. k = M⋅θ + B

(4.4.6)

ka /D = 4 3 2 2 ⎛ ⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ − ⋅ + ⋅ − ⋅ + ⋅ 0.0767 0.1385 0.064 0.0053 ⎜⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − 0.00008 ⎟⎟ ⋅ θ (4.4.7) ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ ⎠ 4

3

2

⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ +23.155 ⋅ ⎜ ⎟ − 33.209 ⋅ ⎜ ⎟ + 17.001⋅ ⎜ ⎟ − 2.0141⋅ ⎜ ⎟ + 2.6216 ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠

7 B 6 5 4 3 2 1 0 0.00

0.004 M

0.003 0.002 0.001

M

B

68

0.000 -0.001 0.20

0.40

0.60

0.80

-0.002 1.00

Cut Depth/Diameter

Figure 4.4.15 Cylinder specimen with intercept (θ = 0) and slope of varied a/D for k (1/in) Vs θ (degrees).

Equation (4.4.7) can be simplified to equation (4.4.8) by removing the -0.00008*θ term with less than 0.3% error. ka / D

4 3 2 ⎛ ⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛ a ⎞⎞ = ⎜ −0.0767 ⋅ ⎜ ⎟ + 0.1385 ⋅ ⎜ ⎟ − 0.064 ⋅ ⎜ ⎟ + 0.0053 ⋅ ⎜ ⎟ ⎟ ⋅ θ ⎜ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ D ⎠ ⎟⎠ ⎝ 4

3

2

(4.4.8)

⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ +23.155 ⋅ ⎜ ⎟ − 33.209 ⋅ ⎜ ⎟ + 17.001⋅ ⎜ ⎟ − 2.0141⋅ ⎜ ⎟ + 2.6216 ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ The predictions from equation (4.4.8) are shown in Figure 4.4.16 and Figure 4.4.17. These figures allow for predictions of cut depth to be made from measurements of geometry factor (k).

69

Geometry factor k (1/in)

7 6 5 4 3 Cut depth/diameter 0 0.25 0.5 0.75 0.88 0.94

2 1 0 0

15

30

45

60

75

90

Angle from cut (o)

Figure 4.4.16 Prediction of cut depth for cylindrical specimen with varied cut depth/diameter for geometry factor (1/in) vs. the angle from the cut to the electrodes (o) (k vs. θ).

7

Geometry factor k (1/in)

6 5

Angle from cut (o) 50

60

75

85

4 3 2 1 0 0

1/4

1/2

3/4

1

Cut depth/diameter

Figure 4.4.17 Predictions of cut depth for cylindrical specimen with varied angle from cut the to electrode (θ) (o) for geometry factor (1/in) vs. cut depth/diameter (k vs. a/D).

70 4.4.5 Influence of Conductivity of Cut This section investigates the influence of the conductivity of the cut on the bulk resistance. For all of the simulations described in the proceeding sections of the chapter, the specimens’ cut has been assumed to have a infinite resistivity. From a practical point of view the crack in a specimen may be filled with air (assumed to have an infinite resistivity) or with water (or some other electrolyte) with a measurable conductivity. The conductivity of the cut was varied from 0 S/in to 10 S/in (where S is a Siemen = ohm-1). The rest of the defined parameters are shown in Table 4.4.5. The specimen geometry was the same as that used in section 0 where the diameter of specimen with a flaw was varied. The data is shown in Figure 4.4.18, Figure 4.4.19, and Figure 4.4.20 where the normalized conductivity of cut is the cut conductivity divided by the conductivity of the specimen.

Table 4.4.5 Parameters of cylinder with varied conductivity. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25*height

Diameter of specimen

in

4

Cut Depth/Diameter of specimen

-

0.375

Cut Width/Diameter of specimen

-

0.01563

Angle

degrees

Variable

Conductivity of cut

S/in

Variable

71

Geometry factor k (1/in)

4 3 2 Normalized conductivity of cut 10000 1000 10 1 0.1 0.01 0.001 0

1 0 0

10 20 30 40 50 60 70 80 90 o

Angle from cut ( )

Figure 4.4.18 Cylinder specimen with varied σ (S/in) for k (1/in) vs. θ (degrees).

4.0 Geometry factor k (1/in)

3.5 3.0 2.5 2.0 1.5 1.0 0.5

Angle from cut (o) 0.874 3.374 5.874 17.124 28.374 39.624 62.124 84.624

0.0 0.001

0.01 0.1 Normalized conductivity of cut

1

Figure 4.4.19 Cylinder specimen with varied θ (degrees) for k (1/in) vs. σ (S/in) less than specimen conductivity.

72

Geometry factor k (1/in)

3.0 2.5 2.0 1.5

Angle from cut (o) 0.874 3.374 5.874 17.124 28.374 39.624 62.124 84.624

1.0 0.5 0.0 1

10

100

1000

10000

Normalized conductivity of cut

Figure 4.4.20 Cylinder specimen with varied θ (degrees) for k (1/in) vs. σ (S/in) greater than specimen conductivity.

By using various material conductivities to fill the cut, and by measuring the bulk resistance at various angles from the electrode to cut, the conductivity of the specimen can be predicted. This is done by plotting either geometry factor k or bulk resistance (Rb) (bulk resistance and geometry factor k are proportional) versus normalized conductivity of cut or cut conductivity. A regression function was applied (R2 = 0.97) in the form of a logarithmic equation for each angle from electrode to cut as seen in Figure 4.4.19 and Figure 4.4.20. These logarithmic equations converge on the conductivity of the specimen at normalized conductivity of cut equal to one (i.e., conductivity of the cut = specimen conductivity). When the conductivity of the cut is very high compared to the conductivity of the specimen (3 orders of magnitude) this convergence breaks down.

73 4.5 Conclusions

In this chapter the geometry factor k was calculated for cylindrical specimens. The aim of this chapter was to develop a prediction equation that could be used to describe the measured bulk resistance as a function of specimen and flaw geometry. Various parameters were varied to determine the effect on the geometry factor including, cut depth, cut width, position of the electrodes relative to the flaw, conductivity of cut, and diameter of the specimen. A geometry factor k as a function of θ,

w a ⎛ w a⎞ , and (i.e., k ⎜ θ, , ⎟ ) was developed D D ⎝ D D⎠

by normalization process. The geometry factors that were developed for each of those ⎛ w⎞ parameters (i.e., k w / D ⎜ θ, ⎟ and ⎝ D⎠

⎛ a⎞ k a / D ⎜ θ, ⎟ ) were normalized by each other as seen ⎝ D⎠

in equation (4.6.1) for a 4” diameter specimen. ⎛ w⎞ k w / D ⎜ θ, ⎟ ⎛ w a⎞ ⎝ D ⎠ ⋅ k ⎛ θ, a ⎞ k ⎜ θ, , ⎟ = a/D ⎜ ⎟ k w / D ( θ) ⎝ D D⎠ ⎝ D⎠

(4.6.1)

a ⎛ w⎞ where k w / D ⎜ θ, ⎟ is equation (4.4.5) when D = 4” and = 0.375”, k w / D ( θ ) is D ⎝ D⎠ equation (4.4.5) when D = 4”,

a w ⎛ a⎞ = 0.375, and = 0.03125, and k a / D ⎜ θ, ⎟ is D D ⎝ D⎠

equation (4.4.8) when D = 4” and

a = 0.375. D

Equation (4.6.1) can be simplified to (4.6.2) by removing the terms accounting for the ⎛ w⎞ width of cut (i.e., k w / D ⎜ θ, ⎟ and k w / D ( θ ) ) with a less than 3.2% error for θ > 50o, ⎝ D⎠ which further validates the conclusion that the cut width does not have significant impact the geometry factor for this setup.

74 ⎛ w a⎞ ⎛ a⎞ k ⎜ θ, , ⎟ ≅ k a / D ⎜ θ, ⎟ ⎝ D D⎠ ⎝ D⎠

(4.6.2)

To further advance this work to a more general case the geometry factor (k) as a function of the diameter of the specimen, cut depth, and angle from electrode to cut (i.e., a⎞ ⎛ k ⎜ D, θ, ⎟ ) was obtained by a normalization process. The geometry factor is a function D⎠ ⎝ of D and θ ( k D ( θ, D ) ) which were used to normalize equation (4.6.2) as shown in equation (4.6.3).

a ⎞ k ( θ, D ) ⎛ ⎛ a⎞ ⋅ k a / D ⎜ θ, ⎟ k ⎜ D, θ, ⎟ = D D⎠ k D ( θ) ⎝ ⎝ D⎠

(4.6.3)

where k D ( θ, D ) is equation (4.4.3) when D < 20” and k D ( θ ) is equation (4.4.3) when D = 4”. Therefore these equations can be used to predict the cut depth from bulk resistance measurements of cylinders of various diameters. These predictions were shown in Figure 4.4.5, Figure 4.4.6, Figure 4.4.16, and Figure 4.4.17. Another method for determining the parent specimen’s conductivity was explored. This was done by filling the flaw with known conductivity and measuring the bulk resistance of the specimen and cut at various angles from cut. The bulk resistance was plotted versus the natural log of cut conductivity at various angles from the cut. Using a regression function, the logarithmic fits for each angle from cut to electrode were extrapolated to their intersection. The cut conductivity at the intersection corresponds to the conductivity of the specimens. After establishing that IS that a method utilizing external electrodes can characterize flaws in concrete, the next phase of research began by characterizing flaws in concrete specimens with geometries and electrode alignments that are more likely to be present in field applications (i.e., only one surface is exposed). The results for this next phase of research are presented in the next chapter.

75

CHAPTER 5: USING IMPEDANCE SPECTROSCOPY TO DETECT AND CHARACTORIZE FLAWS IN SLAB CONCRETE SPECIMENS

5.1 Introduction

In the previous two chapters it has been shown that finite element analysis (FEA) can be used to define a relationship between specimen geometry, flaw geometry, and the measured electrical response. This relationship can be used to correlate experimental measurements to the geometry of physical flaws that are present in a specimen. This chapter describes the next phase of this research (phase III), which attempts to use this technique to describe the severity and geometry of cuts in concrete slabs. In phase III, a geometry factor (m) was determined using FEA for slab specimens in which the electrodes are placed on one surface of the specimen, as shown in Figure 5.1.1. FEA was used to define how the bulk resistance of a slab specimen changes with the introduction of a cut. The concept behind this investigation was to place an electrode on either side of a flaw and to use the measured response to describe the depth and severity of the flaw. It should be noted that the flaw that will be investigated in this chapter has a definite width and depth as seen in Figure 5.2.1, and can be defined as a cut. This differs from the geometry of an actual crack which will have uneven faces that may be separated by a varying width. This approach considered the case in which the electrodes are applied to only one side of the sample that is being tested. The single sided access is an important consideration for many potential field applications as many times only one side is available for testing. Since the EIS technique is nondestructive, it is an attractive method that may be able to be used for monitoring and evaluation of performance of specimens and concrete slabs.

76 In this phase of research the geometry factors associated with each electrode alignment, specimen geometry, and flaw size were assessed using the FEA.

Slab Specimen

Cut Electrode

Figure 5.1.1 Illustration of the Specimen Geometry Investigated in Phase III of research

5.2 Analytical Geometry Functions

The following section describes the relationship between a geometry factor (m) and cut depth (a), cut width (w), slab depth (Dslab), distance from cut to electrodes (Lcut), length between electrodes (L), and cut conductivity (σcut). These parameters are defined in Figure 5.2.1 for a slab specimen. The length between the cut and the electrode is defined using equation (5.2.1). Lcut =

L-w 2

(5.2.1)

77

Distance between electrodes Distance from electrodes to cut Cut inside electrodes Cut outside electrodes

Electrode

Cut depth Cut ~Infinite slab depth

Cut width Distance from electrodes to cut

~Infinite slab width

Figure 5.2.1 Two dimensional view of slab specimen with definitions of parameters.

Since the distance between electrodes was kept constant for the cylindrical setup it never had an effect on the geometry factor (k) or bulk resistance. Since in general, the geometry factor is known to be proportional to the length between electrodes, another geometry factor (m) was introduced which accounted for the distance between electrodes. Instead of using the geometry factor (k) this geometry factor (m) was modified to account for the area of the electrode (A) and distance between the electrodes (L). As such the geometry will be referred to using the variable m which is defined in equation (5.2.2). m=

ρ⋅L Rb ⋅ A

(5.2.2)

78 where ρ is resistivity, L is the distance between electrodes, A is the cross sectional area of the electrode, and Rb is the bulk resistance.

5.3 Experimental Details

Electrodes with a cross sectional area of ¼ in2 (6.4 mm2) (i.e., 1/4” x 1 unit length) were modeled. It was assumed that the surface is perpendicular to the cut depth. Unless stated otherwise, the distance from the cut to each electrode is the same (i.e., the cut was centered between electrodes) as seen in Figure 5.2.1, where the cut is between the electrodes. All data will be taken from FEA and it will be viewed as two dimensional analysis with the third dimension considered to be the unit thickness. It should be noted that this approach is the same as the one used with the cylindrical specimen setup. ANSYS University Advanced 7.1 (ANSYS 2002) was used to create a finite element model for the specimens investigated in this study. This FEM considered the specimen elements, electrode elements, and applied loads. The load in this case was electrical current, in amps. The elements that were used were two dimensional, plane, four node elements. This permitted a temperature and voltage to be defined at each node. The temperature was considered to be constant throughout modeling. While modeling was performed in three dimensions, the solutions considered variations in two dimensions while the third dimension was one unit thick (i.e., the electrodes are one unit long). A current source with a constant amplitude was used to apply the electrical load (I) to the electrode elements. As this electrical load was applied to the specimen the static analysis resulted in a voltage drop across the electrodes and specimen elements. The application of ohms law enabled the voltage drop (V) and the defined current to be used to calculate the bulk resistance (Rb) as seen in (5.3.1). Rb =

V I

where I is the current in amps.

(5.3.1)

79 5.4 Comparison of Model and Experiment

A two dimensional geometry (with a unit thickness) will be modeled in this thesis. These two dimensional results were converted to a true three dimensional model in a step by step approach. Whenever referring to the two dimensional object in this text it is assumed that the two dimensional representation has a third dimension with a unit thickness. For example a “two dimensional” specimen has a defined height and width along with the unit thickness as opposed to a three dimensional specimen that has a defined height, width, and depth. The general case was considered using a semi-infinite half space as seen in Figure 5.4.1 (a). Since it is impossible to model a truly infinite geometry, the term infinite will refer to cases where the boundaries (height, width, or depth) are sufficiently far away from the crack and electrodes such that they have no influence on the bulk resistance measurements for the given electrode alignment up to five significant digits. The image shown in Figure 5.4.1 (b) represents the results from FEA. Grey scale levels, as shown on the right hand side of the Figure 5.4.1 (b) represents different magnitudes of the electric field vector. The lighter levels indicate higher concentration of current, which are analogous to stress concentrations in the mechanical stress solution.

80

(a)

(b) Figure 5.4.1 (a) A graphical illustrated (b) FEM representation 3-d of a slab specimen of semi-infinite width, length, and depth.

81 To provide a procedure for converting the three dimensional system into a two dimensional case, the model was developed with electrodes placed along the top surface as seen in Figure 5.4.1 (b). The nonparallel electric field lines on the surface due to the ends of the electrodes can be eliminated by making the width of the specimen the same as the length of the electrodes as in Figure 5.4.2. This enables a simpler two dimensional modeling geometry as shown in Figure 5.4.3. This procedure is only valid for a 1” long electrodes, but the same procedure could be used for electrodes with various lengths. These results need to be evaluated further to be applied to more general cases (i.e., varied electrode length).

Figure 5.4.2 Three dimensional view of slab specimen of infinite depth and length and finite width.

82

Figure 5.4.3 Two dimensional view of slab specimen of infinite depth and length with specimen width equal to electrode length.

To obtain a valid conversion between the modeled and physical measurements, the nonparallel electric field lines on the top surface (i.e., the effect of the ends of the electrodes) need to be quantified. This is needed because in situ applications, especially on slabs, will generally not have electrode lengths the same as the specimen widths as shown in Figure 5.4.2. This results in an uniform electric field on the top surface. The influence of the ends of the electrodes will be quantified.

Figure 5.4.4 Two dimensional view of slab with infinite width and length and finite depth.

83 A FEA was performed to determine the spacing between the electrode and the edge of the specimen that is required for the geometry to be considered semi-infinite. A portion of the data in this section was taken from section 5.6.1 (Figure 5.4.5) and from Section 5.5. Figure 5.4.5 shows that when the electrodes (distance between electrodes is 2.5”) are moved farther from a cut the bulk resistance decreases. The distance at which the bulk resistance begins to stop decreasing as a function of the distance from the cut is considered to be the distance from the electrodes to the edge that is required for edges to be considered to be an infinite distance away. It was determined that at distances from the cut greater than 2” the bulk resistance remains relatively constant (i.e., the edge has no effect on the bulk resistance). To keep sources of error to a minimum the length of the

Bulk resistance (Ohms)

specimen was at least 10” plus length between electrodes (i.e., greater than 12.5”).

3000 2500 2000 1500 1000 500 0 0

1 2 3 4 5 6 Distance from cut (in)

Figure 5.4.5 Bulk resistance (Ohms) vs. distance from cut to closest electrode (in)

Table 5.4.1 shows that as the slab depth increases it’s influence on the bulk resistance decreases until finally it has no influence. There is only a 0.07% difference between 15” and 20” bulk resistance and a 0.02% difference between 20” and 50” slab bulk resistance. Because of this a slab width of 2*20” plus the length of the electrodes and a slab depth of 20” is considered to be a semi-infinite geometry.

84 Table 5.4.1 Influence of slab depth on bulk resistance Slab depth (in) Bulk resistance (Ohm*in) 15

2368.8

20

2367.1

50

2366.7

100

2366.7

The next goal of this research was to produce a two dimensional model (similar to that shown in Figure 5.4.3) with correction factors so that it is an accurate representation of a three dimensional system. Results from a model made using the three dimensional configuration with a finite depth and infinite width and length will be compared to two dimensional model with finite depth and infinite length by using parameter “d”, which is a function of the distance between electrodes, L and depth of slab, Dslab as seen in equation (5.4.1). d (L, D slab ) =

R 3− d ⇒ R 3− d = d (L, D slab ) ⋅ R 2 − d R 2−d

(5.4.1)

It will be shown that for slab depths smaller than 1” the relationship between the geometry factor and distance between electrodes becomes nonlinear. As result, it becomes increasingly difficult to extend the correction function below 1”. It should be noted that this correction factor is an approximation rather than an exact solution. Slabs of various depths and lengths were investigated as shown in Table 5.4.2. In each of these simulations it was assumed that the electrodes were 1 inch long and ¼” wide. Simulations of the 3-dimensional and 2-dimensional geometries were used to calculate the value of a dimensional correction factor (d), which takes into account the effects of the ends of one inch long electrodes on the bulk resistance. The data from the FE simulation of the 3-dimensional and 2-dimensioanl specimens are shown in Table 5.4.2, Figure 5.4.6, and Figure 5.4.7. As the distance between the

85 electrodes increases the bulk resistance increases for both the 2-dimensional and 3dimensional cases. The increase in bulk resistance in the 2-dimensional case is greater than in the 3-dimensional case. The ratio of 3-dimensional bulk resistance/2-dimensional bulk resistance is then graphed vs. distance between electrodes in Figure 5.4.6. This graph indicates that as the distance between electrodes increases the influence of the ends of the electrodes becomes greater. The ratio of 3-dimensional bulk resistance/2dimensional bulk resistance is then graphed vs. slab depth in Figure 5.4.7. This graph indicates that as the slab becomes thinner the influence of the ends of the electrodes becomes greater.

Table 5.4.2 Data for conversion between 3-d and 2-d Depth of slab (in)

1.000

2.000

5.000

Distance between electrodes (in) 3-d Bulk resistance (ohms) 1.000

995.3

824.09

815.53

2.000

1021.7

992.16

885.47

4.000

1382.4

1137.4

932.17

6.000

1561.3

1251.0

982.12

2-d Bulk resistance (ohms) 1.000

2188.9

2033.7

1804.4

2.000

3192.4

2607.0

2087.4

4.000

5199.4

3753.6

2653.4

6.000

7206.4

4900.2

3219.4

86

Bulk resistance 3-d Bulk resistance 2-d

0.5 0.4 0.3 0.2 0.1 Depth of slab (in)

1.0

2.0

5.0

0.0 0

1

2

3

4

5

6

7

Distance between electrodes (in)

Figure 5.4.6 Slab specimen with varied D (in) for bulk resistances 3-d/2-d vs. L (in).

Bulk resistance 3-d Bulk resistance 2-d

0.5 0.4 0.3 0.2 Distance between electrodes (in)

0.1

1

2

4

6

0.0 0

1

2

3

4

5

6

Slab Depth (in)

Figure 5.4.7 Slab specimen, intercept and slope of varied L (in) for bulk resistances 3d/2-d vs. D (in).

87 A regression was applied to the data from the FEA as shown in Figure 5.4.6 using the form of the linear equation (5.4.2). The values for the resulting coefficients are shown in Table 5.4.3. The fit parameters were determined as a function of the depth of the slab in Figure 5.4.8. The value for M was reasonably consistent for different geometries for an average of M = -0.0290 in-1, which is noted in Figure 5.4.8. The value for B could be fit using a regression function in the form of a linear fit. Equation (5.4.3) was formulated to obtain the geometry correction function, d(L, Dslab), which provides the conversion function between 2-d and 3-d models. Bulk resistance 3-d = M⋅L + B Bulk resistance 2-d

(5.4.2)

d ( L,Dslab ) = -0.029 ⋅ L + 0.0246 ⋅ Dslab + 0.3641

(5.4.3)

Table 5.4.3 Data for slab specimen, B of varied Dslab (in) for bulk resistances 3-d/2-d bulk resistance vs. L (in). Depth of slab (in)

M

B

1

-0.0258

0.3709

2

-0.0311

0.4370

5

-0.0302

0.4811

Average

-0.0290

88

0.000

0.6

Parameter B

M

-0.005 -0.010

0.4

-0.015 0.3 -0.020 0.2

-0.025

0.1

-0.030

0

-0.035 0

2

4

Parameter M

B

0.5

6

Depth of slab (in)

Figure 5.4.8 Slab specimen, B of varied Dslab (in) for bulk resistances 3-d/2-d bulk resistance vs. L (in).

Equation (5.4.3) quantifies the differences of the two-dimensional and the threedimensional modeling approaches as a function of length between electrodes and the depth of the slab for electrodes on the same exposed surface of a slab. The implications of this approach are that the two-dimensional analysis for various flaw sizes and slab depths that will be conducted in the rest of the chapter can be used to predict three-dimensional results with electrodes not of the same length as the specimen width by simply multiplying 2-d results by the conversion function d(L,Dslab). An example of the application of this equation can be illustrated as follows. A 2dimensional FEA was done on 3” deep slab (Dslab). A cut was placed in the specimen with a width of 1/8” and a cut depth that was varied from 0 to 2 7/8”. The length between the electrodes was varied from 1/4” to 11”. The specific case of L = 3” and a = 1.5” results in the bulk resistance (Rb-2-d) of 3561.0 Ohms. Multiplying the correction factor by

89 the Rb-2-d results in a prediction for the 3-dimensional case of the bulk resistance (Rb-3-d) for 1” long by ¼” wide electrodes on an effective infinite width and length as seen in equation (5.4.4). R b-3-d = ( R b − 2 −d ) × ( d ( L,Dslab ) ) = ( 3561.0 ) × ( -0.029 ⋅ 3 + 0.0246 ⋅ 3+ 0.3641) = 1249.6 Ohms

(5.4.4)

5.5 Unflawed Specimen – Slab Depth Determination

In the ideal situation all concrete that is placed would be provided with complete documentation, reporting the dimensions of the members. Unfortunately, this is not always the case. One important aspect of the concrete that is placed, that may not be able to be discerned by visual inspection, is slab depth. Impedance spectroscopy with the setup described here is an attractive option to determine the depth of a slab with only access from one surface because it is nondestructive and external electrodes can be used, allowing for use on almost any concrete. This section shows the influence the depth of the slab has on the geometry factor m. Table 5.5.1 shows the parameters that were used in FEA simulation to develop the relationship between geometry factor m and the depth of the slab. The data from the FEA are shown in Figure 5.5.2 and Figure 5.5.1.

90 Table 5.5.1 Parameters of slab for varied slab depth. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Depth of specimen

in

Variable

Cut Depth

in

N/A

Cut Width

in

N/A

Length between electrodes

in

0.125 to 9.50

Conductivity of cut

S/in

N/A

14 Length between electrodes (in) 0.250 1.25 2.25 2.50 3.25 5.25 8.25 9.50

Geometry factor m

12 10 8 6 4 2 0 0

5

10

15

20

Slab depth (in)

Figure 5.5.1 Slab specimen with varied distance between electrodes, L, (in) for m vs. Dslab less than 10”.

91 14 Slab Depth (in)

Geometry factor m

12 10

1 5

3 10

8

15 50

20 100

6 4 2 0 0

2

4

6

8

10

Distance between electrodes (in)

Figure 5.5.2 Slab specimen with varied slab depth, Dslab, (in) for m vs. L (in).

A regression function was applied to the data in Figure 5.5.2 to produce linear fits in the form of equation (5.5.1) for electrodes greater than 2” apart. The parameters (M and B) for those fits are shown in Figure 5.5.3. A regression function was applied in Figure 5.5.3 producing equation (5.5.2) where mD is the geometry factor due to slab depth and length between electrodes.

m = M⋅L + B

(5.5.1)

⎧ ( 0.3579 ⋅ Ln ( Dslab ) + 0.2355 ) ⋅ L − 0.7555 ⋅ Ln ( Dslab ) + 3.3954 ⎪ ⎪ 2" < Dslab < 10" ⎪ mD = ⎨ ⎪ ⎪( 0.00003 ⋅ Dslab + 1.0347 ) ⋅ L − 0.0003 ⋅ Dslab + 1.6576 ⎪⎩Dslab ≥ 10"

(5.5.2)

92

3.0

M and B

2.5 2.0 1.5 1.0 0.5 0.0 1

10

100

Depth of slab (in)

Figure 5.5.3 Slab specimen M and B of fits of varied slab depth, Dslab, (in) for m vs. L (in) where L is greater than 2 inches.

Equation (5.5.2) can be simplified to equation (5.5.3) by removing the 0.0003*Dslab term resulting in less than 0.3 % error and the -0.0003*Dslab with less than 0.15 % error. ⎧( 0.3579 ⋅ Ln ( Dslab ) + 0.2355 ) ⋅ L − 0.7555 ⋅ Ln ( Dslab ) + 3.3954 ⎪ ⎪2" < Dslab < 10" mD = ⎨ ⎪ ⎪ 1.0347 ⋅ L + 1.6576 Dslab ≥ 10" ⎩

(5.5.3)

This removes the dependence of mD on Dslab for Dslab ≥ 10" and is thus only depenedent on length between electrodes. This equation results in the predictions shown in Figure 5.5.4 and Figure 5.5.5 It should be noted that the distance between electrodes is a maximum of 9.5”, which is roughly the limit of slab depth that can be determined.

93 14

Geoemtry factor m

12 10 8 6 4 Length between electrodes (in) 2 4 6 8

2 0

0

10

10

20 30 Slab depth (in)

40

Figure 5.5.4 Predictions for Slab specimen with varied distance between electrodes (in) for m vs. Dslab (in).

14 Slab depth (in) Geoemtry factor m

12

2 5 10

10

3 8 40

8 6 4 2 0 0

2

4

6

8

10

12

Length between electrodes (in)

Figure 5.5.5 Predictions for slab specimen with varied slab depth (in) for m vs. L (in).

94 Further analysis of the data looked at the possibility of determining slab depths greater than 10” deep. When the geometry factor (m) is graphed verses the depth of slab (Dslab) is graphed where Dslab is normalized by the length between electrodes (L), a bilinear relationship is seen in Figure 5.5.6.

Geometry factor m

12

Slab depth (D) /in 10 D/L = 1 1 3 5

10 8 6 4 2 0 0

3 6 Length between electrodes (in)/Depth of slab (in)

9

Figure 5.5.6 Slab specimen with varied slab depths (in) for geometry factor m vs. length between electrodes normalized by the slab depth (in)

As the L/D increases the length between electrodes dominates the geoemtry factor m and at approxiamtely D/L = 1 the second linear relationship develops as the bottom surface of the slab starts to dominate the goemtry factor (m). It can be inferred that the intersection of the bilinear relationship can be used to determine the slab depth by graphing geometry factor m vs. the length between the electrodes. The length between the electrodes at the intersection of the bilinear relationship corresponds with the depth of specimen. As the slab depth increases the influence of the bottom surface of the slab (i.e., the second linear portion (D/L > 1)) decreases causing the slope of (D/L > 1) → slope of (D/L < 1) making the intersection of the bilinear relationship difficult to determine. This validates the conclusion that as the slab depth increases too far (D > 10”) as seen in Figure 5.5.3, the slab depth becomes difficult to predict.

95 5.6 Flawed Specimen

The influence that a flaw has on the bulk resistance was simulated using FEA. The flaw in this case was a cut made perpendicular to the surface that the electrodes were placed on. Changes in characteristics of the cut (width, depth, and conductivity) were linked to changes in bulk resistance. The cut had a defined width and depth and (unless otherwise noted) has a conductivity of zero. The cut can be viewed as a prismatic void space as seen in Figure 5.2.1.

5.6.1 Influence of the Position of Cut (Lcut)

The influence of the position of the cut was simulated in this section. The implications of this technique are that the location of the cut can be determined from access to a single surface. The distance between the electrodes was kept constant at 2.5” while the relative position from the electrodes to the cut was varied. The cut was positioned in-between and outside the electrodes as defined in Figure 5.2.1. The parameters of the data that was taken is shown in Table 5.6.1. The data that was taken is shown in Figure 5.6.1. Statistical regression fits were applied to develop a relationship between the cut position and the geometry factor (m) as seen in equations (5.6.1) and (5.6.2). The geometry factor mInbetween is the geometry factor as the function of the distance from cut to electrode where the cut is between the electrodes as seen in Figure 5.2.1. The geometry factor mOutside is the geometry factor as function of the distance from cut to electrode where the cut is outside of the electrodes as seen in Figure 5.2.1. 3

2

m Inbetween = 0.3714 ⋅ L cut − 1.1965 ⋅ L cut + 1.3912 ⋅ L cut + 2.7934

(5.6.1)

m Outside = 0.141 ⋅ Ln(⋅ L cut ) + 4.0557

(5.6.2)

96 By moving a pair of electrodes across a surface with a flaw while taking electrical measurements incrementally, the position of the flaw can be deduced by the spikes in the bulk resistance or dips in the geometry factor (m) as the electrodes near the flaw.

Table 5.6.1 Parameters of slab for varied position of cut. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Depth of specimen

in

~Infinite

Cut Depth

in

1.5

Cut Width

in

0.125

Length between electrodes

in

2.5

Conductivity of cut

S/in

0

Position of cut

in

Variable

Geometry factor m

5 4 3 2 Cut is Outside 1

Cut is Inbetween Electrodes

0 0

1

2

3

4

5

Distance from cut (in)

Figure 5.6.1 Slab specimen for m vs. distance from cut (in).

6

97 5.6.2 Influence of Cut Width (w)

This section quantifies the relationship between geometry factor m and the width of the cut. Table 5.6.2 shows the parameters that were used to develop the relationship between geometry factor m and the width of the cut as defined in Figure 5.2.1. The data is shown in Figure 5.6.2 and Figure 5.6.3. It can be seen in those figures that varying the cut width has relatively little influence on the geometry factor m. Thus it would be extremely difficult to predict cut width using the geometry factor m.

Table 5.6.2 Parameters of slab for varied cut width Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Depth of specimen

in

~Infinite

Cut Depth

in

1.5

Cut Width

in

Variable

Length between electrodes

in

Variable

Conductivity of cut

S/in

0

98 12

Geometry factor m

10

Length between eletrodes 0.1 0.2 0.5 1.0 2.0 4.0 6.0 10.0

8 6 4 2 0 0

1

2

3

Cut width (in)

Figure 5.6.2 Slab specimen with varied length between electrodes (in) for m vs. w (in).

Geometry factor m

12 10 8 Cut depth (in)

6

0.001 4

0.125 0.7

2

1.95 0 0

2 4 6 8 10 12 Distance between electrodes (in)

Figure 5.6.3 Slab specimen with varied w (in) for m vs. L (in).

99 5.6.3 Influence of Cut Depth (a)

This section determines the effect of changing the cut depth has on the geometry factor m. Table 5.6.3 shows the parameters that were used to develop the relationship between geometry factor m and the depth of the cut as defined in Figure 5.2.1. The data that was taken is shown in Figure 5.6.5 and Figure 5.6.4.

Table 5.6.3 Parameters of slab for varied cut depth. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Depth of specimen

in

~Infinite

Cut Depth

in

Variable

Cut Width

in

0.125

Length between electrodes

in

Variable

Conductivity of cut

S/in

0

100 14 Distance from cut to electrode (in)

Geometry factor m

12 0.0075

10

0.0375 8

0.1875

6

0.4375 0.9375

4

1.9375

2

2.9375 4.9375

0 0

5

10 Cut depth (in)

Figure 5.6.4 Slab specimen with varied Lcut (in) for m vs. a (in).

12 Cut depth (in) Geometry factor m

10 8

0

0.25

0.5

1.0

1.5

2.5

5.0

10

6 4 2 0

1

2

3

4

Distance from cut to electrode (in)

Figure 5.6.5 Slab specimen with varied a (in) for m vs. Lcut (in).

5

101 A regression function was used to develop linear fits for each set of cut depth data in Figure 5.6.5 for Lcut > 2” in the form of equation (5.6.3). The data for the slope and vertical axis intercept parameters (M and B) are shown in Figure 5.6.6. The polynomial fits were applied to that data for a ≥ 0.5”. Combining equation (5.6.3) and the fits produced equation (5.6.4) where ma is the geometry factor related to cut depth in a slab. m = M ⋅ L cut + B

(5.6.3)

m a = ( 0.0011⋅ a 3 − 0.0268 ⋅ a 2 + 0.1307 ⋅ a + 1.9436 ) ⋅ L cut

(5.6.4)

−0.0033 ⋅ a + 0.0989 ⋅ a − 0.9486 ⋅ a + 2.7409 3

2

where cut depth (a) and distance between cut and electrode (Lcut) are in inches. Equation (5.6.4) can be used to predict the cut depth by measuring the bulk resistance, which are shown graphically in Figure 5.6.7 and Figure 5.6.8.

2.5 M

M and B

2.0

B

1.5 1.0 0.5 0.0 -0.5

0

2

4

6

8

10

12

Depth of cut (in)

Figure 5.6.6 Slab specimen, M and B of varied a (in) for m vs. Lcut greater than 1 ½”.

102 14

Geoemtry factor m

12 10 8 6 4 Distance from electrodes to cut (in)

2

2

3

4

5

0 0

2

4

6

8

10

12

Cut depth (in)

Figure 5.6.7 Prediction of slab specimen with varied Lcut (in) for m vs. a (in).

14 Cut depth (in)

Geoemtry factor m

12

10 10

7

8

5

6

4 2

4 2 0 0

1

2

3

4

5

6

Distance from electrode to cut (in)

Figure 5.6.8 Predictions of slab specimen with varied a (in) for m vs. Lcut (in).

103 5.6.4 Influence of the Conductivity of Cut, σcut

This section looks at the relationship between a cut conductivity and specimen conductivity and how that affects the geometry factor (m). Table 5.6.4 shows the parameters that were used to develop the relationship between geometry factor m and the conductivity of cut. The data taken is shown in Figure 5.6.9 and Figure 5.6.10.

Geometry factor m

Table 5.6.4 Parameters of slab for varied cut conductivity. Resistivity of specimen

ohm*in

1000

Current

amp

1

Electrode area

in^2

0.25

Depth of specimen

in

~Infinite

Cut Depth

in

2.5

Cut Width

in

0.125

Length between electrodes

in

Variable

Conductivity of cut

S/in

Variable

Distance from cut to electrode (in) 0.0075 0.0375 0.1875 0.4375 0.9375

0.000001

1.9375

0.0001

2.9375

4.9375

0.01 Conductivity of cut filling

20 18 16 14 12 10 8 6 4 2 0 1

Figure 5.6.9 Slab specimen with varied Lcut (in) for m vs. σcut (S/in).

100

104

Geometry factor m

14 12 10 8

Conductivity of cut (S/in)

6

100 0.1 0.001 0.000001

4 2

1 0.01 0.0001 0

0 0

1

2

3

4

5

6

Distance from electrodes to cut (in)

Figure 5.6.10 Slab specimen with varied conductivity of cut (S/in) for m vs. Lcut (in)

Taking closer look at the data in Figure 5.6.9 the conductivity of the specimen (0.001 S/in) can be estimated by knowing the conductivity of the cut. This is done by noted that the intersection of the bilinear relationship between the geometry factor (m) and cut conductivity corresponds to the conductivity of the specimen as seen in Figure 5.6.11. It can be seen that when cut conductivity is greater then the specimen’s conductivity the distance between electrodes has little influence on the geometry factor (m). In the case of cut conductivity less than that of the specimen’s conductivity the distance the electrodes begins to influence on the geometry factor (m). The cut conductivity was divided by the conductivity of the specimen to produce the normalized conductivity of cut. The geometry factors (m) were graphed versus the normalized conductivities for each different distance from electrode to cut in Figure 5.6.11. To quantify the predictions of Figure 5.6.11 the equations of the natural logarithmic fits were used to determine the exact intercept, which correspond to an approximation of the conductivity of the specimen. These intercepts are shown in Figure 5.6.12 for each length from cut to electrode. As the electrodes near the cut, the approximation breaks down, predicting a much larger specimen conductivity. Because the slopes of the log fits of the same distance from electrodes to cut are very close, small

105 discrepancies in slopes result in large discrepancies in intercepts. The estimate of 0.0008 S/in of the specimen conductivity from Figure 5.6.12 is close to the actual value of 0.0010 S/in.

14 12 Geometry factor m

Distance from electrodes to cut (in) 10

0.0075 0.9375

0.0375 1.9375

0.1875 2.9375

0.4375 4.9375

8 6 4 2 0 0.001

0.01

0.1

1

10

100

Normalized conductivity of cut

Figure 5.6.11 Slab specimen with varied Lcut (in) for m vs. normalized σcut with bilinear log fits.

106

Length from cut to electrodes (in)

6 Actual specimen conductivity

5 4 3 2 1 0 0.1

1

10

100

1000 10000

Normailzed specimen conductivity prediction

Figure 5.6.12 Specimen conductivity prediction from varied cut conductivity data.

5.7 Conclusion

In this chapter a FEA was used to model the values of bulk resistances for external electrodes on the same surface of a slab specimen. The slab specimen was flawed with a cut centered between the electrodes and orientated perpendicular to the surface. These bulk resistances, along with resistivity of the material, cross sectional area, and distance between electrodes were used to calculate the geometry factor m. The aim of this exercise was to relate the geometry factor m to characteristics of the cut (conductivity, width, and depth), distance between electrodes, and depth of the slab. Since all FEA was done on a pseudo two-dimensional specimen, a conversion procedure was presented that allows to convert a two-dimensional FEA to a threedimensional FEA. This was done by producing a conversion function d(L, Dslab), equation (5.4.3), for 1” long electrodes. The value of which is a function of distance between electrodes (L) and the depth of the slab (Dslab).

107 A procedure for determining the position of the cut relative to electrodes was demonstrated. This demonstration was for electrodes 2.5” apart as a function of distance from cut of the nearest electrode. The response of geometry factor m to the position of the electrodes is shown in Figure 5.7.1 for three cases; the electrodes moving closer to the cut (point A), the electrodes moving farther away from cut and then moving closer to the cut while the cut is between the electrodes (Point A to Point B), and the electrodes moving away from the cut (to the right of Point B).

5 Geoemtry factor m

4

Point A

3

Point B

2 1 0 -6

-4 -2 0 2 4 6 Midpoint of electrodes distance from the cut (in)

Figure 5.7.1 Prediction for the position of cut.

The depth of a slab (Dslab) can be determined from the bulk resistance using one surface by measuring the bulk resistance from electrodes placed at various distances apart. This was demonstrated for slabs between 2” and 10” deep with electrode spacings between ¼” and 8 ¼”. By varying the conductivity of the cut and the spacing between electrodes, the specimen conductivity can be approximated. Practically speaking this can be done by filling the cut with a liquid. There is a bilinear relationship between the geometry factor

108 m and log of conductivity of cut. The conductivity of the cut at the intersection of the bilinear fits is approximately the conductivity of the specimen. A geometry factor ( m ( L, a ) ) that is a function of distance between electrodes (L) and cut depth (a) was formulated and shown equation (5.6.4). A more general case would be were the geometry factor would also be a function of cut width. Since the cut width was shown to have very little impact on the geometry factor m, it was not incorporated into m ( L, a ) for simplicity sake. The power of this equation is that from simple bulk

resistance measurements the cut depth can be predicted, which is shown in Figure 5.6.7 and Figure 5.6.8.

109

CHAPTER 6: SUMMARY AND CONCLUSIONS

6.1 Summary

This work provides an overview of a preliminary research study that was aimed at determining whether the existence and geometry of a flaw could be predicted in a concrete specimen using electrical property measurements. Specifically, this work presents experimental results and numerical simulations (using finite element analysis) to illustrate how the electrical response of concrete elements varies as a function of specimen and flaw geometries. FEA was used in conjunction with physical measurements to enable different specimen geometries and electrode alignments to be simulated quickly and with a minimal amount of experimental variability. Preliminary results were used to show the applicability of the FEA approach by comparing these results with physical measurements. After this objective was completed two specific specimen geometries were investigated in greater detail. The first specimen geometry consists of a cylindrical specimen with a flaw at the outer circumference of the cylinder that exists along the length of the cylinder. Simulations were performed using diametrically opposed electrode pairs that were ‘rotated’ to take measurements at different orientations around the specimen. The measurements were described in terms of the angle from the flaw to the position of the nearest electrode pair. The geometry factor (k) was computed for the specimens with various flaw geometries. In addition to the cylindrical geometry, this thesis provided a preliminary investigation of the use of IS to describe the geometric characteristics of a slab where only single-sided access is possible. The geometry that was investigated in this study

110 assumed that the flaws started at the surface of the concrete. It was assumed that the cut protruded into the specimen perpendicular to the surface of the specimen. The difference between the measurements in the slab and cylindrical specimens (i.e., the bulk resistance) is that while the measurements in the cylinder are a function of the angle of the measurement and the flaw size, the measurements in the slab specimens are a function of not only the flaw size, but also the electrode alignments (i.e., distance between electrodes), distance from electrodes to flaw, and the slab geometry.

6.2 Findings and Conclusions

This work began by correlating results of FEA with physical measurements. FEA was then used to determine geometry factors for flawed specimens by applying a current (I) across two electrodes and measuring the subsequent voltage drop. This enabled the bulk resistance (Rb) to be calculated using (6.2.1) for each specimen. Rb =

V I

(6.2.1)

This bulk resistance was then related to the geometry factor k for the cylinder specimen or geometry factor m for the slab specimen using the material property, resistivity (ρ), as shown in equation (6.2.2). k=

Rb ρ

(6.2.2) a

m=

ρ⋅L Rb ⋅ A

(6.2.2) b

where A is the cross sectional area of the electrode and L is the distance between electrodes. The geometry factors determined using FEA were compared with physical measurements. A reasonable agreement was observed between the physical

111 measurements and FEA models for both unflawed and flawed specimens. With the verification of the FEA completed, the next phases of the research (determining flaw characteristics for specific cylindrical specimens and slab specimens using FEA) could be implemented. For cylindrical specimens the width of the cut, the depth of the cut, and the diameter of the cylinder were varied along with the positions of the electrodes relative to the location of the flaw. It was shown that as the cut depth, cut width, and specimen diameter increased, the magnitude of the geometry factor increased. In general, as the electrode moves closer to the cut, the influence of the cut on the bulk resistance increases. In comparison to the cut depth, the cut width has very limited influence on the geometry factor (k). The geometry factor (k) was determined as function of cut depth and position of electrodes in relation to the cut as seen in Figures 4.4.16 and 4.4.17 and equation (4.4.8.) for a 4” diameter specimen. This relationship can be further generalized to make the geometry function dependent on the diameter of the specimen by a normalization procedure shown in equation (4.6.8.) using equations (4.4.8.) and (4.4.3.). One interesting observation from this research is that the resistivity of a specimen can be predicted by filling the cut with various substances of known resistivities (i.e., filling the cut with some electrolyte solution). The geometry factor can be plotted as a function of the log of the conductivity of the solution in the cut. If the solution is varied, the variability of the geometry coefficient is dramatically reduced as the conductivity of the cut equals that of the specimen. As a result, when the conductivity of the cut equals the specimen conductivity the measurements converge on a single geometry coefficient that is measured irrespective of the orientation of the cut. For the slab specimen, the geometry factor (m) was determined to be a function of the width of the cut, the depth of cut, the distance between electrodes, the conductivity of cut, the depth of slab and the distance between electrodes. The bulk resistances were found for each case (varied slab depth, varied cut depth, varied cut width, varied conductivity of cut) while the distance between electrodes was also varied.

112 Because the FEA was performed in a pseudo two-dimensional specimen a procedure involving a correction function, d(L, Dslab), to convert two-dimensional bulk resistance measurements to three-dimensional bulk resistance measurements as shown in equation (6.2.3) was demonstrated for the case of a 1” long electrode. d(L, Dslab ) =

R 3− d R 2−d

(6.2.3)

where R3-d is the bulk resistance measured in the true three-dimensions and R2-d is the bulk resistance measured in the pseudo two-dimensional environment. A prediction can be made for the depth of a slab by measuring the bulk resistance with various electrode spacings. By plotting the resulting geometry factor (m) versus the length between the electrodes, a bilinear relationship was observed. The length between the electrodes at the intersection of the bilinear relationship corresponds to the depth of the slab. A mathematical equation was also established for a geometry function (m) as a function of the length between electrodes and the depth of the slab (equation 5.5.3) The cut width has a small effect on the geometry factor (m) relative to the influence of the cut depth for cut widths less then ½” wide. As a result, the geometry factor (m) was not considered a function of cut width but a function of cut depth and electrode spacing. This is shown in equation (5.6.4) and Figures 5.6.7 and 5.6.8 for the case of a virtually infinite slab depth, which can be used to predict the cut depth from a measured bulk resistance response. This work has shown that electrical resistance measurements can be a useful technique to describe flaw characteristics in concrete. Specifically, this work showed that the bulk resistance is a function of cut depth, specimen resistivity, specimen geometry, and electrode alignment. The bulk resistance measurements were shown to be insensitive to the cut width (less than ½”). Equations were formulated for two specific specimen cases, cylinder specimens with external electrodes (diametrically opposed) and slab specimens with electrodes on the same surface. By varying the conductivity of cut and the position of the electrodes, a prediction about the resistivity of the specimen can be made. Predictions can also be made about the depth of the slab by varying the electrodes’

113 position relative to each other. It was also shown that the finite element model used in this research showed reasonable correlation with physical measurements. The next step of this research should be to further generalize the geometry functions for different types of flaws (i.e., flaws not perpendicular from surface, flaws not parallel to electrodes, inclusions). Developing a rigorous link between electrical measurement and the mechanical response would allow the vast amount of research done for mechanical responses of specimens with various flaws as well as fracture mechanics to be applied to the electrical impedance case. Further work with slab depth determination is necessary to explore the hypothesis that increasing the electrode width increases the sensitivity of the geometry factor (m) for deeper slabs, which would allow for deeper slab depths to be predicted.

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