Forward models for extending the mechanical damage evaluation capability of resonant ultrasound spectroscopy

Accepted Manuscript Forward Models for Extending the Mechanical Damage Evaluation Capability of Resonant Ultrasound Spectroscopy B.R. Goodlet, C.J. Torbet, E.J. Biedermann, L.M. Jauriqui, J.C. Aldrin, T.M. Pollock PII: DOI: Reference:

S0041-624X(17)30099-9 http://dx.doi.org/10.1016/j.ultras.2017.02.002 ULTRAS 5476

To appear in:

Ultrasonics

Received Date: Revised Date: Accepted Date:

25 October 2016 1 February 2017 3 February 2017

Please cite this article as: B.R. Goodlet, C.J. Torbet, E.J. Biedermann, L.M. Jauriqui, J.C. Aldrin, T.M. Pollock, Forward Models for Extending the Mechanical Damage Evaluation Capability of Resonant Ultrasound Spectroscopy, Ultrasonics (2017), doi: http://dx.doi.org/10.1016/j.ultras.2017.02.002

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Forward Models for Extending the Mechanical Damage Evaluation Capability of Resonant Ultrasound Spectroscopy B.R. Goodleta,∗, C.J. Torbeta , E.J. Biedermannb , L.M. Jauriquib , J.C. Aldrinc , T.M. Pollocka a Materials,

University of California, Santa Barbara, 93106, USA Corporation, Albuquerque, NM 87113, USA c Computational Tools, Gurnee, IL 60031, USA

b Vibrant

Abstract Finite element (FE) modeling has been coupled with resonant ultrasound spectroscopy (RUS) for nondestructive evaluation (NDE) of high temperature damage induced by mechanical loading. Forward FE models predict mode-specific changes in resonance frequencies (∆fR ), inform RUS measurements of mode-type, and identify diagnostic resonance modes sensitive to individual or multiple concurrent damage mechanisms. The magnitude of modeled ∆fR correlate very well with the magnitude of measured ∆fR from RUS, affording quantitative assessments of damage. This approach was employed to study creep damage in a polycrystalline Ni-based superalloy (Mar-M247) at 950 °C. After iterative applications of creep strains up to 8.8%, RUS measurements recorded ∆fR that correspond to the accumulation of plastic deformation and cracks in the gauge section of a cylindrical dog-bone specimen. Of the first 50 resonance modes that occur, ranging from 3 to 220 kHz, modes classified as longitudinal bending were most sensitive to creep damage while transverse bending modes were found to be largely unaffected. Measure to model comparisons of ∆fR show that the deformation experienced by the specimen during creep, specifically uniform elongation of the gauge section, is responsible for a majority of the measured ∆fR until at least 6.1% creep strain. After 8.8% strain considerable surface cracking along the gauge section of the dog-bone was observed, for which FE models indicate low-frequency longitudinal bending modes are significantly affected. Key differences between historical implementations of RUS for NDE and the FE model-based framework developed herein are discussed, with attention to general implementation of a FE model-based framework for NDE of damage. Keywords: RUS, creep, damage, evaluation, NDE, finite element, resonance, modeling

∗ Corresponding

author Email address: [email protected] (B.R. Goodlet)

Preprint submitted to Ultrasonics

January 31, 2017

1. Introduction Demand for fast, reliable, and affordable nondestructive evaluation (NDE) techniques for mechanical components has existed for decades [1], with the aerospace industry often at the forefront due to the everincreasing complexity and cost of turbine engine components [2]. Resonant ultrasound spectroscopy (RUS) 5

falls under the broad field of ultrasonics, and saw early implementations by Fraser and LeCraw of Bell Laboratories in 1964 for measuring elastic properties [3]. However, resonance-based ultrasonic methods often receive less attention compared to pulse-echo, transmission, or phased array ultrasonic methods, especially for NDE of structural and mechanical components. The intent of this work is to demonstrate the utility of a combined framework of RUS measurements and simple finite element (FE) models for NDE of mechanical

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damage, and to broadly discuss the opportunities and limitations of such a framework as it pertains to manufacturing process control, damage evaluation, and materials science research. Ultrasonic methods provide the most accurate characterization of elastic properties of solid media [4, 5] and they accomplish this through nondestructively propagating low-energy elastic waves through the test specimen. Pulse-echo ultrasonic methods rely on accurately measuring the time required for an elastic wave

15

to propagate through a known volume of material in order to determine the multiple, but at a minimum two, independent elastic wave speeds (c0 ) intrinsic to the material. As required by the elastic wave equation [4, 6]: s c0 =

C∗ , ρ

(1)

where ρ is the density and C∗ is an ‘effective’ elastic constant comprised of a linear combination of Cijkl . These Cijkl define the constitutive relationship between stresses (σij ) and strains (kl ) in a 3D Cartesian 20

coordinate system as: σij = Cijkl kl .

(2)

Resonance methods rely on the same fundamental principle outlined in Eq. (1), but instead of time-offlight measurements resonance frequencies (fR ) of the specimen are used for NDE. In a simplified treatment, the natural vibrational modes of the specimen with free-free boundary conditions are linked to c0 and the resonance mode wavelength (λ) by: c0 n fR = = λ 2L 25

s

C∗ ρ

(3)

where dimensions of the specimen geometry: L = nλ/2 and n is an integer representing mode-order [7]. Even with this simplified treatment, general relationships valuable for NDE can be demonstrated. For example, 2

with all other factors held constant an increase in C∗ would increase c0 and fR . Rigorous treatment of the physics described by Eq. (3) requires solving the elastic wave equation in order to yield fR , often referred to as the forward problem. Unfortunately a general analytical solution does not 30

exist for the forward problem [8]. While approximate numerical methods developed by Visscher et al. [9] paved the way to indirect solutions to the inverse problem (calculating elastic properties from measured fR ), Visscher’s xyz-algorithm for tackling the forward problem is still limited to relatively simple specimen geometries by the requirement: geometry must be described with a set of continuous polynomial functions [9]. By contrast, FE methods as described by Liu and Maynard [10] rigorously solve the forward problem

35

over simple discretized (finite element) domains that are combined to describe the resonance of simple or arbitrary specimen geometries [10, 11, 12]. FE methods, implemented in a straightforward manner by commercial FE modeling packages like Abaqus CAE [13], provide a generalized forward modeling capability that can be combined with RUS measurements of fR to create a powerful NDE framework. Forward modeling capabilities provide the means to deconvolve multiple concurrent damage mechanisms affecting resonance

40

through one-factor-at-a-time FE modeling studies; while empirical deconvolution via stringent experimental controls is often impractical or impossible.

2. Background Excitation and measurement of elastic waves for resonance-based ultrasonic methods can be achieved through various means, including: laser pulses [14], magnetic fields [15, 16], or contacting piezoelectric 45

transducers [4, 6, 17, 18]. These elastic waves propagate throughout the specimen, reflect off free surfaces, and interfere with one another as they traverse the specimen. Only when the drive frequency of the excitation source nears a natural vibrational mode frequency of the test specimen will two opposite-traveling elastic waves constructively interfere in such a manner as to greatly amplify the deflections imparted by the excitation force, bringing the specimen into a state of mechanical resonance [6]. While all of the RUS

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measurements collected as part of this study are excited and measured exclusively via contacting piezoelectric transducers, the theory and many of the practical limitations discussed herein also apply to resonance methods that use alternative excitation sources. In the broader context of ultrasonic methods for NDE, RUS has many advantages over pulse-echo methods. Notable advantages include the potential to fully characterize elastic properties from specimen that

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are: smaller [4, 6, 19], irregular in shape [9], or cut with a misaligned crystallographic orientation [20]. Even complex geometry samples are feasible with inclusion of FE methods [10, 12], and all from a single 3

broadband measurement [4, 6, 19]. For a detailed discussion of the merits of RUS as compared to pulse-echo methods, particularly with the aim of accurately measuring elastic properties, the reader is directed toward the works of Heyliger, Ledbetter, and Austin [19], Leisure and Willis [4], and Migliori and Sarrao [6]. 60

A well understood limitation of RUS is the fact that a broadband resonance spectrum contains detailed information about resonance mode frequencies, but little if any information about the resonance mode shape. This limitation is particularly troublesome to efforts evaluating damage or material properties because both endeavors rely on proper mode identification. Laser Doppler vibrometry has been demonstrated as a powerful tool for solving the mode assignment problem by mapping the deflection character of the specimen

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surface as it resonates [21, 14], but is also prohibitively expensive for most NDE efforts. Without additional measurements of mode shape, tracking frequency changes of individual modes will always have a degree of ambiguity; but a strategy for identifying instances of disagreement in mode order between measured and modeled resonance data is discussed in Section 5.1. 2.1. Historical Use of RUS for NDE

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RUS-based approaches for NDE to date have primarily focused on comparing a single component to a population of peers—described herein as population statistics sorting (PSS). NDE frameworks based on PSS methods rely on large databases of fR collected from a population of similar components that are ultimately sorted as acceptable or unacceptable based on prior knowledge of the component history or purposefully imparted damage [22]. Using this teaching set of components with known condition, the fR landscape is

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divided into acceptable and unacceptable regimes against which parts with unknown condition are judged. PSS methods are the founding principle behind one commercially relevant NDE with RUS technique: Process Compensated Resonance Testing (PCRT) [23, 24]. While commercialized techniques for NDE are still being developed with expanded capabilities, many of the obstacles first encountered by simple PSS methods continue to hinder NDE efforts today. For example, PSS methods are prone to reject components that

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exhibit anomalous resonance characteristics—including benign anomalies that may arise from a change in the component manufacturing process [6]. Quantitative correlations between damage accumulation and RUS measurements is often difficult based on measurements alone, and the complex nature of a resonating 3D body can lead to complex changes in resonance with damage [4]. Ultimately these limitations of PSS methods make it difficult to predict how multiple concurrent damage mechanisms will affect resonance, or

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even how a similar component with slight differences in geometry would behave when subjected to similar damage—requiring systematic damage be conferred to each unique component design in order to create a PSS database necessary for NDE. 4

Beyond PSS methods that essentially sort components as either acceptable or unacceptable, RUS has been employed for NDE of silicon nitride ceramic ball bearings in a semi-quantitative manner by taking 90

advantage of inherent symmetries [25, 26, 27]. The high degree of material and geometric symmetry exhibited by a defect-free ball bearing results in degeneracy where multiple resonance modes occur at the same fR . When damage such as cracks or scratches disrupt the geometric symmetry of the bearing, degenerate-mode splitting is observed in the RUS spectrum. Degenerate-mode splitting occurs because the fR of certain modes are affected by damage to a greater extent than others modes, making once degenerate peaks distinguishable

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[17, 15]. The magnitude of degenerate-mode splitting has been correlated with flaw size [17], demonstrating the utility of RUS for NDE of highly symmetric components [24]. Unfortunately, principles developed for these high-symmetry cases do not transfer to complex geometry components because they have few if any reliably degenerate modes. 2.2. Integrating FE forward Models with RUS for Expanded NDE capability

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In this research an NDE with RUS framework is presented wherein RUS measurements are integrated with forward finite element models capable of predicting mode-specific changes in resonance frequency (∆fR ) as a function of ‘damage’, defined as:

∆fR =

fRdamaged − fRbaseline ∗ 100%. fRbaseline

(4)

These modeled ∆fR will be directly compared to measured ∆fR on a mode-matched basis in order to: • Validate FE models capture the dominant damage mechanisms affecting resonance. 105

• Identify diagnostic modes, or mode-types, that are sensitive to damage and reliably measured. • Establish RUS measurement sensitivity for quantitative NDE. Broadly this work will demonstrate how relatively simple forward models can be practically employed to study multiple damage mechanisms associated with creep on a complex geometry specimen. By varying the location, type, and severity of damage, the unique resonance response of the specimen can be assessed. Ulti-

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mately these model results will be compared to measured ∆fR from a crept dog-bone specimen with multiple concurrent high-temperature damage mechanisms operable. Note that all high-temperature mechanisms affecting resonance herein are referred to as ‘damage mechanisms’, however not all are strictly degradative to component life in a service environment. One such example is the formation of an adherent oxide layer 5

on the surface of Ni-based superalloys. Oxide formation may affect resonance, but is often encouraged to 115

passivate the underlying substrate against further oxidation and corrosion [2]. 2.3. Mechanical Behavior and Creep of Ni-based Superalloys Ni-based superalloys maintain remarkable mechanical properties at high fractions of their melting temperature, even exhibiting an anomalous increase in yield strength with temperature [2]. Superalloys also resist creep, which is a time-dependent plastic deformation process that occurring at significant fractions of

120

the homologous temperature of the alloy [28]; making them desirable for high-temperature applications in turbine engines. Creep deformation will ultimately lead to fracture, and is accomplished through a multitude of mechanisms that depend on diffusion and plastic deformation mechanisms [28]. An example of an uninterrupted creep curve for polycrystalline Ni-based superalloy Mar-M247, loaded to 300 MPa at 950 °C in air, is provided in Fig. 1. The figure also delineates the three distinct creep regimes, with a majority of

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the creep life under the test conditions residing in the secondary regime where the creep rate is constant. Creep damage can manifest in a number of forms, from coarsening or dissolution of precipitate structures used to convey strength, to grain structure changes as a result of grain growth or dynamic recrystallization, to grain boundary damage such as sliding, void formation, and cracking [29]. Mar-M247, along with most polycrystalline Ni-based superalloys, utilizes small additions of carbon to promote formation of grain

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boundary metal carbides to inhibit grain boundary sliding [2]. Nevertheless, voids still nucleate and coalesce on grain boundaries, leading to cracks that ultimately cause failure of the specimen [28]. A fully-aged two-phase γ–γ’ microstructure of Mar-M247 is shown in Fig. 2 (a), along with an etched micrograph of the cast dendritic structure in Fig. 2 (b) which elucidates the polycrystalline nature of the material. The cuboidal γ’ precipitates are an ordered FCC phase that resist glide and climb of dislocations necessary for

135

high temperature deformation. Understanding the onset and progressive growth of these types of damage and their likely influence on resonance is important to consider when FE models are developed to predict changes in resonance. Since low-order resonances cause deflections over large volumes of the resonating body, these modes effectively probe the average elastic response of the body. This fact is important to consider when building

140

a FE model for predicting fR of a specimen comprised of a polycrystalline multi-phase material such as Ni-based superalloys. A FE analysis by Nygards demonstrated that as long as the specimen consists of a hundred or more discrete grains without any preferred orientation, i.e. texture free, then the macroscopic elastic behavior of the specimen is effectively isotropic [30]. Indeed, many engineering applications involve structural alloys in polycrystalline form that are assumed to be elastically isotropic [31]. The assumption 6

145

of homogeneous isotropic elastic properties greatly simplifies FE model design, but it is always important to consider the validity of such an assumption when developing a virgin specimen model as there are many cases where isotropy cannot be assumed, particularly directionally solidified and single crystal materials [2].

3. Creep Experimental Methods and Results 3.1. Experimental Methods 150

The cylindrical dog-bone specimen used for this study is depicted in Fig. 3. Specimens were machined from investment cast polycrystalline bars of Ni-based superalloy Mar-M247 with nominal composition given in Table 1. To impart creep deformation to the specimen a uniaxial tensile load was applied via specimen grips to the recessed ends of the specimen shoulder, detailed in Fig. 3. The threaded end of the dog-bone specimen was not used for this study, but serves to affix the dual-purpose specimen to an ultrasonic fatigue

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test rig as discussed in [32]. All creep iterations were conducted with the specimen loaded to 300 MPa inside of a clam-shell furnace at a constant 950 °C in air to simulate temperatures experienced by aerospace turbine engine components [2]. Elongation of the specimen during creep was monitored via a creep-resistant scaffolding that aligns with the specimen grips and translates the change in specimen length to two digital extensometers outside of the clam-shell furnace.

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In order to study how evolving creep damage affects resonance, a single dog-bone specimen was iteratively crept six times—denoted C1, C2, ..., C6—for a total creep strain of 8.8% after the C6 iteration as detailed in Table 2. After each creep iteration the specimen was cooled to room temperature, removed from the creep frame, and nondestructively evaluated via RUS. The RUS setup was developed by the Vibrant Corporation, with further details about the equipment and measurement conditions discussed in ASTM Standard Practice

165

E2001 [22] and E2534 [23]. Depicted in Fig. 4, the dog-bone specimen rests directly on four custom-built omnidirectional piezoelectric transducers (PT) with hemispherical SiC tips that together are referred to as the ‘PT cradle’ or ‘nest’. The cradle is configured such that a dummy and drive PT are parallel to each other on one side of the cradle, with two receive PTs on the opposite side of the cradle to support the specimen. No coupling exists between the specimen and the PT cradle beyond the force of gravity. The drive PT is

170

excited with a swept sinusoidal signal generated by the transceiver in 3 Hz steps. When the drive frequency nears a fR of the specimen it will begin to resonate and generate amplified deflections that are measured by the two receive PTs contacting the specimen, indicated as Rec. 1 PT and Rec. 2 PT in Fig. 4. The receive PTs convert vibrations back into an electrical signal that is returned to the transceiver, recorded,

7

and plotted as a function of the drive frequency to yield a broadband resonance spectrum plot as depicted 175

at the bottom of Fig. 4. A list of fR that are characteristic of the measured specimen were collected from the combined broadband signal of the two receive PT. Three broadband measurements were collected at each creep increment with the average peak frequency used for all comparisons to modeled data. For the PT cradle configuration used in this study the signal amplitudes are not reliably measured from one scan to the next because the

180

relative position of the specimen may vary slightly between measurements and the dog-bone is free to deflect from the PT cradle. For this reason, and to minimize the chance of missing a low-amplitude peak due to any particular specimen placement, the specimen was removed and replaced on the cradle between every scan. Measurements of fR were very repeatable and routinely agree to within 0.02% for all modes, which is consistent with values in the literature for similar equipment and testing conditions [27].

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High temperature exposure of a Ni-based superalloy in air, even in the absence of an applied load, will evolve the γ–γ’ microstructure and form an adherent oxide scale. To examine the ∆fR as a result of high temperature exposure in air the virgin dog-bone was subjected to an isothermal pretreatment for 13 hours at 1080 °C prior to the first creep iteration. The isothermal pretreatment was conducted at a significantly greater temperature than the creep test iterations to accelerate the kinetics of oxidation and microstructural

190

evolution, effectively delineating an upper bound to the affects of a single isothermal heat treatment of the Mar-M247 superalloy specimen. 3.2. Experimental Results The isothermal pretreatment and iterative creep tests conducted on the dog-bone specimen affected ∆fR as shown in Fig. 5 (a). The first five of the 50 lowest-frequency modes have been omitted from further analysis

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due to measurement variability exceeding 0.02%; such variability is often observed [6], and is likely the result of specimen-cradle coupling effects [33]. The expanded inset Fig. 5 (b) demonstrates how several resonance modes exhibited a slight increase in fR as a result of the 13 hour 1080 °C pretreatment. Additionally, mass measurements of the specimen before and after the isothermal pretreatment indicated no appreciable change that would affect ∆fR . Thus of surface oxidation and microstructural evolution have an insignificant effect

200

on resonance under the investigated conditions—especially with respect to creep damage mechanisms—and are therefore omitted from further investigation. Fig. 5 (a) shows how greater amounts of creep strain result in a more-negative ∆fR for most modes, consistent with a bar-like geometry specimen that increases in length due to plastic straining. Generally low-frequency modes are more affected than modes higher in the frequency regime; modes 12–13 and 16–17 8

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appear to be most affected. Other modes e.g. 23–24 and 37–38 are affected very little by the evolution of creep strain in the dog-bone specimen. One unexpected observation highlighted in Fig. 5 (b) is the measured ∆fR after the C1 creep iteration whereby the specimen experienced primary creep of nearly 0.3% strain before the test was halted. The average response of modes 5–50 was an increase in ∆fR by approximately 0.4%. This global positive shift

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in fR was also observed to occur in sibling Mar-M247 dog-bones after the first creep iteration. While a mechanism responsible for this global positive ∆fR has not been identified, some possibilities are discussed in Section 5.1. Beyond the geometry change induced by creep deformation, surface cracking is an additional mechanism of damage expected to affect resonance. Figs. 6 (a)–(d) are light-optical micrographs that show the

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gauge section of the dog-bone. Fig. 6 (a) shows the blue-gray adherent oxide formed during the isothermal pretreatment (13 hour 1080 °C pretreatment), while Fig. 6 (b) shows the same region after the C6 (8.8% creep strain) iteration. Note the development of surface cracks perpendicular to the loading direction, horizontal for all micrographs, after the C6 creep iteration. After collecting the C6 RUS data the dog-bone was sectioned longitudinally to assess the penetration depth of the surface cracks and the extent of internal

220

damage. Fig. 6 (c) provides a cross-sectional view of almost the entire length of the gauge section. With the exception of a few large circular casting pores, the small internal features in dark contrast are metal-carbides with blocky or script morphology. No metal-carbide interface separation was observed. Finally Fig. 6 (d) provides a magnified view of the near-surface damage whereby many cracks are observed penetrating 200 µm into the gauge section and likely contribute to the accelerated tertiary creep rate observed during the

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C6 iteration.

4. FE Modeling Methods and Results 4.1. Model of the Dog-Bone Specimen The commercial finite element modeling package Abaqus CAE 6.12 [13] was used to model the dogbone and predict the fR of the specimen before and after damage. The virgin dog-bone model, with major 230

dimensions specified in Fig. 3, was discretized into a FE mesh containing approximately 150,000 3D linear wedge and hexahedral elements [13]. Then the Lanczos eigensolver was used to determine the first 50 lowestfrequency resonance modes to establish the baseline response. After the baseline response was characterized, damage was imparted to the model to determine ∆fR as a function of damage.

9

Previous analysis of the cast Mar-M247 bar material confirmed the presence of hundreds of grains 235

throughout the casting, many of which are visible in Fig. 2 (b). As grain-size features were deemed too small to necessitate modeling as discrete grains per the Nygards analysis discussed in Section 2.3, the much smaller γ–γ’ microstructure is also omitted in favor of a homogeneous elastically isotropic representation. Isotropic elastic properties can either be: measured, referenced from a reputable source in the literature, or calculated from single crystal elastic constants (Cijkl ) via a polycrystalline averaging scheme like Voigt-Reuss-Hill [34]

240

or Gairola-Kroner [35]. For this study a rule-of-mixtures (ROM) average was made from referenced Cijkl values of Mar-M247’s constituent γ [36] and γ’ [37] phases assuming a 0.55 volume fraction of γ’. Then these ROM average Cijkl were converted to an isotropic Young’s modulus: E = 216 GPa and Poisson’s ratio: ν = 0.313 via Gairola and Kroner’s 3rd order polycrystalline average [35]. With a density of 8700 kg/m3 per [2] the virgin dog-bone model is fully defined for predicting the baseline fR .

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4.2. Results from Virgin Dog-Bone Model The model predicted mode-shapes are similar to those of a resonating bar or rod, and are described in detail by Migliori and Sarrao [6] and Zadler et al. [38]. The mode-shape of the first 50 resonances can be classified as one of four distinct mode-types: longitudinal bending, extensional, torsional or transverse bending; with Fig. 7 depicting the four fundamental mode-types. Longitudinal and transverse bending modes

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consist of non-axisymmetric flexural motion of the longitudinal or transverse axis of the dog-bone, respectively. While torsional and extensional modes are generally axisymmetric, with torsional modes consisting of pure shear motions while extensional modes exhibit predominately longitudinal breathing motions [6, 38]. All resonance modes are either a fundamental mode, i.e. the first-order mode of a specific mode-type, or a higher-order resonance of fundamental modes. Note that when the FE models contain orthogonal symmetry

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about the long axis of the dog-bone, bending modes will occur in degenerate pairs with identical fR . Such degeneracy is often not observed in measurements due to natural material inhomogeneities and geometry imperfections that disrupt the necessary symmetry for degeneracy. Instead, measured bending modes often occur as a pair of discrete peaks in close proximity. 4.3. Methods of Modeling Creep Damage

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The dog-bone shape change due to creep was modeled with a nonlinear plastic analysis which entails a straightforward FE implementation and yields a similar deformed geometry as creep. Specifically, a power-

10

law hardening elastic-plastic constitutive law was employed, and is represented in one dimension with: σ ασ = + E E



|σ| σy

m−1 ,

(5)

where σy is the yield strength, σ is stress,  the total mechanical strain (elastic and plastic), α yield offset, and m is the hardening exponent [13]. The three dimensional constitutive law additionally requires a Poisson’s 265

ratio and fully defines the material with nonlinear elastic-plastic constitutive behavior [13]. Since the RUS measurements are collected from an unloaded dog-bone specimen with no elastic strains, the yield strength in the constitutive law was set at an arbitrarily low value while E = 216 GPa, α = 0.002, m = 10.0, and ν = 0.313. Once populated with material properties the dog-bone model was loaded in uniaxial tension to produce

270

a strained geometry corresponding to the six strain levels from the experimental work detailed in Table 2. Simulating the conditions of the experimental creep setup, the dog-bone was loaded with a non-deforming ring around the recessed shoulder grips as illustrated in Fig. 8. As the non-deforming ring contacts the dogbone deformation is imparted according to the constitutive material behavior per Eq. (5). Fig. 9 depicts a cross-sectional view of the specimen before and after 8.8% strain. Displacement magnitude maps overlay

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the deformed dog-bones (center and right), as measured with respect to the loading direction (LD) and transverse direction (TD). The LD dog-bone shows uniform elongation within the gauge section, while the TD dog-bone depicts contraction of the gauge cross-section. 4.4. Results from Creep Damage Model The deformed dog-bone model was used as the input geometry for the Abaqus Lanczos eigensolver [13].

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As before, the first 50 resonance modes were calculated and the ∆fR with respect to the virgin dog-bone model determined as a function of creep strain. FE models were run for each of the six creep increments outlined in Table 2. Plots reporting the ∆fR for the C2 and C3 creep iterations, 1.2% and 2.6% strain respectively, are provided in Fig. 10. Mode-type is classified based on the model predicted mode-shapes and denoted by the color of the symbol. Note the different abscissa label between Fig. 10 (a) and (b).

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Frequency order plotting highlights mode-order switching in the frequency regime while mode-type order plots correct for mode-order switching and focus attention on how varying levels of damage affects ∆fR . Both plotting methods are valid because ∆fR —by definition—are calculated with respect to the baseline frequency for each distinct resonance mode. Ultimately mode-type order is preferred for depicting how varying levels of damage affect a given resonance mode and will prove useful for comparing measured and 11

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modeled ∆fR data as addressed in Section 5.1. 4.5. Methods of Modeling Surface Cracks The FE model representing surface cracks is depicted in Fig. 11. The blue elements represent crack surfaces where tie constrains were removed from neighboring elements across the crack face to simulate a half-penny shaped crack. The first FE model contained 20 cracks that were distributed in five planar

295

cross-sections normal to the loading direction, four cracks per plane, and a planar spacing of 2.5 mm. Each of the 20 cracks were identical and covered 4.8% of the gauge cross-section. An orthogonally symmetric crack distribution like the 20 crack model described above ascribes a high degree of symmetry to the crack distribution. Indeed a more-random distribution of cracks would certainly be expected to form experimentally, supported by Fig. 6. Therefore a second model for investigating the

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∆fR associated with an asymmetric distribution of cracks was also created by randomly removing four of the 20 cracks described previously. One crack was removed from 4 different cross-sections as demonstrated in Fig. 11, providing a second 16 crack FE model with an asymmetric crack distribution. 4.6. Results from Surface Crack Models The effect of cracks on the resonance behavior is evaluated as before, by submitting the cracked dog-bone

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model to the eigensolver and comparing the results to the baseline dog-bone model without cracks. Fig. 12 details the 20-crack model results with square symbols and the 16 crack model with circles. A cursory observation indicates that the ∆fR between the two different crack scenarios are quite similar in character, except for two predictable differences. First, the magnitude of the ∆fR for all modes of the 16 crack model are smaller than the corresponding 20 crack model—as fewer cracks result in a smaller drop in fR . Second,

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it is clear that the bending modes are not degenerate in the 16 crack model, with modes 9–10 and 16–17 particularly evident of this loss in degeneracy due to the asymmetrical crack distribution. This degeneratemode splitting is also observed in the measured data of Fig. 5, suggesting that damage accumulation is not completely symmetric about the long axis of the dog-bone.

5. Discussion 315

5.1. Comparison of Modeling and Experimental Results Efforts to compare measured and modeled results are important to validate the modeling methodology and to verify that the modeled damage mechanisms capture the predominant ∆fR of the measured specimen. 12

Modeled results are not validated in absolute terms by comparing lists of modeled and measured fR . Instead, ∆fR for the modeled and measured damage mechanisms are compared through a simple residual metric as: eas. odel Residual = ∆fM − ∆fM . R R

320

(6)

Through a progressive process of modeling the predominant outstanding damage mechanism suspected of affecting resonance, the outstanding residual between modeled and measured ∆fR is accounted for until the majority of modes considered have a near-zero residual—validating the modeling effort. Therefore, a significant non-zero residual exhibited by multiple modes indicate additional mechanisms are operable in the measured specimen, or the model over-predicts the ∆fR response. Finally, it is important to remember

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that RUS measurement error, as reported in Section 3.1, can only be responsible for ±0.02% of the residual, while natural variability in the specimen population is made irrelevant by comparing the dog-bone to itself over time. Plotting the residual for the first 50 resonance modes as function of mode number yields Figs. 13 (a)–(c). The uncorrected residual in subplot (a) depicts the difference between the measured ∆fR of Fig. 5 and the

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modeled ∆fR from only the creep deformation model described in Section 4.3. In Fig. 13 (a), the C1– C5 residuals are generally centered around +0.4–0.5%, with additional scatter of ±0.5% for select modes. Had the shape change associated with creep been the single predominant mechanism affecting resonance, the residuals for C1–C6 would be centered around 0%. This does not occur because the C1 residual, displayed individually in Fig. 13 (b), is the result of an anomalous global +0.4–0.5% ∆fR as first mentioned

335

in Section 3.2. To produce the corrected residual subplot (c), the relatively small global C1 anomaly was removed from the data while baseline mode order disagreement was corrected between the measured and modeled data. In fact, agreement of baseline mode order would only occur if the two baselines had the same frequency order. Correcting for disagreement between measured and modeled baseline orders is straightforward and jus-

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tified by a reduction in the sum of squared residuals (SSR), a metric of the total disagreement between measured and modeled ∆fR . The SSR is calculated as the squared residual for each mode, summed over all modes (6–50), and all creep iterations. The first occurrence of disagreement identified involves modes 27–30 in Fig. 13 (a). All four modes occur within ±1% of each other in fR and the SSR is reduced from 0.780% to 0.686% after switching the resonance number of measured modes: 27–28 with 29–30. According to the

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FE models modes 27–28 are a degenerate transverse bending pair, so it follows that they would behave

13

similarly and appear in close proximity in fR . The only additional occurrence of disagreement is between modes 48 and 49, which also occur within ±1% of each other in fR and reduce the SSR after correcting the disagreement. The anomalous positive ∆fR after C1 would be consistent with either an increase in modulus or a decrease 350

in density, but is not consistent with a change in geometry. While changes in the γ–γ’ phase fraction could affect the modulus, a 0.8% increase in Young’s modulus would be required for a 0.4% increase in fR . However, a minimum 10% change in phase fraction would be required due to the γ [36] and γ’ phases [37] exhibiting such similar elastic moduli. Such a significant change in microstructure was neither expected nor observed, but was considered as a mechanism to explain the anomalous behavior. Instead of a change in modulus, it

355

is likely that a combination of mechanisms including changes in residual stress and/or dislocation density are responsible. Ultimately, the mechanism/mechanisms responsible for the anomalous C1 behavior remain unclear—warranting further investigation. Returning to the corrected residual plot, Fig. 13 (c), it is clear that just the creep deformation model sufficiently predicts the ∆fR for most modes of C2–C5, agreeing particularly well up to resonance number 25.

360

However, the C6 residual indicates that the deformation model alone significantly under-predicts the ∆fR observed in the measured data after 8.8% strain. To explain the negative residual that remains an additional damage mechanism must be considered. Notably, this negative residual coincides with the observation of extensive surface cracking depicted previously in Fig. 6. To demonstrate that the surface cracks are indeed responsible for the significant negative C6 residual that remains, an overlay of the C6 corrected residual

365

from Fig. 13 (c) was plotted with the 16 crack FE model from Fig. 12 to yield Fig. 14. When combined, the similarity of the C6 residual to the 16 crack model ∆fR in both character and magnitude affirms that surface cracks are a dominant mechanism affecting resonance after 8.8% creep strain. 5.2. Expanded Capabilities of Proposed NDE Framework The NDE framework proposed by this work combines RUS measurements and FE models for expanded

370

NDE capabilities beyond historical PSS methods in three fundamental ways. First, unlike other RUS efforts cited in the literature [39, 40, 38, 41], the intent of this study was not to minimize the absolute error between measured and modeled fR . Instead, the dog-bone was compared to itself in order to demonstrate how fR changed over time—rendering natural population variability that confounds PSS methods inconsequential. Such part-to-itself inspection procedures increase measurement sensitivity by eliminating the requirement

375

that damage affects a ∆fR greater than the combined natural population variability and measurement uncertainty. 14

The second point distinguishing this work is that inclusion of FE models make it possible to inform measured resonance spectra with mode-type information and identify specific resonances that are particularly sensitive to individual damage mechanisms. For example, consider modes 14–17 in Fig. 10 (c) and Fig. 12 380

that include: a torsional mode (14), an extensional mode (15), and a longitudinal bending mode pair (16–17). Longitudinal bending modes are affected greatly by the creep elongation and surface crack formation, with degenerate peak splitting consistent with damage disrupting the symmetry of the specimen. Alternatively, modes 14 and 15 are both relatively insensitive to the accumulation of creep strain, while mode 14 is substantially affected by surface cracks. Identification of diagnostic modes that are sensitive to individual or

385

multiple concurrent damage mechanisms can be particularly useful for NDE of large component populations, as selectively measuring diagnostic modes from the broadband spectrum can greatly reduce RUS scan time. The third major distinction of the proposed NDE framework is the potential for quantitative evaluation of damage. Even with optimal experimental controls, quantifying how damage affects resonance based solely on empirical methods is—and will likely remain—difficult. This is because damage like creep manifests as

390

multiple mechanisms that each affect resonance in a complex manner. However using FE models and virtual studies, each damage mechanism can be modeled and evaluated for its individual impact on resonance behavior. These individual mechanisms can then be combined with other concurrent mechanisms to study their combined effect, while always having a precise description of the current damage state. Also experimental validation of virtual studies should: require fewer specimens, fewer increments of damage, and ultimately

395

lead to a generalizable quantitative framework. Consider modes 12–13 from the 2.6% creep strain model in Fig. 10 that exhibit a ∆fR of -1.30%. This is 65 times the ±0.02% measurement precision discussed in Section 3.1. Such measurement precision would translate to a dog-bone creep strain detectability limit of 0.04% elongation if the residual for these modes were zero. The residual metric is effectively an aggregate value of the damage affecting resonance in the measured

400

specimen which has not been addressed by the current slate of FE damage models. Residuals hinder NDE potential by requiring damage mechanisms exhibit ∆fR with a magnitude larger than that of the residual, on a per-mode basis. For example, consider the band of modes 11–24 which exhibit relatively small residuals and multiple modes sensitive damage. Scanning such a band of frequencies would likely be desirable as part of an NDE framework. Considering modes 11–24 of C2–C5, residuals vary between a low of -0.09% (for C5 mode

405

24) to a high of 0.24% (for C3 mode 17), giving a range of 0.33%. With the conservative assumption that the ∆fR of a diagnostic mode must exceed 0.33% before it is reliably detectable, the detectability limit for creep in the current study would equate to a creep strain of 0.66%. According to the FE models, elongation of the 15

10 mm dog-bone gauge section by 0.66%—an increase in gauge length of only 0.066 mm—would produce a -0.33% ∆fR for modes 12–13. Therefore, even by conservative estimates the overall dog-bone specimen need 410

only increase in length from 59.400 to 59.466 mm before creep elongation is reliably detected. As for potential avenues of future work, RUS holds great promise for acquiring difficult-to-measure data that is integral to the evaluation of damage mechanisms in highly engineered materials. For example, recrystallization has been observed to occur during creep of single crystal Ni-based superalloys, and is understood to deleteriously increase tertiary creep rates [42]. While physical models have been developed that predict

415

the accelerated tertiary creep rates as a function of recrystallized volume fraction (VRX ) [42], the inability to nondestructively evaluate VRX limits the practical implementation of such physical models. Interestingly, RUS has recently been shown to be an effective approach for detection and NDE of recrystallization in single crystal Ni-based superalloys [43, 44]. In fact, all indications suggest it should be possible to measure recrystallization rates in-situ with a high-temperature RUS apparatus similar to those described in [45]. Therefore,

420

coupling the NDE capabilities of RUS with physical damage models can: further our understanding of RUS for NDE and our understanding of the evolution of life-limiting damage mechanisms that obstruct the use of highly engineered materials.

6. Conclusions Creep damage in polycrystalline Ni-based superalloy Mar-M247 has been evaluated non-destructively 425

with RUS measurements as part of a broader FE modeling framework of resonance for damage detection. FE models afford the opportunity to isolate the individual creep damage mechanisms affecting resonance, e.g. surface cracks and geometry changes, while also informing RUS measurements with valuable mode-type information. The methodology established herein overcomes many historical limitations of RUS for NDE, and leads to the following conclusions.

430

• The deformed shape of a crept specimen was the dominant mechanism affecting resonance, while surface cracking contributed significantly to changes in resonance after 8.8% strain. • Lower-frequency longitudinal bending modes were the most effective (diagnostic) modes for NDE due to their sensitivity (magnitude ∆fR ) and measurement reliability. • For NDE, FE models only need refinement until fR align to within ±1% of an average virgin measure-

435

ment, as residual analysis will indicate if mode-order disagreement is present when damaged.

16

• Once diagnostic modes are identified, measurements can focus on small segments of the broadband to: reduce scan time and exclude regions with large residuals that may hinder detectability. • RUS for NDE of difficult-to-measure material parameters, coupled with physical damage models, holds great potential for investigating the mechanistic underpinnings and evolution of life-limiting damage 440

mechanisms.

Acknowledgments This work was supported by the U.S. Air Force Research Laboratory (AFRL) through Research Initiatives for Materials State Sensing (RIMSS) Contract FA8650-10-D-5210, Universal Technology Corporation.

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Figure 1: Primary, secondary and tertiary creep regimes depicted for a dog-bone specimen of Mar-M247 loaded to 300 MPa at 950 °C in air, crept until failure.

Figure 2: (a) Aged γ–γ 0 microstructure and (b) investment cast grain structure of Mar-M247.

Figure 3: Dual-purpose creep and ultrasonic fatigue cylindrical dog-bone design, all dimensions are in mm.

Figure 4: Experimental RUS setup utilized in this study to measure fR of dog-bone specimens, detailed in [46]. The drive piezoelectric transducer (PT) excites the specimen to resonate with amplified deflections that the receive PTs (Rec. 1 PT and Rec. 2 PT) measure, yielding a broadband resonance spectrum.

Figure 5: Measured ∆fR for a dog-bone specimen subjected to: an aging pretreatment (PT1), and six subsequent creep iterations (C1–C6); all plotted in (a), with the first three increments detailed in (b) with an expanded ordinate axis.

Figure 6: Light optical micrographs from the dog-bone gauge section after PT1 with 0% strain (a), or after 8.8% creep strain (b)–(d); the loading axis being horizontal. (a) and (b) show the evolution of surface oxide and cracks—marked with white arrows. Micrograph (c) depicts nearly the entire cross-section of the gauge which emphasizes the superficial nature of the cracks. Higher magnification of the surface is provided in (d), where except for a few large and highly circular pores, the remaining small features in dark contrast are identified to be script or blocky metal-carbides.

Figure 7: Exaggerated FE model depictions of the four low-frequency dog-bone mode-types, with the resonance mode number, fR , and mode-type to the right of each mode. Note that all bending modes occur in degenerate pairs.

Figure 8: FE dog-bone model with a non-deforming ring of material above the recessed end of the dog-bone shoulder. The nondeforming ring is lowered onto the shoulder, in the direction the arrows indicate, to load the specimen similar to experiments.

Figure 9: Deformation maps from the dog-bone cross-section showing the magnitude of displacement before (left) and after (center and right) 8.8% strain. Elongation along the loading direction (LD) is mapped on the center dog-bone, while the gauge constriction in the transverse direction (TD) on the right. The deformed dog-bone shape is shown to scale, note the key values are in mm and are an order of magnitude smaller in the TD.

Figure 10: Modeled ∆fR for a dog-bone with a C2 (1.2%) and C3 (2.6%) creep strain geometry. Plots (a) and (b) show the difference between frequency order and mode-type order respectively, while (c) demonstrates the ∆fR trends with increasing creep strain for the first 50 resonances in mode-type order.

Figure 11: FE model of dog-bone specimen surface cracks, each 4.8% of the cross-sectional area of the gauge. Cut-away views on the left detail three or four cracks modeled per crack plane. The five planes of cracks normal to the loading direction are depicted on the right, with 2.5 mm separation between planes of cracks along the 10 mm gauge.

Figure 12: FE model predicted ∆fR due to a symmetrical distribution of 20 cracks (square points connected by a dashed line), as compared to an asymmetrical distribution of 16 cracks (circle points connected by a solid line).

Figure 13: Residual plots showing (a) the initial residual values for each mode with the average residual for C1–C5 centered around 0.5% due to (b) the anomalous positive shift of C1. When this C1 residual is set as the new baseline for C2–C6, and mode-order switching corrected for modes 27–30 and 48–49, a corrected residual plot (c) elucidates the good agreement between measured and modeled results for creep iterations C2–C5, and marked divergence of C6.

Figure 14: FE model of ∆fR from 16 cracks along the dog-bone gauge section overlaid with the C6 corrected residual.

20

Figure 1

Figure 2

Figure 3

shoulder grips 10 mm gauge 4 mm diameter 32.4 mm fillet 59.4 mm specimen

threaded end

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Table 1: Nominal composition (wt%) of Mar-M247

Cr 8.2

Co 9.8

Mo 0.75

W 9.9

Al 5.6

Ti 0.90

Ta 3.1

Hf 1.6

C 0.16

Ni Bal

Table 2: Experimental creep strain increments

Step ID C1 C2 C3 C4 C5 C6

Strain increment (%) 0.27 0.93 1.43 2.09 1.41 2.68

Total strain (%) 0.27 1.20 2.63 4.72 6.13 8.81

21

Highlights

Highlights    

A novel framework for nondestructive evaluation with resonance is established Multiple damage mechanisms, including creep deformation and cracks, are evaluated FE models capture the dominant damage mechanisms affecting resonance Experiments on Ni-based superalloy samples validate FE models and NDE methods