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Analytical Study Guideline for Structural Health Monitoring of Steel Canister in Dry Cask Storage System by Using Multimodal Ultrasonic Lamb

Waves

Ayman Kamal, Victor Giurgiutiu Department of Mechanical Engineering, University of South Carolina

300 Main St. Columbia, SC 29208 Tel:(803)777-1535, Email:kamal@email.sc.edu,victorg@sc.edu

Abstract – Guided Lamb waves have been successfully used for some years as a monitoring technique for damage in plates and pipes. Multimodal Lamb waves are typically observed in thick structures, e.g. steel canisters in dry cask storage systems. Different modes have different sensitivity for certain types of cracks, and it is usually desired to excite specific symmetric or antisymmetric Lamb wave modes. To quantify the participation of different modes, it is important to know the power that each mode shares when exciting multiple modes. This study presents an analytical development and numerical simulations of wave power distribution between different ultrasonic Lamb wave modes using normal mode expansion (NME) technique. Piezoelectric wafer active sensor (PWAS) transducers were used to excite ultrasonic Lamb waves in the structure. The interaction between the PWAS and the structure was studied. Numerical simulations were performed for (1) thin aluminum plate (1-mm) in which one symmetric and one antisymmetric modes exist, (2) thick steel plate (½") in which three symmetric and three antisymmetric modes exist. In the later case study, power partition between the 6 modes was presented, and a parametric study was performed for different transducer sizes and excitation frequencies.

I. INTRODUCTION Nuclear power plants and radioactive waste systems

are safety-critical facilities. These systems are in need of continuous scheduled inspection and preferably continuous structural health monitoring (SHM). It is crucial for monitoring structure performance and integrity, and detecting the initiation of flaws and damage in order to predict structural life and avoid catastrophic failures.

There are many well-established techniques currently in operation in industrial facilities, for instance: ultrasonic testing/imaging and phased array technology, conventional and pulsed eddy current technique, vibration techniques, acoustic emission monitoring, and infrared thermography [1]. Ultrasonic testing/imaging is becoming one of the main practices for nondestructive evaluation (NDE) programs in aerospace, civil and nuclear industries [2]. Ultrasonic testing is used for inspecting weld flaws in solid structures [3], where imaging flaws using phased arrays and guided waves was presented. Comprehensive guidelines about ultrasonic guided waves for NDE and SHM for nuclear power plants can be found in [1, 2, 4]. Developed inservice and preservice inspection programs are driven by requirements in the American Society of

Mechanical Engineers Boiler and Pressure Vessel Code (ASME Code) and the requirements of the U.S. Nuclear Regulatory Commission (NRC) [4]. Many studies were performed for testing and monitoring pipeline in nuclear power plants. Ultrasonic wave propagation was used for detecting weld flaws, and wall thinning in stainless steel pipes and elbows [5]. Laser Doppler vibrometer was used for imaging wave propagation excited by lead zicronate titanate transducers. Another method for detecting wall thinning in stainless steel pipes is pulsed eddy current [6]. It was shown that pulsed eddy current testing is a promising technological approach to the NDE, and it has been principally developed for the measurement of surface flaws, subsurface flaws, and corrosion. Creep–related damage in nuclear power plants were studied in [7]. Stress corrosion cracking (SCC) is another major flaw that initiates often in containers and piping systems with high corrosive materials or salt-water cooling systems. SCC mitigation and advanced detection methods were studied in one of the recent Los Alamos reports [1]. SCC is also a major concern for dry cask storage systems, and was observed in stainless steel canisters in dry cask storage systems with high levels of chlorine [8].

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(a)

(b)

(c) Figure 1. The various ways in which PWAS are used for structural sensing includes (a) propagating Lamb waves, (b) standing Lamb waves and (c) phased arrays. The propagating waves methods include: pitch-catch; pulse-echo; thickness mode; and passive detection of impacts and acoustic emission, [9]. PWAS: piezoelectric wafer active sensor.

Dry cask storage systems are being used in the US and around the world for storing used nuclear fuel. It is essential to develop SHM systems that can assess the structure conditions in a timely manner without a human factor on site. The high levels of radiation occurred with Fukushima nuclear reactors, and the expected (40 years of clean up [2]), would require such systems for monitoring both nuclear facilities and high level waste storage systems. The Hanford site in Washington state has early-built tanks made of carbon steel and not stainless steel; continuous monitoring of these tanks is very important.

Although we stated earlier that ultrasonic guided waves is currently one of the lead techniques in NDE and SHM systems, there are many challenges using them, such as: (1) dispersion : every excited wave packet comprises waves with frequency spectrum and not a single frequency wave, these waves with different frequencies propagate at different speeds. Another challenge is (2) existence of multiple propagating modes at given exciting frequency from the source, e.g. function generator to generate ultrasonic Lamb waves into the structure.

The objective of this study is to show the guidelines when multimodal Lamb waves exist in structure under test, which is typically the case for thick structures, e.g. steel canisters in dry cask storage systems.

Piezoelectric wafer active sensor (PWAS) transducers have emerged as one of the major structural health monitoring (SHM) technologies because, with the same installation of PWAS transducers, one can apply a variety of damage detection methods (Figure 1) (a) propagating ultrasonic guided waves, (b) standing waves (electromechanical E/M impedance ), as well as (c) phased arrays. PWAS is small, lightweight, unobtrusive, and inexpensive transducer. PWAS can act as a passive sensor, i.e. without interacting with the structure, and/or (3) an active sensor, where it interacts with the structure to detect the presence and intensity of damage. A characterization of PWAS transducers was studied in [10]. PWAS also helps achieving tuning of guided Lamb waves [11].

It was shown that energy loss of guided waves occur by existing flaws in the structure [12]. Impedance mismatch is considered an “energy-stealing” agent. Other studies [13, 14] gave more insights on how different defects interact with Lamb waves, and how the severity of impact damage can be predicted from the transmitted power. Generally, failure theories based on energy methods are more robust in predicting failure.

In this study, we are analytically modeling Lamb wave modes power and presenting how modes share power. Then, we conclude by presenting parametric studies as optimization tools for power efficient SHM systems.

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II. PWAS-STRUCTURE INTERACTION Normal mode expansion (NME) method is used to (a)

find directly the amplitudes of given mode in terms of loading parameters, (b) evaluate the contribution factor of every mode to the total wave power and energy. Normal modes of the guided waves in the structure serve like the eigenfunctions [15].

Considering a PWAS of length 2a bonded to a structure of thickness 2d . Assuming pin-force model for PWAS-structure interaction, i.e. perfect bond (Figure 2), the developed shear stress by the PWAS is

0

0

( )= [ ( ) ( )]( )[ ( ) ( )]

xt x a x a x aF x a x a� � �� � �

� � �� � � �

(1)

where 0 ( )F � is the force acting at the PWAS ends, and it is a function of excitation frequency � applied to the PWAS. The model is a 1-D analytical model, and straight crested Lamb waves are assumed propagating in positive x-direction (forward waves) and negative x-direction (backward waves). Denoting structure particle motion by

,x yv v for particle velocity in x and y directions, and ,xx xyT Tfor axial and shear stresses respectively; power flow can be defined as [15, 16]

� 1 ( ) ( ) ( ) ( )2

dn n n n

mn x xx y xyd

P v y T y v y T y dy�

� � � n n n nx xx y xyv y T y v y T y dy( ) ( ) ( ) ( )) ( ) ( ) (n n n n( ) ( ) ( )( ) ( )( ) ( )x xx y xyx xx y( ) ( ) ( )( ) ( )) ( ) (v ( ) ( )) (n ( ) ( ) (2)

where ( ) ) indicates the complex conjugate of the quantity, and m , n indicate Lamb wave mode number 1,2,3…

Figure 2. Pin force model for structurally-bonded PWAS, (a) PWAS pin forces at the ends on the upper surface, (b) developed shear stresses.

Amplitudes of Lamb wave modes were normalized with respect to power flow and modal participation factor for the nth mode was defined as

�

0( ) ( )[ ]4

n n n

nx i a i a i xPWAS

nnn

v da x F e e e

P� � �� � �� �

� (

� nxv d�nx (�

( (3)

where a is half the PWAS length. � is the propagating wave number, and it is the ratio between angular frequency and wave velocity ( )c� �� . Eq. (3) shows that the modal participation factor depends on characteristics of Lamb wave modes as well as transducer dimension.

II. A. Electromechanical Admittance

Impedance spectroscopy is commonly called for

electromechanical (E/M) impedance of the transducer. However, the electrical admittance is commonly expressed to calculate the electrical power of the transmitter PWAS. The PWAS admittance when attached to the structure, i.e. the constrained PWAS admittance [9] is

� � 2

0 3111 1

( )cot ( )Y i C k

r� �

� � � � �

�� �� � �� �� �� ��� �� �� �

(4)

where � is the angular frequency, 0C is the capacitance,

31k is the E/M coupling factor, a.k.a. � 2 231 31 11 33/ E Tk d s �� ;

where 31d is the piezoelectric coupling coefficient in 31 direction, 11

Es is the transducer compliance, 33T� is the

electrical permittivity of the transducer, ( ) ( )a� � � �� ; where � is the wave number, a is half the PWAS length,

� r � is the dynamic stiffness ratio, � � /str PWASr k k� �� ; where strk was calculated by dividing the pin-force by the displacement field xu at the PWAS end. Displacement was determined from the NME solution with consideration of modal participation factor, Eq. (3).

PWAS stiffness is defined as 11/ EPWAS ak t b s a� ; where

at is the PWAS thickness, b is the width.

II. B. Electrical Active power and Wave Power

Triggering the PWAS with electrical signal of voltage amplitude V̂ ; and substituting the admittance from Eq. (4), the electrical power is evaluated as

2 2

Re Im1 1ˆ ˆ 2 2active reactiveP Y V P Y V� � (5)

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The active power is the power that is converted to the mechanical power at the interface. The reactive power is the imaginary part of the complex power that is not consumed and is recirculated to the power supply. Mechanical power excites both forward and backward propagating waves initiating from the two end tips at x a�and x a� � . Due to symmetry, we only need to consider the forward wave, which will contain only half of the mechanical power converted from the electrical active power. The time averaged wave power is

� 12 xx x xy y

A

p T v T v dA� � � xx x xy yT v T v dAxx x xy yx xyT v (6)

where A is the structure cross sectional area. Defining Sij as Lamb wave modes strain components, the time-averaged kinetic and potential energies are

�

�

1 . . 41 24

e x x y yA

e xx xx yy yy xy xyA

k v v v v dA

v T S T S T S dA

�� �

� � �

x y y..v v v dA. ...v

xx yy yy xy xyS T S T S dA2xx yy yy xy xyxx yy yy xyS T ST (7)

It is worth mentioning that Equations (6) and (7) are defined for one single mode. Full length formula with modal participation factors of each mode and all involved parameters were reported in the Ref. [17]. In the next section we show numerical simulations for the developed wave power model.

III. NUMERICAL SIMULATIONS We examplify power transaction between the PWAS

and the structure with simulation of two plates, (a) 1-mm aluminum up to 2000 kHz, (b) 12.7-mm (½ in) steel up to 500 kHz frequency of excitation. For both cases, harmonic excitation of 10 volts was applied on a 7-mm PWAS. Complete listing of simulation parameters are in Table I, and Table II.

For 1-mm aluminum case, impedance analyzer was used for measuring the transducer E/M impedance in complex form; then the admittance was calculated. The experiments were performed on multiple beams and plates to show the constrained PWAS resonances. Also the test was performed for constrained PWAS on the square aluminum plate of dimensions 1240 mm x 1240 mm x 1 mm. The frequency sweep was up to 2000 kHz.

As a validation for our developed analytical model, FEM simulations were performed using coupled field physics, where the PWAS was excited by voltage and the piezoelectric coupling introduces induced strains. Damping

was defined using complex compliance and complex electric permittivity with imaginary coefficients of 0.04.

TABLE I

Structure Simulations Parameters

symbol 2024 AL alloy steel-AISI-4340

Length L ∞ ∞ thickness h 1 mm 12.7 mm (½ in) Width b 7 mm 7mm Young’s Modulus E 72.4 GPa 200 GPa

Poisson ratio � 0.33 0.29 density � 2780 7850 Harmonic input voltage amplitude

V̂ 10 V 10 V

Frequency sweep f 1-2000 kHz 1-500 kHz

TABLE II

Transmitter PWAS (PZT850) properties (as from the company website www.americanpiezo.com)

symbol PZT850 Length l 5-25 mm thickness ta 0.2 mm Width b 7 mm Young’s Modulus E 63 GPa Elastic compliance 11

Es 15.8e-12 m2/N Relative dielectric constant 33 0/T� � 1750

Coupling coefficient k31 0.353 Piezoelectric coefficient d31 -175 nm/kV

IV. RESULTS

Comparison between analytical, experimental and FEM simulations are shown in Figure 3. The predicted first resonance was 340 kHz by analytical model and FEM in which both do not account for the width dimension. The experimental result for the beam case was 250 kHz, while it was 400 kHz for the plate. The analytical model overestimated the admittance amplitude of the first resonance with respect to other resonances. Second resonance observed by FEM was 1130 kHz, (1200 kHz analytically and it was off experimentally� 1000-1100 kHz). Third resonance by FEM was 1432 kHz and it was missed by the analytical model; experimental results were � 1330-1370 kHz.

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Figure 3. E/M admittance (a) analytical prediction, (b) experimental and FEM results [18]

IV. A. Thin Structures (One Symmetric and One Antisymmetric Lamb wave modes)

A parametric study was performed based on the

analytical model of electrical active power for different PWAS sizes (5-25 mm) and frequencies (1-1000 kHz).

The results are presented as a 3D mesh plot (Figure 4). It indicated that active power generated from the PWAS to the structure contains the tuning effect of transmitter transducer size as well as the excitation frequency. The maximum active power in the simulation was 8.3 mW. This can be achieved by different combinations of PWAS and excitation frequencies, e.g. (5mm PWAS size and 610 kHz, 9-mm PWAS and 890 kHz). These combinations provide guidelines for the design of transmitter size and excitation frequency in order to obtain maximum wave power into the SHM structure.

Figure 4. Parametric study for electrical active power converted to wave power, for 1-mm aluminum simulation.

IV. B. Thick Structures (Multimodal Lamb waves)

IV. B. 1. Power portioning between modes (7-mm PWAS size)

Thick steel plate ½" – thick and excited up to 500 kHz

was used as a case study in our predictive model, in which three symmetric and three antisymmetric modes exist. The structural particle velocities in x and y directions were reported in Ref. [17, 19] Power partitioning between the 6 modes is presented (Figure 5). Results show that: (a) S0 mode (the fundamental symmetric mode) has a rejection points 175 kHz and 400 kHz. (b) At 175 kHz, A0 mode (the fundamental antisymmetric mode) absorbs all the energy leading to 0% for S0 mode. (c) The reported results represent the ratios of power between modes (Figure 5a). At 175 kHz, S0 is rejected and A0 consumes all the power; however, at 200 kHz, A0 actually consumes more power (than at 175 kHz) (Figure 5b); this is due to the fact that our model is based on constant supplied electrical voltage. (d) A0 mode has a rejection point 400 kHz. (e) At 400 kHz, S1 power is dominant, i.e. it withdraws mostly all the energy, with a little share (less than 10%) by A1 mode, (f) At 280 kHz, power is distributed almost equally between S0, A0, and A1 modes.

IV. B. 2. Parametric study for multimodal Lamb waves

(different PWAS sizes) A parametric study was performed on ½" – thick steel

plate. Different PWAS sizes (5-25 mm) were used, and excitation frequencies were (1-1000 kHz). Figure 6 shows similar judicious combination of PWAS size and excitation frequency. A maximum value of electrical active power

1.6 mW could be obtained by 7-mm PWAS size and 576 kHz excitation frequency, or 15-mm PWAS and 662 kHz, or 24-mm PWAS size and 666 kHz excitation.

0 500 1000 1500 20000

1

2x 10-4

frequency [kHz]

Adm

ittan

ce [1

/�]

(a) (b)

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Figure 5. Power partitioning between modes for Lamb wave propagation in ½"- thick steel beam (a) normalized scale, (b) actual wave power in mWatt.

Figure 6. Parametric study for different PWAS transducer sizes and excitation frequencies for multimodal Lamb waves in thick steel structure.

V. CONCLUSIONS

Analytical investigation of power and energy transduction between PWAS and host structure was performed considering the exact Lamb waves. Predictive models were developed for Lamb wave power and energy using normal mode expansion technique. Input electrical active power is converted to mechanical power at PWAS interface; which in turn excites the guided waves.

The PWAS power and energy study indicates that electrical active power, wave power and energies have the

same tuning effect. Optimal energy coupling is a function of structure properties, geometry, PWAS size, and excitation frequency. The parametric studies give guidelines for maximum wave energy production for a given source of wave generation.

ACKNOWLEDGMENTS This work is supported by Air Force Office of

Scientific Research #FA9550-11-1-0133, Program Manager, Dr. David Stargel; and the National Science Foundation under Grant # CMS-0925466.

S0 A1

A0

S1 S2

A2

S0 A1

A0

S1

S2

A2

S0

(a) (b)

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NOMENCLATURE

an(x) = modal-participation factor for nth mode d31 = piezoelectric coupling coefficient in 31, m/V F0(ω) = pin-force at PWAS ends kstr = dynamic stiffness, N/m k31 = electromechanical cross-coupling coefficient

ek = time-averaged kinetic energy Pmn = power factor. Measure of average power flow p = power, W p = time-averaged power

r(ω) = dynamic stiffness ratio ev = time-averaged potential energy

Y = admittance (simens) τ, τa = shear stress at PWAS tip x = a

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