Calibration of potential drop measuring and damage extent prediction by Bayesian filtering and smoothing

Accepted Manuscript Calibration of potential drop measuring and damage extent prediction by Bayesian filtering and smoothing T. Berg, S. von Ende, R. Lammering PII: DOI: Reference:

S0142-1123(17)30149-4 http://dx.doi.org/10.1016/j.ijfatigue.2017.03.033 JIJF 4296

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

12 January 2017 22 March 2017 23 March 2017

Please cite this article as: Berg, T., von Ende, S., Lammering, R., Calibration of potential drop measuring and damage extent prediction by Bayesian filtering and smoothing, International Journal of Fatigue (2017), doi: http:// dx.doi.org/10.1016/j.ijfatigue.2017.03.033

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Calibration of potential drop measuring and damage extent prediction by Bayesian filtering and smoothing T. Berga,∗, S. von Endeb , R. Lammeringa a Institute

of Mechanics, Helmut Schmidt University/ University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany b Rolls-Royce Deutschland Ltd & Co KG, Eschenweg 11, Dahlewitz, 15827 Blankenfelde-Mahlow, Germany

Abstract Fatigue related damage growth without feasibility of optical assessment can be monitored conveniently by means of the direct current potential drop method in laboratory experiments. By estimating the unknown damage extent of a structure indirectly via observed measurements, the need to relate both quantities, i.e. a calibration of damage extent and measurements, arises. In recent years, Bayesian inference has been applied with a special focus to such inverse problem formulations. In the present paper, a novel approach to the calibration issue is proposed by employing Bayesian filtering and smoothing. A probabilistic state space model incorporating prior information about the damage extent and calibration parameters as well as process describing models is defined and subsequently used to infer the damage extent of fatiguetested specimens from potential drop measurements. First, the obtained results in the form of joint conditional posterior distribution functions are exploited to facilitate an evaluation of a direct model calibration on the one hand and direct damage extent estimation on the other hand given persistent uncertainties. In a further step, the inferred damage extent estimations and associated uncertainties are propagated in time as to allow an assessment of decision-making-feasibility within the extended scope of structural health monitoring and damage prognosis. A thorough performance analysis in the light of actual damage extend data is undertaken, revealing accurate results. Keywords: Probabilistic analysis, Fatigue crack growth, Fatigue test methods, Bayesian model calibration, Parameter estimation 1. Introduction

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In certain engineering domains like aviation, materials that are to be employed have to meet various requirements regarding their properties. In order to specify the fatigue capability of a material, crack propagation testing is - among other testing procedures - commonly used. The direct current potential drop (DCPD) method, see Figure 1, is widely accepted as means to monitor fatiguerelated crack initiation and growth, especially where an optical assessment of the defect is not possible. Within DCPD measuring, a direct current injected into a specimen is utilised to quantify the potential drop over a structural damage. Growing damage extents lead to an increased electrical resistance which in turn yields a higher potential drop. Unfortunately, for real-time measuring the DCPD method presupposes a calibration that facilitates the linkage of the measured potential drop to the actual damage size. ∗ Corresponding author. Phone: +49 40 6541 3425, Fax: +49 40 6541 2034 Email address: [email protected] (T. Berg) URL: http://www.hsu-hh.de (T. Berg)

Preprint submitted to International Journal of Fatigue

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There are various ways to obtain a calibration which encompass experimental, analytical and numerical approaches: An experimental calibration can be achieved by obtaining reference potential changes via manually introduced, pre-defined damages and a consistent probe and electrode setup for following specimens [1] or by crack front marking techniques like heat tints [2] and beachmarks (frequency or stress ratio shifts) [3, 4] to allow a calibration after the test. Theoretical calibrations are based on solving the Laplace equation of an electrical potential for certain geometry and boundary conditions. For simple geometries like plane single-edge, double-edge and center cracks, [5] provides an analytical solution. However, it is not applicable to more complex geometries so that numerical methods like FEM have been employed [6, 7]. In model-based structural health monitoring and prognostics, Bayesian filtering has been widely applied in damage diagnosis and damage prognosis with the aim of predicting a system’s remaining useful life, e.g. [8–12]. In a Bayesian framework, dynamic state estimation problems can be solved in a probabilistic fashion that allows to account for different kinds of variabilities and uncertainties [13, 14] which is why Bayesian filtering appears perfectly suited to the task of inferring the unknown extent of a March 21, 2017

Nomenclature a, b, c x, y F , H, P

Scalars Vectors Matrices

p(x)

Probability distribution of a random variable x Conditional probability distribution of a random variable x given y Random variable x is distributed according to p(x) p(x) is proportional to q(x) Set of vectors {x0 , . . . , xk }

p(x|y) x ∼ p(x) p(x) ∝ q(x) x0:k δ θ θi θ true θˉ MAP

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H x (∙) k ΔK Kk s mk , m− k , mk MG , MB , M S n nms npg N Nθ

Dirac function Vector of unknown parameters i-th element of the discretised parameter space set True or actual parameter vector Conditional marginalised parameter expectation

θˆ λ σmin , σmax Δσ

Maximum a posteriori parameter estimate Walker’s law parameter Minimum/maximum applied stress Stress range

a, ak Afp b1 , b 2 c1 , c 2 C f (∙) F x (∙) g(a) Gk h(∙)

Crack size, at time step k Fracture plane Surface crack size Calibration curve parameters Crack growth constant Dynamic transition function Jacobian matrix of f (∙) Crack-geometric correction factor Smoother gain at time step k Measurement function

structural damage from measurements that are linked to it in a specific, but unknown and uncertain way. In the present paper, Bayesian model calibration is proposed as a novel approach to address the issue of relating potential drop measurements to actual damage extents. By means of Bayesian filtering and smoothing, the estimation of the damage size of fatigue-tested corner crack specimens that are monitored utilising the DCPD method is accomplished. In this recursive Bayesian framework, the aforementioned essential calibration, the current damage extent estimation and the prediction of the damage extent propagation are obtained while taking into account physical and modelling-related uncertainties. The advantages of a Bayesian approach to the calibration issue are apparent: Neither are marking techniques and the accompanying specimen destruction or an elaborate preparation in terms of an accurate probe positioning needed nor exist

N (∙) s P k, P − k ,Pk qk , q k rk , r k Rσ Sk U, Uk U (∙) vk wki W xk , xk xtrue k ˉ x ˆk (yk , θ) yk , y k

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Jacobian matrix of h(∙) Time step number Stress intensity factor range Filter gain time step k Updated, predicted, smoothed mean at time step k Correction factor shape functions Paris exponent Total number of measurements Time step of starting power growth behaviour Cycle number Number of discretised parameter space elements Normal distribution Updated, predicted, smoothed covariance at time step k Scalar or vectorial Gaussian process noise at time step k Scalar or vectorial Gaussian measurement noise at time step k Stress ratio Innovation covariance at time step k Potential drop measurement, at time step k Uniform distribution Innovation vector at time step k Weight of the i-th element of the discretised parameter space at time step k Side of the quadratic cross section Scalar or vectorial random variable of the state at time step k True or actual damage extent State estimation for given yk , θˉ Scalar or vectorial random variable of the observation at time step k

limitations regarding the restriction on simple specimen geometries. In contrast to numerical simulation methodologies, actual measurement data is utilised to estimate the damage extent of the specific specimen that is observed. The paper is organised as follows: Section 2 gives an overview of the experimental procedure providing time dependent measurements while Section 3 recalls the fundamentals of recursive Bayesian filtering and smoothing. In Section 4, the general Bayesian approach is applied to the problem at hand, formally defining the specific probabilistic state space model. Section 5 presents and discusses the results obtained from the test series and Section 6 concludes the paper.

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V 2 105 4

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σ(t) Figure 1: Experimental setup: Applied stress, geometries, crack propagation orientation as well as electrode and probe positioning

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b1

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b2

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Figure 2: Experimental setup: Specimen cross section with crack sur- 125 face (quarter circle), crack propagation (dashed lines) and unwanted crack propagation (dotted lines).

2. Experimental procedure 130

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The fatigue test setup consists of notched Udimet 720Li superalloy specimens that are subjected to uniaxial dynamic stress by a hydraulic machine in a convection oven, see Figures 1 and 2. Udimet 720Li is a nickel-base polycrystalline superalloy widely used in aeronautics and the 135 aerospace industry. Its excellent resistance to oxidation and corrosion as well as its high temperature strength and excellent workability rationalize the usage in gas turbine blade and disc applications [15]. The elongated specimens have a quadratic cross section and are notched in the respective plane at one corner halfway between the lower and upper ends. The crack initiation starting from the aforementioned notch is carried out by a sinusoidal initiation loading at f = 10 Hz normal to the cross section as to allow Mode-I crack propagation. The usage of an initial loading aims at overcoming non-continuum mechanicsgoverned crack growth [16] up to a stage of stable crack growth. The specimen is then subjected to a trapezoidal shape loading at f = 0.25 Hz, allowing for an expedient crack growth monitoring. The temperature in both cases is 400 ◦C. The crack propagation is assumed to be quartercircular so that the crack sizes at the surface b1 and b2 as 3

well as the average crack plane radius can be condensed to a single quantity a characterising the damage extent. This is a reasonable simplification as long as crack tunnelling [17] can be ruled out, e.g. from past experience under similar test conditions or via ensuing crack surface investigation. If such effects are to be taken into account, the dynamic and measurement models introduced in Section 3 and specified in Section 4 have to be adjusted accordingly. In order to quantify the damage extent, direct current potential drop measuring is employed. For this purpose, direct current is injected through electrodes located near the lower and upper ends of the specimen with a maximal distance as to harness field uniformity in the inspection area [18]. Potential probes are spot welded closely on either side of the defect, though only roughly at predefined positions. Neither symmetry in the respective distances nor accurate positions can therefore be assumed. However, the probe positioning next to the defect permits precise potential drop measurements at any given time. With specific cycles N the corresponding potential drop U (N ) can be measured. Since the potential drop is caused by the ever growing damage, crack size a and potential drop U (N ) can be related, yielding a cycle-dependent crack size a(N ). The required calibration curve U (a) is unknown and has to be defined. The obtained data consists of six sample trajectories (specimens #1 to #6) obtained as U (N )-curves with 26 to 52 measurement points each subjected to various stress ranges Δσ = σmax −σmin and stress ratios Rσ = σmin /σmax , see Tab. 1. The number of measurement points, i.e. the temporal resolution, can be increased arbitrarily up to observations for each cycle, although in real-time applications, like for example certification testing of turbine blades to obtain material parameters, the processing time has to be kept in mind. In order to be able to benchmark the proposed Bayesian model calibration, reference damage propagation curves have to be provided. Therefore, an additional experimental calibration U (a) is accomplished after every test by cutting the specimen, measuring the respective fracture plane Afp and averaging the crack sizes a0 and aend at the beginning and the end of the crack

Table 1: Experimental set up: Test parameters

Measurements nms #1 #2 #3 #4 #5 #6

40 39 26 35 41 52

Stress range Δσ MPa 816 550 592 592 718 860

Stress ratio Rσ 0 0.1 0.1 0.1 0.1 0.1

Width W m 8 × 10−3 8 × 10−3 7.71 × 10−3 7.71 × 10−3 7.8 × 10−3 7.8 × 10−3

propagation process as a=

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4Afp , π

• Markov property of states. Given the current state, the future state is independent of all state and measurement history:

(1)

where the specific cycle count can be linked to the damage extent, associating a0 and aend with U0 and Uend and interpolating the intermediate values. The assumption of a linear relation holds in good approximation [4, 19]. However, if the crack propagates beyond one or both of the corners as indicated by the dotted line in Figure 2, the aforementioned linking cannot be done. Bearing in mind that the damage extent is unknown, the aim of obtaining as many as possible progressed measurements and an as far as possible propagated crack growth without causing the experimental calibration to fail leads to a variable test duration. The reference data are hereafter denoted as 185 actual or true parameter and damage extent values. 3. Bayesian approach

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The issue at hand can be considered a statistical in- 190 version problem, where hidden states that evolve in discrete time as well as unknown model parameters are to be inferred from observed measurements with due regard to existent uncertainties. It can be resolved by Bayesian filtering and smoothing [13, 14]. Hereinafter, the fundamentals of Bayesian inference are recalled to allow for a transfer of the conceptual idea to DCPD model calibration. Additionally, practical issues are considered. 195 Within a Bayesian approach, the joint posterior distribution of all states and parameters given all measurements is formed by a prior probability plus a likelihood function and derived from a probabilistic state space model (SSM). For each time step k = 1, 2, . . . it can be written in the form θ x0 xk yk

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∼ p(θ), ∼ p(x0 |θ), ∼ p(xk |xk−1 , θ), ∼ p(y k |xk , θ),

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(2)

where θ ∈ Rdθ are the unknown parameters, xk ∈ Rdx is the hidden state and y k ∈ Rdy is the measurement, p(xk |xk−1 ) is the dynamic model stochastically describing the evolution of the state in time and p(y k |xk ) is the measurement model stochastically describing the relation between measurements and states. Depending on the ob- 205 servable physical process that is to be modelled, the state, measurement and parameter vectors can either be utilised to represent a multidimensional system, e.g. a complex structure with multiple damage locations and sensors, or be reduced to scalars, for example in structural health 210 monitoring on component level. Herein, p(∙) is used to denote a probability distribution function. The sequential data formed in this manner are assumed to be Markovian, which exhibit two significant properties. 4

p(xk |x1:k−1 , y 1:k−1 , θ) = p(xk |xk−1 , θ).

(3)

• Conditional independence of measurements. Given the current state, the current measurement is conditionally independent of all preceding states and measurements: p(y k |x1:k , y 1:k−1 , θ) = p(y k |xk , θ).

(4)

The Markovian assumption significantly simplifies the Bayesian inference of the joint posterior distribution by discarding the necessity to condition on the state and measurement history. It must however be kept in mind that the dynamic models utilised to represent the dynamic processes should exhibit the same memorylessness. Higher order Markov processes are not addressed herein, though conceivable [20]. The joint posterior distribution of the states and parameters given the measurements can then be computed via Bayes’ rule as p(x0:T , θ|y 1:T ) =

p(y 1:T |x0:T , θ)p(x0:T |θ)p(θ) p(y 1:T )

(5)

for a given last time step T , where p(y 1:T |x0:T , θ) is the likelihood, p(x0:T |θ) and p(θ) are the prior information and p(y 1:T ) is the evidence. Since it is often more desirable to infer only the state xk at the current time step k given all measurements and parameters, the conditional independence of the measurements is used to form the marginal posterior distribution recursively: p(xk |y 1:k , θ) =

p(y k |xk , θ)p(xk |y 1:k−1 , θ) . p(y k |y 1:k−1 , θ)

(6)

In order to further recursively predict the marginal posterior distribution of a future state xk at the time step k − 1 given the measurement history and parameters, the Chapman-Kolmogorov equation can be employed by utilising the Markov property of the state: Z p(xk |y 1:k−1 , θ) = p(xk |xk−1 , θ)p(xk−1 |y 1:k−1 , θ)dxk−1

(7)

The update step Eq. (6) and prediction step Eq. (7) together constitute the recursive Bayesian filtering equations [13]. In addition to incorporating new observations to update the current state and to predict future ones, to adjust preceding states might be deemed expedient as well. By conditioning previous states on all observations (including future observations) as in Z p(xk+1 |xk )p(xk+1 |y 1:T ) p(xk |y 1:T ) =p(xk |y 1:k ) dxk+1 , p(xk+1 |y 1:k ) (8)

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the marginal posterior distribution of the state xk given all observations up to time step T with T > k can be obtained. Note that in this backward recursion, called Bayesian smoothing, the unknown parameters have been integrated out. Closed form solutions for Eq. (6) to (8) generally exist for a few classes of filtering and smoothing problems only, giving reason to the usage of numerical approxi- 250 mation methods. With regard to the update (Eq. (6)) and prediction steps (Eq. (7)), the extended Kalman filter (EKF) is a widespread approach for non-linear and nonGaussian probabilistic SSM [21]. The dynamic and measurement models are approximated by linearisation about estimations of the current means and covariances. In combination with assumed Gaussian a priori distributions, this leads to Gaussian approximations to the filtering distributions of low computational complexity where a high performance can be achieved. The dynamic and measurement models with additive noise can be written as xk = f (xk−1 , θ) + q k−1 , y k = h(xk , θ) + r k ,

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(9)255 (10)

where f (∙) is the dynamic transition function, h(∙) is the measurement function, q k−1 ∼ N (0, Qk−1 ) is the Gaussian process noise and r k ∼ N (0, Rk ) is the Gaussian260 noise of measurements. The EKF is then given [13] by the Gaussian approximations p(xk−1 |y 1:k−1 , θ) ≈ N (xk−1 |mk−1 (θ), P k−1 (θ)), − p(xk |y 1:k−1 , θ) ≈ N (xk |m− k (θ), P k (θ)), p(xk |y 1:k , θ) ≈ N (xk |mk (θ), P k (θ)),

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(11)265

− where m− k (θ) and P k (θ) are the predicted mean vector and predicted covariance matrix before measurement y k−1 . The mean vector and covariance matrix mk (θ) and P k (θ) contain the information acquired by measurement y k . These quantities are computed outright in the predic- 270 tion step as

m− k (θ) = f (mk−1 , θ), T P− k (θ) = Fx (mk−1 , θ)P k−1 (θ)Fx (mk−1 , θ) + Qk−1 , (12)275

and in the update step as v k (θ) = y k − h(m− k (θ), θ)

− − T S k (θ) = Hx (m− k (θ), θ)P k (θ)Hx (mk (θ), θ) + Rk ,

K k (θ) = mk (θ) =

T P k (θ) = P − k (θ) − K k (θ)S k (θ)K k (θ),

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− T −1 P− , k (θ)Hx (mk (θ), θ)S k (θ) − mk (θ) + K k (θ)v k (θ),

(13)

where Fx (∙) and Hx (∙) are the Jacobian matrices of f and h, v k (θ) is the innovation vector, S k (θ) is the innovation covariance and K k (θ) is the filter gain. Analogous to the EKF, the extended Rauch-Tung-Striebel smoother 5

(ERTSS) can be employed to approximate the Bayesian smoothing solutions in Eq. (8) as p(xk |y 1:T ) ≈ N (xk |msk , P sk ),

(14)

where msk (θ) and P sk (θ) are the smoothed mean vector and smoothed covariance matrix under consideration of future measurements y k:T . The smoothing step is then given [13] by m− k+1 = f (mk ), T P− k+1 = Fx (mk )P k Fx (mk ) + Qk , −1 Gk = P k FxT (mk )(P − , k+1 )

msk = mk + Gk (msk+1 − m− k+1 ),

T P sk = P k + Gk (P sk+1 − P − k+1 )Gk ,

(15)

where Gk (θ) is the smoother Gain and msT = mT as well as P sT = P T hold at the last time step T . With Gaussian approximations in Eq. (11), (14) and the filtering and smoothing steps in Eq. (12), (13), (15), the high performance of the EKF and ERTSS become apparent, even inspite of the necessary Jacobians: The posterior distributions are characterised by only two quantities which in turn can be calculated by plain matrix operations that can easily be parallelised. Furthermore, the dependency on the unknown parameters θ is worth noticing. If the parameters are not fixed, the implementation of filtering and smoothing steps results in nested functions that become more and more complex with every iteration. 4. Problem statement As outlined in Section 1, the model calibration and damage extent estimation of fatigue-tested specimens that are monitored by means of the DCPD is to be accomplished by a Bayesian approach. Section 2 gives insight into the observable physical process at hand while Section 3 introduces the mathematical machinery that is to be utilised. Hereinafter, the definition of the probabilistic state space is addressed, which encompasses the modelling of the dynamic process, the measurement process and the random variables and their available prior information. When utilising fracture mechanics to characterise the nature of fatigue crack growth in metallic materials, a widely used model to describe sub-critical crack growth is the Paris-Erdogan law [22, 23]. In order to account for non-zero stress ratios, it can be improved by Walker’s law [24], which is one among other modifications of the Paris’ law. Walker’s law relates the stress intensity factor range to the crack growth rate as defined in n  ΔK(a) da , (16) =C dN (1 − Rσ )1−λ √ ΔK(a) = Δσg(a) πa, (17)

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where a is the crack length, N is the number of load cycles, C as well as n are material-dependent scaling constants, ΔK is the stress intensity factor range and Rσ is the stress ratio whose influence on the growth behaviour is governed by λ (typically around 0.5 for metals). The335 crack-geometric correction factor g(a) can be derived numerically for various crack shapes and is given for corner crack specimens by [25] as 2 g(a) = MG (a)MB (a)MS (a) , π

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where the precise expressions of the polynomial correction functions MG (a), MB (a) and MS (a) are omitted for the sake of better readability. Alternative formulations to 340 [25] can for example be found in [26, 27]. Consequently, the crack propagation law in Eq. (16) is a first-order ordinary differential equation which - in a discrete-time (cycle) domain - can be written as recurrence relation, yielding da ΔN (19) ak = ak−1 + dN a=ak−1 at time step k for sufficiently small ΔN = Nk − Nk−1 . Recalling the assumed Markov property in Eq. (3),345 the present Bayesian approach limits the conditioning of xk to xk−1 , i.e. of ak to ak−1 when treating the crack sizes as random variables. Modelling constant-amplitude loading or variable-amplitude loading with non-interactive behavior, for example with blocks of constant-amplitude, 350 the aforementioned limitation does not have an impact. However, when incorporating high overloads, e.g. utilising Wheeler’s retardation model, a conditioning on the crack sizes of the overload cycles is necessary, thereby exceeding the limitations made. A possible way to circumvent this 355 shortcoming lies in the usage of point estimates, e.g. MAP or conditional expectation estimates instead of the appropriate random variable. The corresponding uncertainty is thus neglected, yet an implementation in accordance with 360 the Markov property can be accomplished easily. As emphasised in Section 2, a calibration curve and therefore its parameters are crucial within the DCPD method to relate the measured potential drop at given cycles to the evolution of the damage extent itself. For the case of corner crack specimens and constant ampli- 365 tude loadings, where quarter-circular crack front shapes are presumed as mentioned in Section 2, a linear calibration curve of the form U (a) = c1 a + c2

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(18)

(20)

is assumed [4, 19]. It must be kept in mind that different specimen geometries as well as more complex crack front shapes (e.g. crack tunnelling or tortuous crack fronts as investigated in [28]) can result in significantly differing functional relations. However, with knowledge of the expected form of the calibration curve a parametrised rela- 370 tion can be introduced as measurement function analogous 6

to Eq. (20), whereas the number of unknown parameters is changing. Considering the crack size ak as hidden state xk ∈ R, the aforementioned constants C, n, c1 , c2 as unknown parameters θ ∈ R4 and the potential drop U as measurement yk ∈ R as well as additive Gaussian noise, Eq. (19) and Eq. (20) are utilised to form the dynamic and measurement model of the probabilistic SSM, which read xk = f (xk−1 , θ) + qk−1 ,

(21)

yk = h(xk , θ) + rk ,

(22)

where qk−1 ∼ N (0, Qk−1 ) is the process noise, rk ∼ N (0, Rk ) is the noise of measurements and  θ2 ΔK(xk−1 ) f (xk−1 , θ) = xk−1 + θ1 ΔN, (23) (1 − Rσ )1−λ h(xk , θ) = θ3 xk + θ4 .

(24)

The process and measurement noises account for different sources of uncertainty that are inherent to the modelling of the observable physical process, e.g. the measurement errors or the appropriateness of the selected dynamic model. The determination of the variances qk−1 and rk is accomplished on grounds of the available reference data with fixed values for all test series. This approach is valid because of the homogeneity of the respective tests in fatigue testing. If it is not possible to draw on reference data or if the tests exhibit no such homogeneity, adaptive filter tuning techniques have to be employed, e.g. [29]. The information collected by the measurements is available as sequential data and therefore offers the following: On the one hand, only the most recent estimates of state and parameters are of further interest while the state history inferred from current measurements needs not be known. On the other hand, the provision of measurements in regular intervals makes it desirable to have a constant number of computations per time step and not an ever increasing one implied by application of the full Bayes’ rule. Keeping in mind that the objective is to calibrate a potential drop measuring, to estimate the current damage extent and to predict the damage extent propagation, the recursive Bayesian filtering and smoothing equations (Eq. (6) to (8)) can be utilised. They have to be complemented by the marginalised joint posterior distributions of p(xk , θ|y1:k ), yielding Z p(yk |xk , θ)p(xk |y1:k−1 , θ) p(θ|y1:k ) = p(θ|y1:k−1 ) dxk , p(yk |y1:k−1 ) ∝ p(θ|y1:k−1 )p(yk |y1:k−1 , θ), (25) Z (26) p(xk |y1:k ) = p(xk |y1:k , θ)p(θ|y1:k )dθ. The model parameters θ are fixed but unknown and are estimated anew at every time step in the presence of new observations. Recalling the EKF from Eq. (11) - (13),

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p(θ|y1:k ) ≈

Nθ X i=1

wki δ(θ − θ i ),

da/dN (m/cycles)

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the dependency of the state xk on the parameters θ therefore has to be propagated through every iteration resulting in nested functions that become more complex with every time step. To overcome this issue a combination of EKF and approximate grid-based filtering is employed. In approximate grid-based filtering, the continuous state space is decomposed into predefined cells and the posterior density is approximated by a weighted sum of δ functions, providing a good solution and optimal recursion for the filtering densities [30, 31]. Discretising the parameter space into a finite number of cells {θ i : i = 1, . . . , Nθ }, the posterior density p(θ|y1:k ) is then given [32] as

da dN

(27)

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i p(yk |y1:k−1 , θ i ) wk−1 , Nθ P j wk−1 p(yk |y1:k−1 , θ j )

(28)

x0 ∼ N (μ0 , σ02 ), xk ∼ N (f (xk−1 , θ), Qk−1 ), yk ∼ N (h(xk , θ), Rk ).

(30)

p(xk |y1:k ) ≈

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Nθ X i=1

p(xk |y1:k , θ i )wki .

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θ4 ∼ U (θ4,low , θ4,up ),

p(yk |xk , θ i )p(xk |y1:k−1 , θ i ) . (29) p(xk |y1:k , θ ) = p(yk |y1:k−1 , θ i ) Z p(xk |y1:k−1 , θ i ) = p(xk |xk−1 , θ i )p(xk−1 |y1:k−1 , θ i )dxk−1 i

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ΔK(a) (1−Rσ )1−λ

θ1 = C0 , θ2 = n0 , θ3 ∼ U (θ3,low , θ3,up ),

where the initial weights are denoted as w0i := N1θ . At every time step k the Bayesian filtering equations (Eq. (6) and (7)) are evaluated for each θ i

and the marginalised conditional posterior distribution of the state p(xk |y1:k ) is obtained by summing over the parameters:

30 40 √ ΔK (MPa m)



be normally distributed as well. The general probabilistic SSM from Eq. (2) can then be recasted accordingly as

j=1 385

= C0

Figure 3: Fatigue crack growth rate obtained by experimental calibration for all six specimens and regression curve (ordinates omitted by virtue of proprietary information, logarithmic scales).

where wki represents the conditional probability of θ i given the measurements y1:k that can be computed by wki ≈

#1 #2 #3 #4 #5 #6

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(31)415

For every time step k, an adjustment of the preceding states in the light of new measurements can be achieved by Bayesian smoothing (Eq. (8)) by utilising the ERTSS as given in Eq. (15), yielding more accurate state estimations 420 x0:k−1 . In this paper, the material-dependent scaling constants are assumed to be known and the corresponding uncertainties are captured by the dynamic and measurement model noise without specifically quantifying the extent. The pa- 425 rameters characterising the calibration curve are treated as fixed random variables without prior knowledge, i.e. they are uniformly distributed. The initial hidden state x0 is assumed to be normally distributed with mean μ0 and variance σ02 . The hidden state xk and measurement yk at time step k are - due to the respective additive Gaussian noise in the dynamic and measurement models - assumed to 430 7

(32)

The scaling constants C0 and n0 are chosen as typical values for Udimet 720Li obtained from other test series performed by Rolls Royce through linear regression analysis by fitting the sample trajectories to the logarithmised Walker’s law in Eq. (16). The boundaries for the uniformly distributed parameters are chosen in ample intervals around the values valid for the six data sets by taking into account the form of the measurement function in Eq. (24): Both slope and intercept have to be positive while the latter is bounded by the respective initial measurement y0 . The slope is set to a multiple of the maximum of the actual values θ3true of the six test series (which differ in one order of magnitude). Hence, in practise the setting of the upper boundary θ3,up should be backed by experience on possible extents of the calibration curve. Alternatively, the grid in the parameter space can be extended arbitrarily, however when maintaining a constant resolution only at the expense of increased computational complexity. 5. Results and discussion The fatigue crack growth rate is plotted over the stress intensity factor range together with the corresponding Walker’s law for the whole test series in Figure 3. At low

2

1.8 ˆMAP θ 3 ˉ θ 3

1.8

1.6

1.6

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1.2

θ4 /θ4true

θ3 /θ3true

1.4

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465

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Figure 4: Normalised measurement model parameter θ3 MAP and marginalised expectation over normalised cycles N for all specimens (dotted lines correspond to specimen #5).

Figure 5: Normalised measurement model parameter θ4 MAP and marginalised expectation over normalised cycles N for all specimens (dotted lines correspond to specimen #5).

growth rates, a test-initiation phase and the transition to a subsequent power growth regime can clearly be identified for all specimens. The power growth behaviour is in general well caught by the material-dependent Walker’s law, 470 although specimen #5 displays a growth rate stagnation for the latter fifth of measurements. However, no specimen exhibits behaviour of instable crack growth at the end of the crack propagation that could not be covered by the selected dynamic model in Eq. (21). 475 In Figures 4 and 5, the convergence of the estimated measurement model parameters θ3 and θ4 is depicted as a result of Eq. (27) by means of the respective maximum MAP a posteriori (MAP) estimates θˆ and the marginalised ˉ Given480 conditional expectations given the measurements θ. a posterior distribution, MAP and conditional expectation estimates are typical Bayes estimators [13, 33–35] that best characterise it following differently defined loss functions. The former corresponds to the mode of the posterior distribution, i.e. the most probable point, while the latter 485 minimises the mean squared error. The components of θˉ are obtained by marginalising the outcome of Eq. (27) and averaging over the respective parameter. Initially assigned a uniform distribution at N0 = 0, both parameter estimates vary noticeably at the beginning and ap- 490 proach with distinct precision the respective true underlying parameters θ true throughout the test. In doing so, MAP estimates and expectations display a differing behaviour: The former (blue dashed lines) remain (almost) constant for up to 80 % of the overall-cycles and there- 495 after approach the actual parameter value with a constant or steepening slope. The latter (red lines) diverge from their initial values in direction of the respective MAP estimates for up to 70 % of the cycles and approach the true parameter value subsequently. However, both MAP esti- 500 mates and expectations converge to one another as is indicated by one pair of estimates (dotted lines). The reason

why no direct convergence behaviour of MAP and expectation parameter estimates is observed can be explained by the underlying dynamic model: First of all, the choice of the material-dependent constants C0 and n0 naturally does not cover the response of all specimens but is a compromise to fit experimentally obtained crack growth rates into one model, where C0 and n0 are given as typical values for Udimet 720Li. As a consequence, the definition of the material-dependent dynamic model parameters as random variables and their estimation in the Bayesian filtering steps can be considered. It must however be kept in mind, that the higher dimensionality of the parameter space noticeably increases the computational complexity. Secondly, the test-initiation phase can not be represented by Walker’s law which is why - at first - deviant values are inferred for the unknown parameters. Thirdly, even though no instable crack growth is measured, the aforementioned stagnation of the crack growth rate apparent from specimen #5 can be recognised in specimens #4 and #6 as well. This behaviour can only be covered by a different set of parameters C0 and n0 , thus leading to different estimations. In both Figures 4 and 5 it can be seen that the expectation parameter estimates are partially clearly more accurate than the MAP estimates throughout the test. Both estimates however tend to converge to one another as mentioned before, providing similar results at the end of the test as emphasised in Table 2. Figures 6 to 8 provide a comparison of the true crack propagation over time (cycles) with a set of different estimates for specimens #1, #2 and #5. These test results are explicitly discussed since the respective power growth behaviours exhibit the best and the worst compliance with the applied Walker’s law as depicted in Figure 3. The true crack size xtrue is plotted over k cycles for every measurement yk taken and is complemented by the respective marginal conditional expectation

8

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xtrue k E(xk |y1:k ) E(x[k,l] |y1:k ) 90 % prediction band ˆMAP ) E(xk |y1:k , θ ˉ x ˆk (yk , θ)

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Figure 8: Comparison of actual damage extent with normalised conditional expectations and state estimations including corresponding 90 % smoothing and prediction bands plotted over cycles for #5.

E(xk |y1:nms ) 90 % smoothing band

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Figure 6: Comparison of actual damage extent with normalised conditional expectations and state estimations including corresponding 90 % smoothing and prediction bands plotted over cycles for #1.

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Nk (cycles) Figure 7: Comparison of actual damage extent with normalised conditional expectations and state estimations including corresponding 535 90 % smoothing and prediction bands plotted over cycles for #2.

E(xk |y1:k ) together with a 90 % highest posterior density (HPD) credible interval, the respective conditional expecMAP 540 ) and the respective state estimatation E(xk |y1:k , θˆ ˉ derived from the filtering equations. In tion x ˆk (yk , θ) addition, the respective marginal conditional expectation E(xk |y1:nms ) together with a 90 % HDP is given as a result of the Bayesian smoothing. The expectation E(xk |y1:k ) is obtained by averaging the outcome of Eq. (31) over xk , the545 MAP expectation E(xk |y1:k , θˆ ) by conditioning Eq. (29) on MAP

515

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xtrue k E(xk |y1:k ) E(x[k,l] |y1:k ) 90 % prediction band ˆMAP ) E(xk |y1:k , θ ˉ x ˆk (yk , θ)

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xk /xtrue nms

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xk /xtrue nms

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xtrue k E(xk |y1:k ) E(x[k,l] |y1:k ) 90 % prediction band ˆMAP ) E(xk |y1:k , θ ˉ x ˆk (yk , θ)

θ i = θˆ and averaging over xk as well and the state estiˉ by rearranging the measurement model in mation x ˆk (yk , θ) Eq. (22) with respect to xk , disregarding the process noise 550 (0 mean) and utilising the current measurement yk and the ˉ At each cycle-step Nk , the vector of mean parameters θ. damage extent E(x[k,l] |y1:k ) is predicted at a generic future cycle Nkl = Nk + ΔNpred l with l ∈ {1, 2, . . . , ˉl}. Addition9

ally within the scope of prediction, each credible interval is continued as a 90 % prediction band for the respective E(xk |y1:k ). Since the uncertainty incorporated in each estimation propagates in time, the prediction bands widen for increasing Nkl . In combination with the potential crack growth defined by the dynamic model, the widening intensifies considerably with greater damage extent estimates which can only be circumvented by more frequently taken measurements, i.e. smaller cycle-steps ΔN . A prediction for bigger Nkl is therefore possible, in light of the persistent uncertainties albeit not reasonable. The expectation E(xk |y1:nms ) is computed by straightforward application of the ERTSS in Eq. (15), where the particular terms are provided by the aforementioned different estimates of the filtering process. The corresponding 90 % credible interval is continued as smoothing band. For the sake of completeness, it is referred to Tab. 1 once more to point out the varying stress ranges employed for every specimen. Along with unequal initial crack sizes, this leads to diverging total cycle numbers. MAP In Figures 6 and 7, the expectations E(xk |y1:k , θˆ ) inferred from the measurements for specimens #1 and #2 fail to catch the behaviour of the true crack propagation. MAP The aforementioned parameter convergence of both θˆ only manifests in the damage extent estimations in the latter tenth of cycles, leading to occasional deviations of almost 100 %. Similar results can be observed for specimen #3 to #6. In contrast to that, the damage extent estimations ˉ match the true crack growth in E(xk |y1:k ) and x ˆk (yk , θ) very good approximation whereas the 90 % credible intervals stay within reasonable bounds. By considering future measurements for state estimations in the Bayesian smoothing process, i.e. by computing E(xk |y1:nms ), exceedingly accurate results are achieved. The state xk can be identified at any given Nk , thus allowing for an effective

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xk /xtrue nms

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Figure 9: Comparison of actual damage extent with normalised conditional expectations and state estimations including corresponding 90 % smoothing and prediction bands plotted over cycles for #1 obtained at each measurement after crack initiation phase.

Figure 10: Comparison of actual damage extent with normalised conditional expectations and state estimations including corresponding 90 % smoothing and prediction bands plotted over cycles for #2 obtained at each measurement after crack initiation phase.

indirect model calibration. 590 Basically, the preceding assessment can be assigned to the crack size estimations for specimen #5 in Figure 8. The only drawback is the diverging of true crack propagation and corresponding estimates in the last interval which results from the already mentioned crack growth stagna- 595 tion, confer Figure 3. However, for almost all measurements of the crack growth stagnation phase the true crack size still lies within the associated 90 % credible intervals. Because of the test-initiation phase that cannot be covered by Walker’s law, the noise parameters qk−1 and rk in 600 the dynamic and measurement model (Eq. (21) and (22)) have to be chosen in a manner as to still allow said specific behaviour in the defined probabilistic SMM. Due to partially considerable divergences, qk−1 and rk do not only represent the systems inherent variabilities as mentioned 605 in Section 4 but stochasticity is also utilised to account for those model uncertainties, i.e. the variances Qk−1 and Rk have to accept higher values [13]. This approach yields higher uncertainties in the state and parameter estimations and of course wider credible intervals. A possible way to overcome this issue within the proposed probabilistic SSM is to omit those measurements that correspond to the crack initiation phase and do not 610 represent crack power growth behaviour, thereby implying a model selection approach as in [9]. Hence, the values for Qk−1 and Rk decrease, subsequently reducing the systems uncertainties. It is worth noticing that this can self-evidently only be done in light of reference 615 data, i.e. the true crack propagation, but it emphasises the strong dependency on the utilised respective dynamic model. Figures 9 and 10 show the same comparison of true crack propagation with different estimations provided before with the distinction of the respective underlying mea- 620 surements for specimens #1 and #2. The expectations

MAP therefore change to E(xk |ynpg :k ), E(xk |ynpg :k , θˆ ) and E(xk |ynpg :nms ) as well as the state estimation x ˆ k (yk , θˉnpg ), where θˉnpg is computed as mean of p(θ|ynpg :k ) and npg is the time step of starting power growth behaviour in Walker’s law, cf. Figure 3. As in Figures 6 and 7, the parameter-bound MAP estimations again fall short in comparison with the indirect state estimation x ˆk (yk , θˉnpg ) and the direct state estimation E(xk |ynpg :k ). However, the accuracy of MAP E(xk |ynpg :k , θˆ ) is significantly improved. The filtering damage extent estimations E(xk |ynpg :k ) and x ˆk (yk , θˉnpg ) match the true crack growth in very good approximation and the smoothing outcome provides almost exact accordance, especially for specimen #1. Moreover, the 90 % credible intervals reduce markedly when omitting the crack initiation data with the consequence of being able to predict the remaining life to failure with a predefined failure threshold more precisely. A comparison of all results for time step nms is given in Table 2.

10

6. Conclusion A novel approach to the calibration issue in direct current potential drop measuring has been presented by Bayesian filtering and smoothing. Data obtained from potential drop measurements are utilised to infer the extent of fatigue-driven damage propagation in corner crack specimens by Bayesian filtering and smoothing without the need of conventional experimental, analytical or numerical calibration. Therefore, a state space model that defines the unknown states, unknown parameters and acquired observations as well as physics-based formulations linking these quantities in the form of dynamic and measurement models is introduced in probabilistic terms.

Table 2: Estimations of state xk and parameters θ3 , θ4 for k = nms (normalised with xtrue , θ3true and θ4true , respectively) k

Specimen #1 #2 #3 #4 #5 #6

625

630

635

640

645

650

655

1.025 0.870 1.055 0.948 0.688 0.998

  MAP E xk |y1:k , θˆ 1.123 0.958 1.169 0.990 0.743 1.030

ˉ x ˆk (yk , θ)

E(xk |ynpg :k )

θˆ3MAP

θˉ3

θˆ4MAP

θˉ4

0.993 0.842 0.992 0.910 0.634 0.976

1.006 0.884 1.014 0.973 0.697 0.984

0.839 1.034 0.886 0.997 1.359 0.977

0.980 1.211 1.059 1.075 1.555 1.021

1.211 1.091 0.995 1.052 1.180 1.050

1.094 0.890 0.956 1.027 1.057 1.017

First, a direct model calibration, i.e. estimation of the 665 unknown parameters, is performed by means of Bayesian filtering. The convergence behaviour of these parameters with the distinction of maximum a posteriori estimates and expectations compared to the actual parameter val- 670 ues is presented as a function of time (cycles), exhibiting a rather slow but in the end accurate approach of the true underlying values. Ensuing, the model calibration is accomplished indirectly through direct damage ex- 675 tent estimation via Bayesian filtering and smoothing by integrating out the unknown parameters. Thus, a comparison of the marginalised state expectations with the parameter-bound state estimations is given, yielding good 680 overall approximations to the actual damage propagation throughout the cycles. With the former providing very accurate state estimations for corner crack specimens under constant-amplitude loading, the proposed Bayesian ap- 685 proach is deemed worthwhile for further investigation. Furthermore, the strong influence of Walker’s law defined as the only dynamic model as well as the accordingly chosen dynamic model parameters on the quality of the 690 estimation is emphasised. As a consequence thereof, the probabilistic SSM’s inherent uncertainties are relatively high, hence allowing only short damage prediction intervals. However, it is shown that with an adapted model se- 695 lection approach this issue can be circumvented. Not only do the obtained state estimates almost perfectly match the actual damage propagation, but also the estimations’ respective credible intervals can be reduced markedly, 700 thereby allowing longer prediction intervals. Additionally, in combination with dynamic model parameter estimation and the incorporation of prior information of the unknown parameters within the probabilistic SSM, con- 705 siderably more accurate results should be achievable. References

660

E(xk |y1:k )

710

[1] R. O. Ritchie, K. J. Bathe, On the calibration of the electrical potential technique for monitoring crack growth using finite element methods, International Journal of Fracture 15 (1) (1979) 47–55. doi:10.1007/BF00115908. 715 [2] K. George, H. Reemsnyder, J. Donald, R. Bucci, Development of a DCPD Calibration for Evaluation of Crack Growth in Corner-Notched, Open-Hole Specimens, Journal of ASTM International 1 (9) (2004) 19044. doi:10.1520/JAI19044. 720

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[3] G. Belloni, E. Gariboldi, A. Lo Conte, M. Tono, P. Speranzoso, On the Experimental Calibration of a Potential Drop System for Crack Length Measurements in a Compact Tension Specimen, Journal of Testing and Evaluation 30 (6) (2010) 1–9. doi:10. 1520/JTE12346J. [4] C. Schweizer, M. Schlesinger, H. Oesterlin, V. Friedmann, P. Bednarz, C. Meilgen, J. Szwedowicz, Methodology for fatigue crack growth testing under large scale yielding conditions on corner-crack specimens, Engineering Fracture Mechanics 126 (2014) 126–140. doi:10.1016/j.engfracmech.2014.04.032. [5] H. Johnsen, Calibrating the Electric Potential Method for Studying Slow Crack Growth, Materials Research & Standards 5 (1965) 442–445. [6] R. A. Cl´ audio, C. M. Branco, J. Byrne, Numerical calibration of electric potential difference for crack size monitoring, in: 10th Portuguese Conference on Fracture, 2006. [7] L. Doremus, Y. Nadot, G. Henaff, C. Mary, S. Pierret, Calibration of the potential drop method for monitoring small crack growth from surface anomalies Crack front marking technique and finite element simulations, International Journal of Fatigue 70 (2015) 178–185. doi:10.1016/j.ijfatigue.2014.09.003. [8] G. Deodatis, H. Asada, S. Ito, Reliability of aircraft structures under non-periodic inspection: A Bayesian approach, Engineering Fracture Mechanics 53 (5) (1996) 789–805. doi: 10.1016/0013-7944(95)00137-9. [9] X. Guan, R. Jha, Y. Liu, Model selection, updating, and averaging for probabilistic fatigue damage prognosis, Structural Safety 33 (3) (2011) 242–249. doi:10.1016/j.strusafe.2011.03.006. [10] J. Chiachio, M. Chiachio, A. Saxena, G. Rus, K. Goebel, An Energy-Based Prognostics Framework to Predict Fatigue Damage Evolution in Composites, in: Annuel Conference of the Prognostics and Health Management Society 2013, 2013. [11] M. Corbetta, C. Sbarufatti, A. Manes, M. Giglio, Sequential Monte Carlo sampling for crack growth prediction providing for several uncertainties, in: Second European Conference of the Prognostics and Health Management Society 2014, 2014. [12] M. Gobbato, J. B. Kosmatka, J. P. Conte, Developing an integrated structural health monitoring and damage prognosis (SHM-DP) framework for predicting the fatigue life of adhesively-bonded composite joints, in: Fatigue and Fracture of Adhesively-Bonded Composite Joints, Elsevier Ltd, 2014, pp. 493–526. doi:10.1016/B978-0-85709-806-1.00017-3. [13] S. Sarkka, Bayesian Filtering and Smoothing, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139344203. [14] J. B. Nagel, B. Sudret, Bayesian Multilevel Model Calibration for Inverse Problems Under Uncertainty with Perfect Data, Journal of Aerospace Information Systems 12 (1) (2015) 97–113. doi:10.2514/1.i010264. [15] M. Marchionni, High temperature low cycle fatigue behaviour of UDIMET 720 Li superalloy, International Journal of Fatigue 24 (12) (2002) 1261–1267. doi:10.1016/S0142-1123(02) 00043-9. [16] N. Perez, Fracture Mechanics, Vol. 1, Kluwer Academic Publishers, Boston, 2004. doi:10.1007/b118073. [17] L. Zhao, J. Tong, J. Byrne, Stress intensity factor K and the

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[22] 740

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755

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elastic T-stress for corner cracks, International Journal of Fracture 109 (2) (2001) 209–225. doi:10.1023/A:1011016720630. G. Sposito, Advances in potential drop techniques for nondestructive testing, Ph.D. thesis, Imperial College London (2009). M. Schlesinger, C. Schweizer, Y. Brontfeyn, H. Oesterlin, Experimental investigation of the thermomechanical fatigue crack growth and theoretical description by the effective cyclic crack tip opening displacement, in: DGM Conference on material testing, 2013. C. C. Strelioff, J. P. Crutchfield, A. W. Hubler, Inferring Markov Chains: Bayesian Estimation, Model Comparison, Entropy Rate, and Out-of-class Modeling (2007) 1–15arXiv:0703715, doi:10.1103/PhysRevE.76.011106. N. Gordon, D. Salmond, A. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings F Radar and Signal Processing 140 (2) (1993) 107. doi:10.1049/ip-f-2.1993.0015. P. Paris, F. Erdogan, A Critical Analysis of Crack Propagation Laws, Journal of Basic Engineering 85 (4) (1963) 528–533. doi: 10.1115/1.3656900. R. Ritchie, Mechanisms of Fatigue-Crack Propagation in Ductile and Brittle Solids, International Journal of Fracture 100 (1999) 55–83. arXiv:0005074v1, doi:10.1023/A: 1018655917051. K. Walker, The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum, in: Effects of Environment and Complex Load History on Fatigue Life, ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, 1970, pp. 1–1–14. doi: 10.1520/STP32032S. A. C. Pickard, The application of 3-dimensional finite element methods to fracture mechanics and fatigue life prediction, Engineering Materials Advisory Services Limited, 1986. J. C. Newman, I. S. Raju, NASA Technical Memorandum 85793 (April). M. Janssen, J. Zuidema, R. Wanhill, Fracture Mechanics, Second Edition, VSSD, 2006. E. Fessler, E. Andrieu, V. Bonnand, V. Chiaruttini, S. Pierret, Relation between crack growth behaviour and crack front morphology under hold-time conditions in DA Inconel 718, International Journal of Fatigue 96 (2017) 17–27. doi:10.1016/ j.ijfatigue.2016.11.015. M. Shyam Mohan, N. Naik, R. M. O. Gemson, M. R. Ananthasayanam, Introduction to the Kalman Filter and Tuning its Statistics for Near Optimal Estimates and Cramer Rao Bound (February). arXiv:1503.04313. M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing 50 (2) (2002) 174–188. doi:10.1109/78.978374. Z. H. E. Chen, Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond, Statistics 182 (1) (2003) 1–69. V. Klumpp, U. D. Hanebeck, Nonlinear Fusion of MultiDimensional Densities in Joint State Space, in: Proceedings of the 12th International Conference on Information Fusion, 2009. L. Fahrmeir, R. K¨ unstler, I. Pigeot, G. Tutz, Statistik, Vol. 1 of Springer-Lehrbuch, Springer Berlin Heidelberg, Berlin, Heidelberg, 2003. arXiv:arXiv:1011.1669v3, doi:10.1007/ 978-3-662-22657-5. E. Janes, Probability Theory: The Logic of Science, Cambridge University Press, 2003. W. A. Link, R. J. Barker, Bayesian Inference: with ecological applications, Academic Press, 2010.

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Highlights

   

A novel approach to direct current potential drop measuring calibration is proposed. Fatigue-tested corner crack specimens under constant-amplitude loads are monitored. Application of Bayesian filtering and smoothing to infer the unknown quantities. Reasonable estimation of calibration parameters and damage extents.