The state-space model based on the crack propagation law consists of a state equation and observation equation [28], as shown in Eq. (1).(1) {xk=f(xk−1,wk−1)yk=g(xk,vk) where x_k−1 is the crack length at time k, f( ⋅ ) is defined by a crack growth model, through which the crack growth law from time k − 1 to k is started, w_k−1 is the uncertainty of crack growth. y_k is the observation value, which includes the fatigue crack information at the time k. g( ⋅ ) is a model, which is the relationship between the observation value and the crack length. v_k refers to uncertainties during testing.

Paris’ law [29] is often used to describe the growth of fatigue crack, for which crack propagation is shown in Eq. (2).(2) dxdN=C(ΔK)m (3) ΔK=Δσπx

where N is the number of loading cycles, C and m are material parameters and ΔK is the stress intensity factor. The method [25] is simple to calculate. Eq. (2) is rewritten as(4) ln⁡(dxdN)=ln⁡(C)+mln⁡(ΔK)

The specimens are labeled as {L_j, j = 1, …, S}, the M group of crack measurement is collected {xij,i=1,…,M} and the corresponding load cycles are labelled as {Nij,i=1,…,M} , where x0j is the initial crack length. The crack growth rate can then be expressed as(5) (dxdN)ij≈ΔxijΔNij=xij−xi−1jNij−Ni−1j

Given the crack growth rate {(dx/dN)ij,i=1,…,M} and stress change factor ΔK, ln(C_j) and m_j, which corresponds to the specimen L_j can be obtained through linear fitting.

The uncertainty estimate of crack propagation can be written as(6) ln⁡(ΔxΔN)=ln⁡[C(ΔK)m]+w (7) eij=ln⁡(ΔxijΔNij)−ln⁡{C[ΔK(xij)]m}

where difference eij is assumed to be the uncertainty of crack growth, which affects the crack growth rate. Then, the difference between the samples can be used to calculate the variance σw2 of the random variable w.(8) σw2≈Var{eij,i=1,…,M,j=1,…,S}

In order to consider uncertainty during the growth of fatigue crack as well as for simplification [30], Eq. (2) can be discretized as the state Eq. (9).(9) xk=xk−1+C(ΔK)mΔN+w where ΔN is the discrete load step and w is the uncertainty of the growth crack, which follows N(0,σw2) .

Among the acquired monitored signals, the increase in the crack can cause signal distortion, energy attenuation, and phase delay [31]. Accordingly, such characteristic parameters can be used to describe the current state of fatigue cracks. In order to quantify these features, the damage index (DI) was used, which evaluates the difference between the undamaged signal and the signal with fatigue load information. The results demonstrated that the S0 mode is more sensitive to early fatigue cracks compared to that of A0 mode [32]. Therefore, the first wave packet (S0 mode) was used to calculate DI, which is given to Eq. (4).(10) DI=1−CXYσXσY

where C_XY is the covariance of the two signals, σ_X and σ_Y are the mean square deviations of the two signals, X is the baseline signal and Y is the monitored signal during crack propagation.

The corresponding observation is descriptive of the fatigue crack monitoring result, which can be obtained to fit all of the specimen’s DIs under different crack lengths, as shown in Eq. (11).(11) yk=b0+b1xk+b2xk2+b3xk3+vk where {b₀, b₁, b₂, b₃} are the coefficients of polynomial fitting.

The uncertainty of measurement is assumed to be normally distributed and is expressed as v∼N(0,σv2) . The damage factor variance of crack length x_i can be expressed as(12) (σv2)i≈Var{yij,j=1,…,S}

The standard PF was provided by the findings put forward by Gordon et al. [33], which was applied to the nonlinear and non-Gaussian problem. The PF was designed to calculate the posterior pdf p(x_k | y_1:k) with the specimen’s observations [34]. The Bayes theorem is used to calculate the posterior pdf p(x_k | y_1:k), as expressed in Eq. (13).(13) { p(xk−1|y1:k−1)=∫p(xk|xk−1)p(xk−1|y1:k−1)dxk−1p(xk|y1:k)=p(yk|xk)p(xk|y1:k−1)p(yk|y1:k−1)

However, the pdf p(x_k | y_1:k) is actually difficult to calculate. The basic idea of the PF is to approximate the posterior pdf p(x_k | y_1:k) through a set of particles {xki,i=1,…,N} with their normalized weights {w~ki,i=1,…,N} (14) p(xk|y1:k)≈∑i=1Nw~kiδ(xk−xki) where δ( ⋅ ) is the Dirac delta function, N is the number of particles, {xki,i=1,…,N} are sampled from the pdf p(x_k | y_1:k).

When p(x_k | y_1:k) is unknown, the importance density function q(x_k | y_1:k) can then be introduced. Therefore, a posterior estimate can be expressed using Eq. (15), in which the particle weight is defined as in Eq. (16).(15) x^k=∫xkp(xk|y1:k)q(xk|y1:k)q(xk|y1:k)dxk (16) wki∝p(xk|y1:k)q(xk|y1:k)

The importance density function can be derived as(17) q(xk|y1:k)=q(xk−1|yk)q(xk|xk−1,y1:k)

The unnormalized weight can be calculated through the iteration of Eq. (18).(18) wki=wk−1ip(yk|xki)p(xki|xk−1,y1:k)q(xki|y1:k)

The weight normalization is shown in Eq. (19).(19) w~ki=wki∑i=1Nswki where p(yk|xki) is the likelihood value corresponding to the particle. Due to the simplicity of the conversion probability density function (PDF), the particle filter algorithm adopts the conversion probability density function as the importance density. Therefore, the weight update can be simplified into Eq. (20).(20) wki=wk−1ip(yk|xki)

The prognostic result [29] of the crack length x_k is calculated as in Eq. (21).(21) x^k=∑i=1Nxkiw~ki

The following describes three resampling methods:(1)

Residual resampling [27]

ni′=[Nwi] copies of the particle x_i are allocated to the new distribution. Additionally, m=N−∑ni′ particles are resampled from {x_i} by making ni′′ copies of particle x_i where the probability for selecting x_i is proportional to wi′=Nwi−ni′ using one of the aforementioned resampling schemes.(2)

Systematic resampling [35]

N ordered numbers are then generated, as shown in Eq. (22).(22) uk=(k−1)+u~N where u~∼U[0,1) , k = 1, 2, 3…N, which are then used to select xk∗ according to the multinomial distribution.

Here,(23) xk∗=x(F−1(uk)) where uk∈[∑s=1i−1ws,∑s=1iws) , F⁻¹ denotes the generalized inverse of the cumulative probability distribution of the normalized particle weights.(3)

Multinomial resampling [27]

N ordered numbers are generated.(24) uk=uk+1u~k1k (25) uN=u~N1N

Here, u~k∼U[0,1) , which are used to select xk∗ according to the multinomial distribution.

The resampling method involves eliminating particles that have small normalized importance weights and copying upon particles with large weights, setting all the weights to 1/N.

Fig. 1 shows the framework of the prediction process, which is roughly divided into three steps.

Step 1. Record the number of loads and crack length during the fatigue crack growth process, calculate the crack growth parameters, and establish the crack growth state equation.

Step 2. Obtain response signals under different fatigue crack conditions via ultrasonic lamb wave detection technology, extract characteristic values, and establish observation equations.

Step 3. Use particle filter algorithm to realize early fatigue microcrack prediction.

In order to ascertain the energy change of lamb waves passing through different crack lengths, numerical experiments are performed using the finite element method (FEM). An aluminum plate with the encastre boundary condition is modeled using the ABAQUS software, as shown in Fig. 2.

A through-thickness crack with a width of 0.5 mm was modeled onto the side of the plate. In order to simulate early fatigue cracks, at the center of this crack, another size crack was cut out with a width of 0.1 mm and length varying from 0 to 12 mm, which had an increment of 3 mm, as shown in Fig. 3. The pitch-catch configuration was often used for damage detection, and the actuator and receiver were set on both sides of the damaged region.

According to the dispersion curve of Lamb wave propagating in a 6061-T6 aluminum alloy plate with 5 mm thickness, when the frequency thickness product is less than 2 MHz⋅mm , the excited lamb wave possess only two modes, S0 and A0. The excitation frequency of the Lamb wave is then set as be 0.148 MHz, as shown in Fig. 4. The properties of the material are listed in Table 1.

Fig. 5 illustrates the response signal under different crack lengths. Fig. 6 shows that when the lamb wave is reflected when passing through the crack, the energy is attenuated, flight time becomes longer, and the whole waveform is distorted. In light of these three characteristics, fatigue cracks of different lengths can be designated.

With regard to verify the proposed method, a fatigue test for plate specimens consisting of 6061-t6 aluminum was performed. The thickness of all specimens was 5 mm, which were labeled from L1 to L5. A 5 mm notch was then machined at the specimen’s edge in order to initiate the direction of fatigue crack growth. As illustrated in Fig. 1, two sensors were adhered onto the plate specimen surface, and PZT1 was applied as the actuator to activate the lamb wave, while PZT2 was used as the sensor to acquire the wave signal. The crack specimens and sensor layouts are shown in Fig. 7.

The Lamb wave based on the SHM technique was used for fatigue crack detection, which was sensitive to fatigue crack. The change in characteristics was due to discontinuities introduced by the growth of fatigue crack. These discontinuities can disperse and reflect the energy of the original Lamb wave, triggering changes in wave characteristics [36]. The fatigue crack can then be identified by extracting feature changes from the Lamb wave signals.

A Multipurpose Servohydraulic Universal Testing Machine (Serises LFV 250 KN) was used to apply fatigue load. Specimen L1 was initially performed to determine the fatigue load with a fracture load of 35 KN. According to the results, a sinusoidal load with a peak value of 10 KN was chosen for specimens L2 to L5. The stress ratio was r = 0.1, and the frequency of fatigue load was selected as 15 Hz, as shown in Fig. 8a. Accordingly, a 5-cycle sine burst signal with 160 kHz central frequency and ±48 V amplitude was excited by PZT1. The Lamb wave signal was then acquired at a sampling rate of 256 MHz.

During the fatigue test, a digital microscope was used to observe the growth of the crack. When the crack grew to about 1 mm, the specimen was taken down. The system was then used to monitor the fatigue specimen, as shown in Fig. 8b.

Fig. 9 shows the L2 growth of the fatigue crack to 3 mm. The appearance of fatigue cracks caused discontinuities within the material structure as well as the energy attenuation and phase delay of Lamb waves passing through the discontinuous positions. Fig. 10 shows the waveform changes of the S0 mode under different fatigue crack lengths, while Fig. 11 shows the changes of the three typical characteristic parameters of DI, Normalized amplitude, and Time delay with the length of the fatigue cracks. The results demonstrated that the DI and phase increase with a rise in fatigue cracks, while the amplitude decreases with the increase of fatigue cracks. The results are noted to be consistent with the simulation analysis.

Fig. 12a shows the crack growth curve of specimens L3–L5, in which the uncertainty of the material is evident. Fig. 12b describes the DIs under different fatigue crack conditions. Fig. 12c illustrates the DI curve of the specimens L3–L5 under different fatigue cracks, which was small during early fatigue crack growth. There were a number of uncertainties with the growth trajectories of the fatigue crack. According to the sensor arrangement, when the crack was small, the lamb wave of the S0 mode did not directly pass through the crack, having a lesser signal distortion.

In order to verify the feasibility of the proposed method. L3–L5 specimens were used to establish the state-space model, while L3–L4 specimens underwent prognostic validation. As mentioned in Section 2.1, material parameters log C₀, m and σ_w of specimens L3–L5 were calculated [22], as shown in Table 2.

The value of log C₀ and m were obtained as log C₀ = −39.75, m = 5.406, σw2=0.1692 , which were then applied to Eq. (9), where the discrete load step was set as ΔN = 50 and the particle number was set to 5000. According to Eq. (10), the DI data were calculated from the L3–L5 specimens. The curve fitting tool was then used to fit the observation equation, as shown in Fig. 12c. Furthermore, the RMSE of the fitting curve was selected as the observation uncertainty, which was 0.01² and v ∼ N(0, 0.01²).

By substituting the calculation parameters into Eq. (1), the state and observation equation were obtained, as shown in Eq. (26).(26) {xk=xk−1+C(ΔK)mΔN+wyk=−0.000211xk3+0.0076xk2−0.0688xk+0.1846+v