# Wave Interaction with Defects in Pressurised Composite Structures

## Abstract

There exists a great variety of structural failure modes which must be frequently inspected to ensure continuous structural integrity of composite structures. This work presents a finite element (FE) based method for calculating wave interaction with damage within structures of arbitrary layering and geometric complexity. The principal novelty is the investigation of pre-stress effect on wave propagation and scattering in layered structures. A wave finite element (WFE) method, which combines FE analysis with periodic structure theory (PST), is used to predict the wave propagation properties along periodic waveguides of the structural system. This is then coupled to the full FE model of a coupling joint within which structural damage is modelled, in order to quantify wave interaction coefficients through the joint. Pre-stress impact is quantified by comparison of results under pressurised and non-pressurised scenarios. The results show that including these pressurisation effects in calculations is essential. This is of specific relevance to aircraft structures being intensely pressurised while on air. Numerical case studies are exhibited for different forms of damage type. The exhibited results are validated against available analytical and experimental results.

## Keywords

Composite structures Pressurisation Wave finite element Wave interaction with damage Wave propagation## Nomenclature

- \(+, -\)
Positive and negative going waves properties

- \(\mathbf {a}\)
Wave amplitude

- \(\mathbf {c}\)
Wave scattering coefficient

- \(\mathbf {D}, \mathbb {D}\)
Dynamic stiffness matrices of the waveguide and the coupling joint

- \(\mathbf {k}\)
Wavenumber

- \(\mathbf {K}, \mathbf {M}, \mathbf {C}\)
Stiffness, mass and damping matrices of a waveguide’s modelled periodic segment

- \(\mathbf {q}, \mathbf {f}\)
Physical displacement and forcing vectors for an elastic waveguide

- \(\mathbf {S}\)
Wave scattering matrix

- \(\mathbf {T}\)
Wave propagation transfer matrix

- \(\mathbf {z}\)
Physical displacement vector for the coupling joint

- \(\varvec{\phi }, \varvec{\varPhi }\)
Eigenvector and grouped eigenvector

- \(\intercal \)
Matrix transpose

- \(\lambda \)
Propagation constant and eigenvalue of the wave propagation eigenproblem

- \(\mathbb {i}, \mathbb {n}\)
Property of interface and non-interface nodes

- \(\mathbb {J}\)
Property of a coupling joint

- \(\mathbb {K}, \mathbb {M}, \mathbb {C}\)
Stiffness, mass and damping stiffness matrices of the coupling joint

- \(\omega \)
Angular frequency

- \(\mathfrak {R}\)
Real operator

*b*,*h*Width and depth of a cross-section

- \(E, G, \nu , \rho \)
Elastic modulus, shear modulus, Poisson’s ratio and density of an elastic waveguide

*g*Global coordinate index

*j*Number of DoFs on each cross-section of the periodic waveguide segment

*k*,*n*,*N*Waveguide indices and total number of waveguides existing in the considered system

*L*Length

*L*,*R*,*I*Left, right sides and interior indices

- \(q, f \)
Displacement and forcing indices

*s*Periodic segment positioning index

*t*Time

*w*,*W*Wave eigenvector index and total number of waves accounted for in the waveguide

*x*,*y*,*z*Property in the

*x*,*y*, or*z*direction

## 1 Introduction

Composite structures are increasingly used in modern aerospace and automobile industries due to their well-known benefits. However, they exhibit a wide range of structural failure modes, which include delamination, notch, crack, fibre breakage and fibre-matrix debonding [1], for which the structures have to be frequently and thoroughly inspected in order to ensure continuous structural integrity. Approximately, 27% of an average modern aircraft’s lifecycle cost [2] is dedicated on inspection and repair. The use of ’offline’ structural inspection techniques currently leads to a massive reduction of the aircraft’s availability and significant financial losses for the operator. Structural health monitoring (SHM) combines non-destructive evaluation (NDE) technologies with new modelling methodologies and robust sensing technologies to detect, identify and monitor the integrity of structures and predict their remaining lifetime. The non-destructive detection and evaluation of damage in industrial structural components during service, is of pertinent importance for monitoring their condition and estimating residual life. This evaluation has been widely studied using the ultrasonic guided wave techniques. These techniques are more sensitive to gross defects compared to micro damage. However, acousto-ultrasonic techniques [3, 4], which are excellent for both forms of defects, have been receiving increasing attention during the last decade.

Non-destructive ultrasonic wave distortion during propagation in structural media has been studied as early as in [5]. It has been demonstrated that ultrasonic waves can be successfully employed in non-destructive detection of structural defects and deterioration (such as fatigue) [6, 7, 8]. The developed NDE approaches can be classified into matrix formulation techniques: in which ultrasonic waves in layered media are defined by coupling the matrix formulation of each of the layers which constitute the media, and wave propagation techniques: which strongly rely on the calculation of dispersion curves and wave interaction reflection and transmission coefficients to inspect and evaluate structural media. The wave propagation NDE inspection techniques can furthermore be categorised into two steps, namely response and modal steps [9]. The former measures the wave reflection and transmission characteristics of the structure, while the latter determines the wave dispersion and propagation characteristics, such as the wave phase and group velocities as well as the wavenumber. These techniques have been successfully demonstrated in various structural media such as truss [10, 11], beams [12], 3-D solid media [13] and composite structure [14]. It has also been applied to calculate wave interaction coefficients from structural joints such as curved [15], spring-type [16], welded [2], adhesive [17], angled [18] and liquid-coupled joints [19].

Implementing a suitable modelling technique is as important as selecting an appropriate NDE method for SHM. The finite element (FE) method [20] is one of the most common ones employed to analyse the dynamic behaviour of structures. The structure is split into a number of elements to form a mesh and equilibrium relationships which are applied to relate the entire structure and boundary conditions to arrive at a unique solution for a specific problem. Finite element based wave propagation NDE technique for periodic structures was first introduced in [21]. It was shown that the wave dispersion characteristics within the layered media can be accurately predicted for a wide frequency range by solving an eigenvalue problem for the wave propagation constants. The work was extended to 2-D media in [22]. The wave finite element (WFE) method was introduced in [23] to facilitate the post-processing of the eigenvalue problem solutions and the improvement of the computational efficiency of the method was presented in [24]. The method is considered as an expansion of Bloch’s theorem and its main assumption being the periodicity of the structure to be modelled. It couples the periodic structure theory to the FE method by modelling only a small periodic segment of the structure, thereby saving a whole lot of computational cost and time. WFE method has been successfully implemented in 1-D [23, 25] and 2-D [26, 27] wave propagation analyses. The method has recently found applications in predicting the vibroacoustic and dynamic performance of layered structures [28]. The variability of acoustic transmission through layered structures [29, 30], as well as structural identification [31] have been modelled through the same methodology. The same FE based approach was employed to compute the reflection and transmission coefficients of waves impinging on linear joints in [25, 32].

The principal contribution of the work hereby presented is to investigate wave propagation and interaction with defects in periodic structures, and examine the effect of pre-stressing on the wave interaction coefficients. The structure can be of arbitrary complexity, layering and material characteristics as an FE discretisation is employed. The defective structure is discretised into a number of healthy waveguides coupled through a defective coupling joint. Free wave propagation properties of the periodic waveguides are computed through a wave finite element method. A hybrid WFE–FE methodology is then developed to quantify interaction of the WFE computed waves with defect within the full FE defined coupling joint. In general, the structure is pre-stressed by subjecting it to a uniformly distributed surface pressure. The pre-stress effect is evaluated by comparing the wave response (dispersion and reflection properties) of the pressurised structure to that of non-pressurised structure. This is exhibited through presented numerical case studies.

## 2 Stiffness Property of a Pressurised Structure

## 3 Finite Element Modelling of Structural Damage

A system of N waveguides connected through a coupling joint (Fig. 4) is considered in this study. In the general case, waves travel from one of the waveguides to other waveguides through the joint. Scattering coefficients are calculated from interaction of the waves with structural inconsistencies (such as damage). Composite structures are prone to a number of structural failure modes which range from microscopic fibre faults to large, gross impact damage. Among these failure modes, notch, cracks, delamination and fibre breakage are important modes of failures commonly found in composites [1, 35].

Simplified FE methods can be used to simulate the effect of the damage on the mechanical behaviour of the coupling joint. Some of these methods include element deletion, stiffness reduction, duplicate node and kinematics based methods. Descriptions of each of these methods and their applicability are given in the following sections.

### 3.1 Element Deletion Method

This method is mainly applicable for modelling notches such as holes (fibre fractures) and rectangular notches in composites. Here, an element or a number of elements along the axis of the defect is/are deleted from the structure to simulate the effect of the defect. This leads to a reduction in the overall mass and stiffness of the structure. It is one of the simplest FE damage modelling methods as it doesn’t require mesh modification.

### 3.2 Stiffness Reduction Method

*P*is the reduced material property, \(P_0\) the original magnitude of the property (which can be elastic modulus, shear modulus or density). \(\beta \) being the reduction factor, equals unity for a pristine structure. This method is applicable to model cracks and delamination, but it is limited to wave interaction problem where mode conversion is not expected.

### 3.3 Node Duplication Method

The node duplication method is applicable for modelling various damage types such as single and multiple delamination and cracks, and fibre breakages.

In this method, nodes along axis of the crack, within the structural segment, are disconnected by adding duplicate nodes, which have the same nodal coordinates but different nodes numbers, to the nodes being disconnected. Each duplicate node is assigned to an adjacent element such that when a tensile force is applied, the nodes along the crack front are separated. In this respect, if the original nodes are connected to the left side elements, the duplicate nodes will be connected to the elements on the right side.

As an illustration of this method, a structural segment with six plane strain FEs is considered. Elements and nodes numbering of the segment are as shown Fig. 2. For the damage depth considered, nodes 6, 7 and 8, which are along the damage axis, are disconnected by adding duplicate nodes 13, 14 and 15 of same respective nodal coordinates. In a pristine state of the segment, nodal arrangement of finite element 2 is [2, 6, 7, 3] in that order, while that of element 5 is [6, 10, 11, 7]. But, in a damaged state, nodal arrangement of element 2 remains [2, 6, 7, 3] while that of element 5 becomes [13, 10, 11, 14] to model defects at the interface of the two FEs. Similar node ordering holds for elements 3 and 6 with nodal arrangements [3, 7, 8, 4] and [14, 11, 12, 15] respectively in the damaged state of the structural segment.

Although a 2D structural segment is used to illustrate the procedure of this method, extending the procedure to model damage in a 3D structure is quite similar and straightforward.

### 3.4 Kinematics Based Method

This approach has a lot of similarities to the node duplication method. It involves enforcing kinematics to the nodes surrounding the damage. The structural segment is segmented into multiple domains along the crack front. The stiffness and mass matrices of each domain are generated and coupled to obtain the overall matrices of the structural segment. More details on the approach can be found in [36]. The method is applicable to model delamination, cracks and fibre breakages.

## 4 Free Wave Propagation in an Arbitrarily Periodic Structure by WFE Method

*x*direction of the arbitrary periodic structural waveguide of Fig. 3. A FE model, of a periodic segment of the structural waveguide, is meshed using commercial FEA software.

*L*, right

*R*and internal

*I*DoFs of the periodic segment as

*s*and \(s+1\) are given as

*x*direction, its amplitude should be decreasing, or that if its amplitude is constant (in the case of propagating waves with no attenuation), then there is time average power transmission in the positive direction. Then the wavenumbers of the waves (at a specified angular frequency) in the positive \({\mathbf{k}^+_{\varvec{\omega }}}\) and the negative \({\mathbf{k}^-_{\varvec{\omega }}}\) directions can be determined from the propagation constants as

## 5 Elastic Wave Interaction Modelling by Hybrid WFE-FE Approach

In the general case, a system of *N* healthy periodic waveguides connected through a structural coupling joint as shown in Fig. 4 is considered. The coupling joint could exhibit arbitrarily complex mechanical behaviour such as damage, geometric or material inconsistencies and is fully FE modelled. As already stated, each waveguide can be of different and arbitrary layering and can also support a number (*W*) of propagating waves at a given frequency.

*w*with \(w \in [1\cdots \ W]\) for waveguide

*n*with \(n \in [1\cdots \ N]\) in the system can be grouped as

*n*th waveguide. These give rise to reflected waves of amplitudes \(\mathbf {a^{-}_{n}}\) in the

*n*th waveguide and transmitted waves of amplitudes \(\mathbf {a^{-}_{k}}\) in the

*k*th waveguide (and vice versa as shown in Fig. 4) expressed as

## 6 Numerical Case Studies

This section presents case studies to demonstrate the application of the developed methodology. The case studies are divided into two; validation and test case studies. The validation cases are presented for models whose analytical and experimental wave dispersion and scattering properties can be obtained. The analytical and the experimental results are compared to the numerically predicted results in order to illustrate the validity of the presented methodology. The test cases present the application of the proposed scheme in computing waves propagation constants and quantifying waves interaction with defects within damaged layered structures subjected to pressurisation. Effect of pre-stress (due to pressurisation) on these waves properties is also examined. In all cases, finite element size is chosen to ensure that mesh density is fine enough to represent the structure accurately at a reasonable computational cost. All properties and dimensions are in SI units, unless otherwise stated.

### 6.1 Validation Case Studies

#### 6.1.1 Two Collinear Bars Coupled Through a Finite Bar

Two collinear bars connected through another bar (the coupling joint) of a different material characteristics is considered. The bars are of uniform circular cross-section and undergo longitudinal vibration. Arrangement of the bars is presented in Fig. 5. Each waveguide is made of aluminium (\(E_1 = E_2 = 70 \times 10^9\), \(\rho _1 = \rho _2 = 2600\)) and the joint is made of steel (\(E_J = 210 \times 10^9\), \(\rho _J = 7500\)). Cross-sectional areas \(A_1 = A_2 = A_J = 0.003\), lengths \(L_1 = L_2 = 0.2\) and \(L_J = 0.003\).

Incident wave of amplitude \(\mathbf {a^{+}_{1}}\) impinging on the coupling joint from waveguide 1 will give rise to reflected and transmitted waves of amplitudes \(\mathbf {a^{-}_{1}}\) and \(\mathbf {a^{+}_{2}}\) in waveguides 1 and 2 respectively. Standing wave is present in the joint since both forward and backward moving waves of amplitudes \(\mathbf {a^{+}_{J}}\) and \(\mathbf {a^{-}_{J}}\) are simultaneously present.

#### 6.1.2 Delaminated Beam

The coupling joint is undamped, i.e. it is of real-valued elastic and shear moduli. As a conservation of energy condition, the algebraic sum of reflection and transmission efficiencies of a lossless (undamped) structural segment equals unity. As observed in Fig. 10, conservation of energy condition is satisfied for all presented waves as sums of reflection and transmission efficiencies are ones. This further establishes the validity of the presented methodology. Also observed in the waves transmission and reflection results is the fact that the incident waves in waveguide 1 is transmitted or reflected through the coupling joint into waves of the same type without any form of mode conversion. This is expected in waveguides collinearly connected through a joint as waves will be fully transmitted without reflection and modes conversion. Reflection observed is solely as a result of damage in the coupling joint.

#### 6.1.3 Notched Plate

Validity of the presented methodology is further proven using notched plate of thickness 2d \(= 0.003\) and length \(L = 0.6\). The plate is made of mild steel (\(E = 210 \times 10^9\), \(\rho = 7850\) and \(\nu = 0.29\)) and has uniform area throughout its cross sections.

Based on the presented methodology, the plate can be discretised as a system of two pristine waveguides (\(L_1 = L_2 = 0.295\)) connected through a notched coupling joint (\(L_J = 0.01\)) as shown in Fig. 11. Plane strain condition is assumed.

ANSYS is used to model a segment (of length \(\varDelta = 0.001\)) of each waveguide with PLANE 182 (4-noded quadrilateral finite elements with two translational DoFs per node) FEs. The segment of each waveguide is meshed across its width using 12 elements. Similar element size used for the waveguide segments is repeated for the coupling joint, thereby meshing the joint using 120 elements.

Waves reflections are obtained from notches, of depths 0.0005, 0.001 and 0.002, which are respectively 17, 33 and 67% of the plate thickness. Uniform notch (of width 0.0005) is used in all cases. The notches are modelled using the element deletion approach presented in 3.1.

In practice, the wave reflection calculation can be made by a full FE transient simulation. The WFE computed eigenvectors can be windowed and then applied as time-dependent harmonic displacement boundary conditions (of excitation frequency \(\omega \)) at one of the extreme cross-sections of the plate. In this case, the entire plate is modelled as one plate instead of a system of waveguides and coupling joint as in the case of the presented WFE-FE approach.

Good agreement is observed among the WFE-FE, full FE and experimental results as shown in Fig. 12. It is also worthy to state that the developed methodology is more efficient (than the full FE approach) for predicting wave scattering (reflection and transmission) from damage within structural waveguides for the following noted reasons. First, model size and computational time. Finite element mesh of the plate consists of 7200 elements and 15,626 DoFs in the full FE model against a total of 144 elements (12 for each waveguide and 120 for the joint) and 390 DoFs in the presented WFE-FE model. Solving the full FE model requires a computational time of about 105 min compared to the WFE-FE model which is solved under 5 min. Therefore, a great deal of computational time and hence cost is saved by the WFE-FE approach. Another noted point is in terms of the complexity of the structural system. Full FE model mostly assume plane strain condition in order to simplify and reduce model size of a structural system. In this manner, some propagating waves especially those along the suppressed axis might not be captured. However, the presented WFE/FE approach can be applied for analysing wave interaction in complex structural systems (such as composite structures and structural networks) with low computational size and cost.

### 6.2 Test Case Studies

#### 6.2.1 Transversely-Isotropic Beam

A segment (of length \(\varDelta = 0.001\)) of each waveguide is modelled in ANSYS with 40 SOLID185 finite elements using cubed sized elements of length 0.001. Using similar element size, the coupling joint is modelled with 160 finite elements. The crack within the joint is modelled using the node duplication approach (Sect. 3.3). Two crack scenarios, of depths 0.001 and 0.002, are considered. These are respectively equivalent to 20 and 40% of the total depth of the beam. The cracks are through-width and located at mid length of the joint.

Dispersion curves for each waveguide are obtained by solving Eq. (12) within frequency range \(\omega = [1.0 \times 10^2{-}3.3 \times 10^4]\) Hz. The dispersion curves are presented in Fig. 14. Four propagating modes at each frequency are obtained for the non-pressurised waveguide. For the pressurised waveguide, there are three propagating modes (y-axis bending wave, z-axis bending wave and longitudinal wave) at low frequency range. Fourth mode (torsional wave) cuts on at \(\omega = 3.8 \times 10^3\), \(8.1 \times 10^3\), \(9.6 \times 10^3\) and \(1.0 \times 10^4\) Hz in the 0.1, 0.5, 1.0 and 1.5 GPa pressurised waveguides. In the low frequency range, the wavenumbers of the pressurised waveguide are significantly different compared to the non-pressurised one. An average difference of about 32% per 0.1 GPa is observed for the bending waves at low frequency range. Differences of about 20 and 11% are observed for the longitudinal and torsional wavenumbers. Increase in the wavenumbers can be attributed to reduction in loss factor of the waveguide due to an increase in strain energy as a result of the applied pressure. The difference in wavenumbers (of the non-pressurised and pressurised waveguides) diminishes gradually as frequency gets higher.

With respect to applied pressure, it can be seen that wave reflection coefficient magnitudes increase with an increase in the magnitude of applied pressure. Average increments of about 45, 20, 25 and 90% per 0.1 GPa are respectively observed for the y-axis bending wave, the z-axis bending wave, the longitudinal and the torsional wave.

Consequently from the presented results, magnitudes of wave constants and interaction coefficients are generally boosted by applied pressure. This can therefore be used to detect micro structural defects which may not be easily detected under non-pressurisation scenario. Application of pre-stressing (through pressurisation) as a damage detection method is further examined using a sandwich laminate with micro delamination as presented in next section.

#### 6.2.2 Sandwich Beam

The delaminated beam is discretised as two healthy beams (\(L_1 = L_2 = 0.2\)) coupled through a delaminated joint (\(L_J = 0.004\)) as shown in Fig. (17). The beams are modelled in ANSYS using SOLID185 elements. Cubed sized elements of length 0.001 are used to model the facesheets, while elements of size \(0.001 \times 0.002 \times 0.001\) are employed for modelling the core. As a result, 40 finite elements are used for the WFE model of a periodic segment (of length \(\varDelta = 0.001\)) of each waveguide and 160 elements for the full FE model of the coupling joint.

Interlaminar delamination, along the interface of the upper facesheet and the core, is considered. Two delamination scenarios (20 and 40% of the beam width) are examined. They are both of length 0.002 (symmetrically located about the mid length of the joint).

The reflection coefficients magnitude becomes greatly significant in the pressurised system as shown in Fig. 20. Compared to the non-pressurised system, there is an average change of about 50–70% in the low frequency range and of about 10–25% in the high frequency range. This can be explained by the fact that structural pre-stressing brings about change in the loss factor of the structure. In general, this consequently affects the magnitude of wave propagation properties.

At 0.8 kHz (Fig. 22), the reflection magnitudes of the x and y axes bending waves increase with the applied pressure up to 1 GPa. Beyond this pressure, reduction in the magnitudes is observed. Longitudinal and torsional waves both show similar trends. There is an increase in their coefficient magnitudes with respect to applied pressure. Increment obtained for the torsional wave is however more than that of the longitudinal wave.

Unlike at 0.2 and 0.8 kHz, the reflection coefficient magnitudes of the bending waves at 6.4 kHz (Fig. 23) increase with respect to applied pressure. The longitudinal and the torsional waves maintain similar trend as in 0.2 and 0.8 kHz.

Similar to the non-pressurised system, it is noted that there is no significant difference observed in the reflection coefficients (with respect to frequency and applied pressure) of the 20% width delaminated joint compared to that of the 40% width delamination.

## 7 Concluding Remarks

- (a)
The presented approach is validated with analytical and full FE transient response predictions. Very good agreement is observed.

- (b)
The approach is able to predict the dispersion properties of an arbitrarily complex structure as well as the reflection and transmission coefficients of the wave interaction with defects within the structure.

- (c)
The approach also successfully examined the effect of pre-stress on the wave properties of pressurised structures. It was shown that pressurisation can be used to detect micro defects which may be too small to detect under no pressurisation.

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