1 Introduction

The structural health monitoring (SHM) has been increasingly used in aircraft^{[1, 2]},large bridges^{[3, 4]},and nuclear power plants^{[5, 6]} due to their advantages such as on-line and real-time monitoring the overall structure or key parts of target structures. A new SHM technology called the intelligent coating monitoring (ICM),which is mainly based on intelligent coating consisted of driving layer,sensing layer,and protective layer,has been developed^{[7, 8, 9]}. Intelligent coating is a new kind of functionally gradient material,which is characterized by a functionally thin film,and the crack size and crack propagation for the health status of the substrate (or structure) can be represented by the variation of the conductive resistance of the sensing layer of intelligent coating. Currently,the ICM presents good capability of monitoring crack initiation and propagation,and it has shown some advantages such as simple,reliable,economic,and practical characteristics in the SHM of aircrafts^[7].

In the intelligent coating system,the crack-detected ability strongly depends on the penetration of the matrix crack at interfaces (matrix/driving layer/sensing layer),which causes the variation of the conductive resistance of the sensing layer. When the matrix crack reaches interfaces,the crack propagation may appear two trends: (1) if the interfacial fracture toughness is higher,the matrix crack penetrates the interface directly; (2) if the interface is weak,it may deflect along the interface. However,if the matrix crack deflects along the interface,it will be not propitious to monitor crack for the ICM^[7]. In this regard,an accurate prediction of the crack propagation path ensures an optimized design of the intelligent coating system. So far, no theoretical investigations have been reported on the crack penetration path in the ICM.

The stress and energy criteria are typically used to determine the crack path. The former is governed by the local asymptotic stress field at the interface. The crack deflection at interfaces was first investigated using stress criterion by Cook and Gordon^[10],who considered possible interfacial debonding occurrence before a main crack tip met the interface lying ahead of a main crack. This concept was extended many years later by Gupta et al.^[11]. In their work,the crack propagation by penetration or deflection was determined by the ratio of the maximum crack-opening normal stress ahead of the incident crack to the maximum crack-opening normal stress along the interface.

The energy criterion is based on the differences of work of fracture along possible alternative crack paths^{[12, 13]},and the investigation of related phenomena has received much interest^{[14, 15, 16]}. The most familiar energy-based approach is that of He and Hutchinson^[12] (with corrections^[13]), who were the first to pay particular attention to the competition between crack penetration and deflection and solved several important crack configurations which approach a bimaterial interface from different angles to the interface. The development of effective techniques to measure the interface toughness^{[17, 18, 19]} has made it possible to use the energy criterion more easily while measuring the interface strength^{[20, 21]} still remains difficult to perform. In this paper,the energy criterion is adopted to investigate the behavior of crack penetration/deflection at interfaces in the intelligent coating system. According to the energy criterion, a three-phase model of the intelligent coating system is established in this investigation. The influence of the elastic mismatch parameters and the thickness of the interface layer on the ratio of energy release rate of a deflected crack to that of a penetrated crack is comprehensively numerically analyzed,and the numerical results are verified by the experiments as well. 2 Analysis 2.1 Problem description

The substrate (Material 1),the driving layer (Material 2),and the sensing layer (Material 3) are included in the three-phase model of the intelligent coating system,as shown in Fig. 1. For the three-phase model under plane strain conditions,two sets of Dundurs’ parameters,namely, α₁ and β₁ for Materials 1 and 2 and α₂ and β₂ for Materials 2 and 3,are defined as^[22]

where v_i and E_i (i = 1,2,3) are Poisson’s ratios and the elastic moduli with the subscripts in Eq. (1) indicating the material number.

In order to ensure the crack-detected ability of the intelligent coating system,two cases of the three-phase model are necessary to consider: (i) the relative tendency to deflect or penetrate when a main crack reaches the interface between Materials 1 and 2; (ii) the competition between deflection and penetration when the crack reaches the interface between Materials 2 and 3. For both cases,the relative tendency for the main crack to deflect or penetrate can be addressed by the energy criterion^{[12, 13]}. A necessary condition to enable a crack to penetrate into the next material instead of deflection along the interface is given by^{[12, 13]}

where R is the ratio of the energy release rate,G_d and Γ_d denote the energy release rate and the surface energy for the case of crack deflection along the interface,respectively,and G_p and Γ_p denote the corresponding quantities for the crack penetration across the interface,respectively. According to Refs. ^[12] and ^[13],for two dissimilar materials,G_d/G_p can be expressed as where c,d,and e are dimensionless complex functions of α and β,a_d and a_p are the putative initial deflection and penetration lengths,respectively,as illustrated in Fig. 2,and λ is the exponent of the stress singularity,which is expressed as^[23]

The stress singularity exponent,λ,varies from 1.0 (when α = −1.0) to about 0.3 (when α = 1.0) and determines the stress distribution in the front of the crack tip via the relation σ ~ r−λ,where r is the distance from the crack tip. Especially,when α = 0,λ becomes 0.5 yielding the familiar σ ~ 1/√r stress singularity in isotropic homogeneous materials^[18].

In Fig. 2(a),the condition of matrix crack penetration can be expressed as

where Γ_d1 and Γ_p2 are the interface critical energy release of the substrate/driver layer and the critical energy release of the driver layer,respectively.

In Fig. 2(b),the condition of matrix crack penetration can be expressed as

where Γ_d2 and Γ_p3 are the interface critical energy release of the driving layer/sensing layer and the critical energy release of the sensing layer,respectively. 2.2 Numerical analysis The model in the present study is a single-edged notch (SEN)-like specimen which is composed of three-phase materials and subject to uniform tensile loading as shown in Fig. 1. The three materials,designated as 1,2,and 3,are assumed to be isotropic and elastic with the elastic properties given by Eq. (7). The thickness values of Materials 1,2,and 3 are initially set to be h₁ = 200 mm,h₂ = 0.5 mm,and h₃ = 10 mm without a specific unit for an elastic prob- lem,while h₂ is changed later to obtain three ratios of h₂/h₃ = 0.05,0.1,and 0.2 to simulate different physical backgrounds.

Previous studies^{[12, 13]} demonstrate that Dundurs’ parameter α is the more important one, and β does not significantly affect the elastic fields of the model. Ignoring the effects of β, Eq. (1) can be simplified as

All the problems are analyzed under the linear plane strain elasticity conditions for the stated geometric thickness and various elastic mismatches. The crack driving forces,G_d and G_p,are calculated through the estimation of J-integral based on the virtual crack extension/domain integral methods. It is assumed that the lengths of deflected and penetrated crack are a_p = ad = 0.002 08h₃ to approach the results of ap → 0 and ad → 0^{[12, 13, 14]}.

Considering the structural symmetry,half of the geometry in Fig. 1 is modeled for the numerical analysis. The symmetry constraint is exerted on the symmetry lines,and the external load is applied on the right edge of the substrate,as shown in Fig. 3. A general purpose finite element analysis package ANSYS is used for the analysis. Due to the similarity of Figs. 2(a) and 2(b),the situation in Fig. 2(a) is only analyzed here.

3 Results and discussion

Figures 4-6 are the relationship between R and α1 for the thickness (h₂) of 0.5 mm,1 mm, and 2 mm,respectively. Note that the ranges for R presented in Figs. 4−6 are different. The effect of α2 on R is also included in Figs. 4−6.

The values of R show a stronger dependence on large positive values of α1,while there is only a weak dependence on large negative values of α1. In addition,a large α2 always produces a large R,which can weaken the ability of crack penetration. Hence,it should select an appropriate elastic modulus so as to obtain an optimized penetrating interface.

However,the ratio of energy release rates R not only depends on the elastic mismatch α1 between the substrate (Material 1) and driving layer (Material 2) but also on the elastic mismatch α2 between the driving layer and the sensing layer (Material 3),as shown in Figs. 4−6. In addition,the thickness of the driving layer also significantly influences R for large positive α1,though there is a slight dependence for large negative values of α1. The tendency is most pronounced at the thickness of 0.5 mm in Fig. 4. For the thickness (h2) of 0.5 mm,1 mm, and 2 mm,α1 only influences R slightly for α1 < −0.4,−0.2,and 0.0. It can be seen from Figs. 4−6 that larger thickness of h2 results in smaller R for α2 > 0 which is beneficial to crack penetration,while larger thickness of h2 also results in larger R for α2 < 0 which is propitious to crack deflection.

Moreover,the results for the three-phase material system gradually converge to that of the infinite bi-materials with increasing values of h2. As shown in Fig. 7,the thickness of the driving layer h2 = 20 mm is presented. For α1 < 0.8,the elastic mismatch parameter α2 has almost no effect on R. While for α1 > 0.8,there is a slight change for the ratio of energy release rates G_d/G_p. It suggests that the influence of α2 on the ratio of energy release rates G_d/G_p reduces gradually with the increase in the driving layer thickness.

From the discussion aforementioned,it can be concluded that the ratio of energy release rates not only depends on the combinations of the substrate and the driving layer but also on the combinations of the driving layer and the sensing layer,together with the thickness of the driving layer. Hence,it is important to select a constituent modulus and a thickness of the driving layer and the sensing layer so as to develop an intelligent coating sensor with desired penetrating properties. 4 Experimental verification

The numerical simulation for G_d/G_p versus the elastic mismatch parameters α_i (i = 1,2) and the thickness of the driving layer are discussed above. Those simulations are verified by some experiments.

The 7050-T7452 alloy/driving layer/sensing layer specimens are fabricated. The fracture evolution during tensile test of the specimens is observed and discussed by the finite element numerical results R.

The intelligent coating including the driving layer and the sensing layer are in-situ fabricated based on the nano-technical process of materials. The driving layer is made of nonconductive materials,and the sensing layer is mainly made of nano-Cu conductive materials. In this investigation,the thicknesses of the driving layer and the sensing layer are approximately 20 mm and 15 mm,respectively,and the width of the sensing layer is 1 mm. Uniaxial tension tests are performed on the substrate (see Fig. 8),and the fracture surfaces are examined for the evidence of crack propagation (see Fig. 9).

The measured elastic and toughness properties of these materials are given in Table 1^[24]. Based on the plane strain conditions,the critical energy release rate or surface energy can be calculated from the toughness KIC via Γ = K_IC² (1−ν2)/E in each case. From Young’s modulus and Poisson’s ratios of the 7050-T7452 alloy,for the driving layer and the sensing layer,α1 and α2,are −0.92 and 0.95,respectively.

It is assumed that the interface toughness between the substrate and the driving layer is Γ_i1, and Γ_i2 represents the interface toughness between the driving layer and the sensing layer. The ratio of Γ_i1/Γ_p2 is about 9.24^{[17, 18, 19]} when the main crack tip reaches the interface between the substrate and the driving layer. It is learnt from the finite element results that R is about 0.36. Hence,R is less than Γ_i1/Γ_p2. It indicates that it is unlikely to occur for the interface debonding between the substrate and the driving layer,and the crack penetration into the driving layer prefers to occur,which is verified by the experiments.

The situation is also investigated when the crack penetrates into the driving layer and the crack tip reaches the interface between the driving layer and the sensing layer. The ratio of Γ_i2/Γ_p3 is about 2.68. Moreover,R is about 1.58 from the finite element results. Thus, R < Γ_i2/Γ_p3. It suggests that the interface deflection between the driving layer and the sensing layer is unlikely to occur,and the crack can penetrate into the driving layer directly. This is in good agreement with the experimental result. Despite the complexity of the problem and unknown factors in materials and geometries,it is believed that the modeling and numerical results are still useful to disclose the crack evolution process of the intelligent coating to some extent. 5 Conclusions

The possibilities of crack penetration in the intelligent coating system are investigated using a finite element analysis based on the energy criterion. From above investigations,the following conclusions can be obtained:

(i) The relative tendencies to deflect and penetrate when a main crack reaches the two interfaces are predicted by computing the ratio of energy release rates R = G_d/G_p for infinitesimal deflecting and penetrating cracks. The numerical analysis indicates that the ratio of energy release rates R not only depends on the elastic mismatch parameters αi (i = 1,2) but also on the relative thickness of the driving layer as well. Further analysis of the numerical results shows that R has a strong dependence on large positive values of α1,while it is not sensitive to large negative values of α1. Moreover,a large α2 always produces a large R,which can weaken the ability of crack penetration. The numerical results also predict that a larger thickness of driving layer can result in a smaller R for α2 > 0 which is beneficial to crack penetration,while a larger thickness of h2 also results in a larger R for which α2 < 0 is not propitious to crack penetration. In addition,the influence of α2 on the ratio of energy release rates G_d/G_p reduces gradually with the increase in the thickness of the driving layer.

(ii) Experimental results show that the substrate crack can penetrate into the driver layer and the sensing layer in succession,which is consistent with the finite element analysis. Hence, it is important to select an appropriate thickness and elastic modulus of the driver layer and the sensing layer for the intelligent coating sensor with better properties.