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A numerical evaluation on nonlinear dynamic response of sandwich plates with partially rectangular skin/core debonding

A numerical evaluation on nonlinear dynamic response of sandwich plates with partially... Curved and Layer. Struct. 2022; 9:25–39 Research Article Tuswan Tuswan, Achmad Zubaydi*, Bambang Piscesa, Abdi Ismail, Rizky Chandra Ariesta, and Aditya Rio Prabowo A numerical evaluation on nonlinear dynamic response of sandwich plates with partially rectangular skin/core debonding https://doi.org/10.1515/cls-2022-0003 1 Introduction Received May 20, 2021; accepted Aug 15, 2021 Abstract: As one of the most dangerous defects in the sand- As one of the unique thin-walled structures, sandwich ma- wich panel, debonding could significantly degrade load car- terial is broadly used in shipbuilding industries due to com- rying capacity and affect dynamic behaviour. The present bining high performance with a lightweight design. Sand- work dealt with debonding detection of the rectangular wich panels are profitable during construction and repa- clamped hybrid sandwich plate by using ABAQUS software. ration because they can reduce the weight by eliminating The influence of various damage ratios on the linear and the stiffeners compared to the stiffened plate structures. nonlinear dynamic responses has been studied. The finite However, weight reduction often leads to an increase in vi- element model was initially validated by comparing the bration problems occurring under manufacture and service modal response with the experimental test. Rectangular conditions. Increased vibration can cause a variety of struc- debonding was detected by comparing dynamic responses tural damage. Face sheet/core debonding is the serious of free and forced vibrations between intact and debonded damage that can reduce the stiffness [ 1, 2, 3] and threaten models. A wide range of driving frequency excitation cor- the safety and service life [4]. responding to transient and harmonic concentrated loads The difficulty in controlling the appropriate bonding was implemented to highlight nonlinear behaviour in the during the manufacturing stage is the main problem [5, 6], intermittent contact in the debonded models. The results in which the debonding location is often invisible in the showed that debonding existence contributed to the nat- interface layer between two adjacent surfaces [7]. Debond- ural frequency reduction and modes shape change. The ing has resulted from manufacturing failure when a small numerical results revealed that debonding affected both region’s adhesive layers have not been sufficiently bonded. the steady-state and impulse responses of the debonded Moreover, this early-stage damage may propagate to create models. Using the obtained responses, it was detected that larger debonding, altering the vibration characteristic and the contact in the debonded region altered the dynamic behaviour. The debonding can lead to frequency reduction, global response of the debonded models. The finding pro- leading to structural failure in the lower mode [8, 9]. Since vided the potential debonding diagnostic in ship structure bonding quality during manufacture determines the per- using vibration-based structural health monitoring. formance and integrity of the structure, an effective and efficient debonding detection in the early stage is necessary Keywords: debonding detection, hybrid sandwich, finite el- to assess structural health and performance. ement analysis, dynamic contact, forced vibration, marine. Tuswan Tuswan: Department of Naval Architecture, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Abdi Ismail: Department of Naval Architecture, Institut Teknologi 60111, Indonesia; Department of Naval Architecture, Faculty of Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya 60111, In- Engineering, Universitas Diponegoro, Semarang 50275, Indonesia donesia; Ship Machinery Study Programme, Faculty of Vocational *Corresponding Author: Achmad Zubaydi: Department of Naval Studies, Indonesia Defense University, Belu, Indonesia Architecture, Institut Teknologi Sepuluh Nopember, Kampus ITS Rizky Chandra Ariesta: Department of Naval Architecture, Institut Sukolilo, Surabaya 60111, Indonesia, E-mail: zubaydi@na.its.ac.id Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Bambang Piscesa: Department of Civil Engineering, Institut 60111, Indonesia Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Aditya Rio Prabowo: Department of Mechanical Engineering, 60111, Indonesia Universitas Sebelas Maret, Surakarta 57126, Indonesia Open Access. © 2022 Tuswan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 26 | Tuswan et al. Assessment of the structure using vibration-based To address this issue, the numerical damage assess- structural health monitoring (VSHM) is widely used by ex- ment of a hybrid sandwich containing a steel faceplate and amining the changes in measured vibration response [10]. polymer core was developed in this study. The essential The basic principle is comparing the change of dynamic contribution of this work is to study the nonlinear effect of characteristics between intact and debonded structures debonding contact between the faceplate and core due to such as natural frequencies [11, 12, 13], mode shapes [14], free and forced vibrations. An initial study was performed frequency response function (FRF) curvatures [15], modal by comparing the numerical model using ABAQUS software damping, and time or frequency domain data [16, 17]. Re- [27] with the experimental test to check the correctness of cently reported VSHM studies based on linear and non- debonding modelling. The three-dimensional model of the linear vibration analysis in the composite sandwich were clamped sandwich plate with rectangular shape debonded comprehensively presented. Burlayenko and Sadowski in- zone was firstly conducted free vibration analysis to extract vestigated the influence of debonding ratio, location, and natural frequency. The general dynamic based on explicit type on free vibration response [18]. It showed that the ra- time-stepping procedure due to transient and harmonic tio, location, and type of debonding influence the vibration load was then undertaken to check the effect of dynamic behaviour of the composite sandwich. Moreover, the pre- response due to contact dynamic under various debonding vious studies found that debonding is easily detected in ratios. The structure of this paper is constructed in sections higher mode than lower mode [19, 20, 21], but it is chal- as follows. Section 1 presents the introduction and objec- lenging to get higher modes in the practical experiment tive of the study. The theoretical backgrounds of model dis- due to restriction of excitation magnitude. Consequently, cretization, contact modelling, and finite element analysis nonlinear vibration utilising time-domain data of structural are reported in Section 2. Section 3 offers the Experimental responses of the composite sandwich is more popular. Com- test preparation and procedure, and Section 4 reports the pared with modal analysis, investigating the damage based result of the free and forced vibration test. In the last, a few on time-domain dynamic responses is easier [22], making concluding remark and summary is provided in Section 5. the result more reliable [23]. Several studies have been re- viewed in detail. Burlayenko and Sadowski have created an explicit 3D finite element model (FEM) to examine the 2 Numerical modelling of sandwich transient dynamic response of an impacted sandwich panel with debonding [24]. Moreover, damage detection using an panels implicit dynamic has been reviewed in [25] to investigate the responses of the debonded sandwich subjected to har- 2.1 Aspects of model discretization monic forces. A recent development study of debonding identification based on the different types of analysis in In the aspect of model discretization, there are several the composite sandwich is comprehensively reviewed by schemes available for modelling sandwich and lami- Burlayenko and Sadowski [26]. nate structure. They are classified into equivalent single Although a majority of previous investigations have layer (ESL), layerwise (LW), and continuum-based three- been reviewed to explore the dynamic behaviour of the dimensional (3-D) elasticity theories. ESL theory is simply composite sandwich, there is limited study in the com- efficient for general analysis with computationally efficient, prehensive assessment of dynamic behaviour and nonlin- but interlaminar stress and transverse compressibility are ear characteristic of hybrid sandwich panels containing ignored [26]. In order to alleviate the shear-locking phe- metal/polymer combinations as part of ship structure due nomenon in shell/plate, mixed interpolation tensorial com- to debonding problems. The issue is crucial since the me- ponent technique (MITC) technique with Carrera Unified chanical properties of two adjacent layers of hybrid sand- Formulation (CUF) framework is proposed by Kumar et al. wich have higher dissimilar properties, leading to initiate [28] to perform modal analysis of delaminated composite a debonding problem more severe. There is an important shell structures. Using this strategy, the shell element has issue investigating the response characteristic of debond- 9-nodes and allows displacement distribution along with ing problems due to the free and forced vibration of this the thickness of the multilayered shell. The implementation type of sandwich under different loading scenarios. More- of refined and advanced delamination model of CUF shell over, there is a limited study concerning the validity and model using various theories and different ordered theories correctness of proposed debonding modelling of hybrid was studied. Moreover, using 2D MITC finite element based sandwiches with the response obtained from the experi- on the CUF framework, Legendre-like polynomial expan- mental test. A numerical evaluation on nonlinear dynamic response of sandwich plates | 27 Table 1: Sandwich configuration and debonding geometry. sions are introduced by Pagani et al. [29] to implement ESL, LW, and variable kinematics theories. Models h −h −h D = D T (mm) t c 1 2 A nonlinear pattern in the height of the core during b (mm) (mm) deformation/ core transverse compressibility should be accounted for modelling modern sandwich panels. This Intact (0%) 4-20-4 0 0 makes it an obstacle to handle this modelling using ESL 2D 5% debonding ratio 4-20-4 67.08 0.00004 elements. In this regard, quasi three-dimensional model 10% debonding ratio 4-20-4 94.90 0.00004 or layerwise (LW) scheme is preferable due to accommo- 20% debonding ratio 4-20-4 134.2 0.00004 dating core compressibility with effective computational 30% debonding ratio 4-20-4 164.3 0.00004 cost. Solid-brick elements for the core and continuum shell elements for faceplates were used in this case. Table 2: Material constants of faceplates and core sandwich mate- rial. 2.2 Sandwich configuration and debonding Parameters Faceplate Core material geometry −3 ρ (kg m ) 7850 1465 11 9 E (Pa) 2.1 x 10 4.4 x 10 Ferry Ro-Ro 6300 Gross Tonnage (GT) stern ramp door was 10 9 G (Pa) 8.07 x 10 1.69 x 10 used as a reference model, as previously described in [21]. σ (MPa) 275 24.8 The sandwich panel geometry of the stern ramp door is σ (MPa) 292 24.8 illustrated in detail in Figure 1. The steel faceplate and υ 0.3 0.3 resin/clamshell core material with dimensions 300 mm in length and width are modelled. The configuration of the sandwich panel is 4 mm in the upper and lower faceplate trated in Figure 1. The rectangular debonded region at the thickness (h = h ) and 20 mm in core material thickness middle span was presented by a small artificial gap ( t = 0.1% t b (h ). The sandwich thickness configuration and debonding c x h ) between the upper faceplate and the core. No treat- geometry are presented in Table 1. The material constants ment of either geometrical entities or material constants is of the sandwich panel used for debonding detection are applied in the debonded area to guarantee a physically real shown in Table 2 and previously reported in [11, 21]. case condition. The ratio of debonding was represented by a damage parameter (D = A /A x 100) expressing the D T percentage of the region of debonding (A ) to the whole region of the interface layer (A = D x D ). In rectangular T 2 debonding, the length and width of the debonding have similar values (D = D ). The model was modelled by a solid/shell layerwise (LW) approach in the model discreti- sation, as previously recommended [30, 31]. In terms of the element type, the faceplate was modelled using a con- tinuum shell element (SC8R), and the core material was modelled using a solid element (3CD8R). Meanwhile, the clamped-free-clamped-free (CFCF) boundary condition, as illustrated in Figure 1, was applied. 2.3 Theoretical basis of modal analysis Free vibration analysis was used to investigate structural dynamic behaviour. In free vibration, there was no external Figure 1: Sandwich geometry with debonding [21]. load causing the motion. The motion has resulted from initial conditions, such as an initial displacement from an A total of v fi e models was created, as presented in Table equilibrium position. The equation of motion for forced 1. Debonding was modelled using the FEM by single rectan- vibration can be assumed in the form [32]. gular damage at the centre of the top adhesive layer, as illus- 28 | Tuswan et al. Then, the accelerations, velocities, and displacements in ¨ ˙ ¨ ˙ [M][U ] + [C][U ] + [K][U] = [F] (1) this time increment are expressed by U , U , U , re- i+1 i+1 i+1 spectively. Therefore, Eq. (5) at t = t + ∆t is expressed i+1 i i+1 where M, C, and K are mass, damping, and stiffness in the form: ˙ ¨ matrices. U is nodal points displacements, and U and U are their time derivatives referring to nodal velocities and ext cont ¨ ˙ MU + CU + KU = F − F (6) i+1 i+1 i+1 i+1 i+1 acceleration. For free vibration analysis, the damping of the structure can be neglected. Thus, the equation can be U = U and U = V are the assumption given at the ini- 0 0 written in Eq. (2). tial boundary conditions. The explicit algorithms determine Eq. (6) using cen- [M][U ] + [K][U] (2) tral difference formulation without iterations and tangent stiffness matrix by explicitly advancing the kinematic state where Eq. (1) shortens to an eigenvalue problem. The known from a previous increment to the next one [34]. Ini- governing equation for free vibration analysis can be writ- tially, the accelerations at the beginning of each increment ten in Eq. (3). ∆t are calculated. Then, Eq. (6) is rewritten: i+1 (︁ )︁ [K] − λ [M] ϕ = 0 (3) [ ] i i −1 ext cont U = M F − F (7) i i where λ is eigenvalue and ϕ is eigenvector correspond- i i ext where F is the vector of the external nodal forces at ing to the eigenvalues. The eigenvalues associated with the int time t and F is the sum of nodal internal F =KU , damp- natural frequency can be written in Eq. (4), as follows: , i i i damp cont ing F =CU and contact force F which are updated √︀ during the previous time increment ∆t . f = λ /2π (4) i i Then, the accelerations computed at t are used to ad- The analysis was conducted to present an insight into vance the velocity solution to t + ∆ t and the displace- i i+1 the oscillation response between intact and face sheet/core ment solution to t +∆ t as follows: i i+1 debonded models. The free vibration analysis in the first six modes using ABAQUS/Standard was carried using the Lanc- ∆t + ∆t i+1 i ˙ ˙ ¨ U = U 1 + U (8) i+1 i i− zos method to extract eigenvalues. The modal responses were used as a damage index to detect the debonding prob- ˙ ˙ U = U + ∆t U 1 (9) lem. i+1 i i+1 i+ The initial half-step lagging velocity U is calculated -1/2 from the initial velocity assuming the initial acceleration. 2.4 Descriptions of explicit time-stepping The computational time in the explicit analysis linearly procedure increases with problem capacity, and the explicit procedure is only stable if the time increment in Eq. (6) is smaller The explicit procedure accomplishes a large number of time than the stability limit of the central difference operator increments. A set of a nonlinear differential equation of [34]. An approximation to the stability limit (∆t ) can be crit motion with the assumption of linear elasticity and small formulated as follows: deformation is written as [33]: ∆t ≈ (10) crit ext cont ¨ ˙ MU (t) + CU (t) + KU(t) = F (U(t)) − F (U(t)) (5) Where L is the smallest element dimension in the mesh and c is dilatational wave speed. where Ü(t), (t) and U(t) is the global vectors of unknown To gain a deeper investigation of debonding diagnostic, accelerations, velocities, and displacements, respectively, ext the comparison of dynamic behaviour due to debonding at each instant of time. F is the vector of the external cont problem was investigated using general dynamic analysis. force, and F is the vector of contact forces. Then, M, C, First, after analysing the modal analysis of the debonded and K are the global mass, damping, and stiffness matri- model for evaluating its natural frequencies, the FE mod- ces serially. Eq. (5) is discretised in a time domain. The els were subjected to the concentrated harmonic load at time interval [0, T] is divided into subintervals in the form L−1 the upper faceplate. The investigations were developed in [0, T ] = ∪ t , t , where t < t , and t = 0, t = 0. [ ] i i+1 i i+1 0 L i=0 ABAQUS/Explicit that incorporated the explicit integration So, Eq. (5) can only be found in a finite number of time steps. For example, the time increment is ∆t = t − t . i+1 i+1 i A numerical evaluation on nonlinear dynamic response of sandwich plates | 29 algorithm for a general dynamic analysis with contact de- 2.5 Descriptions of contact formulation scribed in Section 2.5. The models with rectangular debond- ing were loaded with a concentrated harmonic force F(t) = The fundamental features of the contact formulation em- F0 sin Ω t subjected at the central point of the upper face- ployed in the current study to perform dynamic FEA were cont plate. The dynamic response of the model was investigated described briefly. The vector of F are presented by nor- for fixed excitation amplitude F = 100 N by varying the mal t and tangential t components of a contact traction 0 N T excitation frequency (Ω ). vector. The normal g and tangential g gap functions, N T In this respect, four different driving frequencies ( η=1/3, which describe the relative motions of contacting surfaces η=1/2, η=3/4, and η=2) were analysed so that the applied in the normal and tangential directions, can be formulated load can highlight dynamic phenomena that occur in the in Eqs. (14) and (15) [35, 36]: model. The driving frequency (η) is the ratio between the (︀ )︀ − + + excitation frequency (Ω ) and the first natural frequency g = x − x · n ¯ (14) on the intact model panel (f ), i.e., η = Ω / f . Comparing 0 0 (︀ )︀ dynamic responses between intact and debonded models +α − + + g = g a , g = x − x · a (15) T Tα Tα α were investigated in the different four different measure- 1 2 ment points at the upper faceplate. The dynamic responses where x is slave surface point and x (ξ , ξ ) is an corresponding to transverse displacement, velocity, and orthogonal projection on the master surface parameterized phase portrait were measured in both the inside debonding by x (α = 1, 2) and n is the unit vector normal to the master (N1 and N2) and outside debonding area (N3 and N4), as surface and a (α = 1, 2) is the tangent base vectors at the illustrated in Figure 2b. point x . The rate of the tangential gap function at this Further, the effect of debonding ratio due to transient point may be calculated in the geometrically linear case as: load will be analysed using ABAQUS/Explicit. To check the responses of debonded model, the clamped sandwich (︀ )︀ ˙ α + +α − + + ˙ β ¯ ¯ g ˙ = ξ a = g ˙ a with g ˙ = x ˙ − x ˙ · a = α ξ (16) models on both sides were subjected to the concentrated α α T T T αβ α α force at the centre point on the upper faceplate. The period + + + where a = a · a is the metric tensor at x . The αβ α of the force applied was set in such a way that it was shorter impenetrability criteria known as the Karush–Kuhn–Tucker than the analysis time (1/10 of the total analysis time). inequalities can therefore be formulated as follows [37]: {︃ F , 0 ≤ t ≤ t* F(t) = (11) t ≤ 0, g ≥ 0, and t g = 0 (17) N N N N 0, t > t* where t is the scalar quantity of the normal contact The amplitude of F = 50 kN and the time t = 0.05 0 * pressure, i.e., t = t n . The friction conditions that arise N N s. The dynamic responses corresponding to time histories in tangential directions can be expressed in Eq. (18). of displacement, velocity, acceleration, and displacement trendline within the upper faceplate of the model were compared between intact and debonded models. ‖t ‖ ≤ τ , ‖g ‖ ≥ 0, ‖t ‖ − τ ‖g ‖ = 0 (18) ( ) T crit T T crit T In this case, the damping ratio was estimated at 1% of the critical damping. Therefore, the coefficients of the where τ is a threshold of tangential contact traction crit Rayleigh damping matrix C were determined for each de- when a tangential slip occurs. The Coulomb friction law sired frequency range. The damping material in the system defines τ = µt where µ is the coefficient of friction. crit N was defined by the C matrix represented by Rayleigh damp- Using an analogy between plasticity and friction leads to ing with the following formula: the Eq. (18) along with loading–unloading conditions can be expressed in Eqs. (19) and (20). C = αM + βK (12) ∂ϕ (t ) t slip T T g ˙ = 𝛾 ˙ = 𝛾 ˙ (19) Where α dan β according to damping ratio can be cal- T ∂t ‖t ‖ T T culated as follows: ϕ ≤ 0, 𝛾 ˙ ≥ 0, and ϕ𝛾 ˙ = 0 (20) α βω ξ = + (13) 2ω 2 where 𝛾 ˙ is the slip rate parameter, and the potential function is expressed as ϕ(t ) = ‖t ‖ − µt . T T N 30 | Tuswan et al. In this case, between the surface in part 1 and part core material has been previously developed and tested [41] 2 within the debonded zone (Figure 2a), a contact for- based on the Det Norske Veritas-Lloyd’s Register (DNV-GL) mula was developed. The surface-to-surface contact ap- standard [42]. proach in terms of master and slave formulation in the The casting process manufactured the sandwich ma- ABAQUS/Explicit was utilised for modelling the debond- terial. The first process was cleaning and drying the cavity ing interface. The formulation of pure master-slave contact of faceplate surfaces and releasing them from surface rust. pair was applied due to high dissimilar mechanical con- The minimum surface roughness of 60 microns was cre- stants. Further, in this case, small-sliding displacement ated on the bonding surfaces before injection. The second kinematics to describe small oscillations of the interacting process was a mechanical blending of the core, and the mix- surfaces was assumed. The computational cost of using ture was injected into the steel mould. In the final stage, the a small sliding kinematic is less expensive than a finite blending core was cured, preferably up to 24 hours. After sliding formulation [38]. removal from the mould, the core was visually examined The constitutive behaviour of the contact in the normal from surface defects. For debonded models, the debond- direction between two adjacent surfaces in the debonded ing region was created by inserting the Teflon tape on the region was governed by the hard contact model. It is as- specimen mould. The debonding geometry and thickness sumed that the surfaces transmitted no contact pressure of four debonded models are created based on Table 1. The unless the nodes of the master surface contacted the slave assembly of the debonded model is presented in Figure 3. surface, and penetration was not allowed between them. In the case of tangential direction, the contact was devel- oped by the isotropic Coulomb friction model. The penalty parameter was automatically computed to provide some portion of reversible tangential motion specified by a user. Figure 3: Assembly of the debonded specimen. 3.2 Experimental setup and procedure Figure 2: a) Details of intermittent contact modelling in debonded region b) four different measurement points (N) in the upper face- EMA was conducted on the intact and 30% debonded spec- plate. imens to obtain natural frequencies experimentally. The time-domain response was obtained from the impact load input with the instrumentation setup illustrated in Figure 4. The clamped model of both sides was excited by the im- pact hammer at the centre of the plate. Simultaneously, 3 Experimental modal analysis the natural frequency was measured from the accelerom- (EMA) eter located at the centre of the upper faceplate in three different measurement points, see Figure 5. The range of 3.1 Material and specimen manufacturing accelerometer frequency was 1 – 2000 Hz. The accelerome- ters are affixed with the help of adhesive on the surface of The Sandwich panel is consisted of relatively thin but has the upper faceplate to fix the position [ 43]. Before taking high stiffness faceplates and a thick core with relatively measurements, it is necessary to measure the response of low-density soft material [39]. The compressive and ten- the accelerometer and impact hammer by calibrating it to sile stresses are mainly carried by the faceplates, while the determine the sensitivity. The time-domain response was core carries transverse shear stresses [40]. In this study, transformed using a Fast Fourier Transform (FFT) into the the steel faceplate and UPR/ clamshell core were used. The frequency domain. The peaks on the frequency domain are A numerical evaluation on nonlinear dynamic response of sandwich plates | 31 identified as the natural frequency of the models. The re- The accuracy and efficiency of the proposed numerical sult of the experimental natural frequency of intact and modelling are crucial to study and compare with experimen- debonded specimens will be compared with that of numer- tal data [46]. The results of natural frequencies between ical results. numerical methods and experimental vibration tests on the intact model and the 30% debonding model are pre- sented in Table 3. As seen in the result, it shows that the largest error is 11.22% in mode 1 in the intact model. The comparison results show a small error rate (< 15%), so that it is assumed that the proposed numerical modelling of the sandwich plate has good accuracy. Figure 4: Experimental vibration test setup. 4.2 Result of free vibration analysis To explore the issue of debonding problems on the modal characteristics, the undamaged and damaged models containing debonding were investigated in the first six 4 Numerical results and discussion modes. The model discretisation was similar to the pre- vious convergence study in Section 4.1. A total of v fi e mod- 4.1 Convergence study and validation test els containing different debonding ratios were analysed by ABAQUS/Standard. The natural frequency of the first To be confident in the result, mesh convergence was ini- six modes between intact and debonded models is illus- tially studied to get an accurate solution at the minimum trated in Figure 6a. As presented in Figure 6a, debonding element sizes required to obtain optimum CPU time [44, 45]. reduces the frequencies of the debonded model compared In this analysis, the general mesh size between 0.02 m and to the intact plate, and the frequency reduction is different 0.006 m was evaluated. The convergence study was per- for each mode. The result demonstrates that debonding formed by comparing the first three natural frequencies causes a significant frequency, especially 20% and 30% and mesh sizes of the intact model. All the geometries, ma- debonding ratio. However, the reduction of frequency in terial properties, and the applied boundary condition of the small debonding ratio (5%) is practically no change. the intact model are similar to Section 2.2. Figure 5 shows Figure 6b presents the normalised frequency (ω/ω ) the convergence study of the intact model in the first three as a function of the debonding ratio. The natural frequency modes analysed in the mesh sizes between 0.02 – 0.006 m. of damaged models (ω) has been normalised with respect It can be concluded that mesh element size 0.008 m with to the natural frequency of the intact model (ω ). It can be 7220 elements ensures sufficient accuracy with moderate mentioned that frequency changes are more rapid with the computational efficiency. increase of mode number. However, the natural frequency changes do not exhibit a defined trend as the mode number increases. Moreover, one can also be analysed that the rela- tively small debonding (D < 10%) does not almost change the frequency in the lower modes and only decreases the frequencies in the higher modes. It may be violated by the effect of the local thickening phenomenon [ 47], where the frequency of debonded model is higher than those of the intact model, for instance, in mode 3. It is also noted that mode 6 has the highest frequency reduction of all corre- sponding modes. Hence, its result can be summarised that the damage can influence natural frequencies and depend on the mode number. The frequency reduction increase due to a loss in stiffness and strength of the model is caused by Figure 5: Mesh convergence of the intact model in the first three initial discontinuity [11, 13, 17, 26]. modes. In general, it can be analysed that the natural frequen- cies and associated mode shapes of the debonded models 32 | Tuswan et al. Table 3: The natural frequency between experimental test (EMA) and numerical analysis (FEM). Intact model Debonded model (30%) Mode number FEM EMA Error (%) FEM EMA Error (%) Mode 1 986.05 875.4 11.22 875.79 784.2 10.46 Mode 2 1214.19 1095.2 9.80 1009.06 922.5 8.58 Mode 3 1862.67 1748.6 6.12 1104.06 1045.1 5.34 Figure 6: Comparison of a) natural frequency b) normalised natural frequency between intact and debonded model. shift from the initial model. The eigenmode is introduced responses. The amplitudes of displacement and velocity re- to understand the deformation of models. Figure 7 shows sponses of the debonded models are higher than that of the the eigenmodes between intact and 30% debonded models. intact model. The larger the ratio of debonding, the larger It can be concluded that local oscillations in the debonded the amplitude responses. It is caused by the stiffness reduc- region and the global modes of the entire sandwich lead to tion due to the existence of debonding, where the larger the changes in the mode shapes. debonding ratio, the larger the stiffness reduction. More- over, the displacement and velocity-time responses of the intact sandwich are periodic motions with a pure sinusoidal 4.3 Result of forced dynamic analysis that corresponds to its steady-state motion (see Figure 8a). In contrast, the response of debonded models has periodic Forced vibrations in terms of external harmonic excitation displacement motions with the sinusoidal waveform with a are of great importance as it raises in the practical field, modulated signal (see Figure 9b). The presence of contact which may cause severe damage [48]. The comparison of dy- motion in the area of debonding caused the modulated re- namic behaviour between the intact and debonded models sponse. It can be found that the higher the debonding ratio, was investigated using explicit dynamic analysis. Time his- the bigger the modulated signal of the responses. tory responses corresponding to transverse displacements, Analyzing in more detailed result, the comparison of velocities, and phase portrait are firstly compared at four phase portrait between intact and 5% debonding at dif- measurement points (see Figure 2b) at the upper faceplate ferent measurement points of the upper faceplates is pre- with a different driving frequency η=1/3, η=1/2, η=3/4 dan sented in Figure 10. Analysing these plots, one can see η=2. that the phase plots in the point inside debonding region The debonding detection with frequency ratio (η) of (N1 & N2) and the outside (N3 & N4) of the intact model 1/3 leading to the driving frequency of about 2333 rad/s is are elliptical forms. However, the phase plots of the 5% firstly analysed due to concentrated transverse force load. debonded model measured in the point outside debond- Fig 8a and b present displacement and velocity-time histo- ing (N3 & N4) are close to the elliptical form. In contrast, ries of the models with and without debonding at N1. As the phase plots in the inside debonding (N1 & N2) are sig- shown, the existence of the interfacial debonding in the nificantly disturbed from the elliptical form. Hence, it can model significantly influences all evaluated time history be assumed that the responses calculated in the outside A numerical evaluation on nonlinear dynamic response of sandwich plates | 33 Figure 8: Dynamic response of frequency ratio η=1/3 between intact and debonded models a) displacement b) velocity. Figure 9: Comparison of phase portrait calculated at different mea- surement points at frequency ratio η=1/3. Figure 7: Eigenmodes between intact and debonded model (30%). Figure 10 shows the comparison of displacement and velocity responses at frequency ratio½ (3500 rad/s) at N1. As can be seen, the same phenomenon occurs where the debonding region oscillate periodically with the driving presence of damage significantly influences time responses. frequency, but at the point in the debonding region, it ex- The displacement and velocity-time responses of the dam- periences a general periodic motion which is resulted by aged models are higher than the responses of the intact superposition between driving frequency and motion of ones. Moreover, the signal of displacement and velocity periodic contacts [38]. of the intact model is steady-state motion with excitation Further, the debonding diagnostics using phase por- frequency. Still, the displacement and transverse velocity trait between intact and small debonding with driving fre- response of debonded models is a periodic motion with quency 1/3 measured in the various points are depicted in a modulated signal that is much more complicated than Figure 9. From the result, it can be analysed that the dif- that in the previous driving frequency. As shown, the phase ference of phase plot between intact and debonded model plots of the debonded model are significantly disturbed by measured in the inside debonding area (N1 and N2) is more the elliptical form. The surfaces in the debonded zone run significant. The phase plot in the 5% debonding model into an aperiodic contact manner due to such interactions has a larger size than the intact one. Meanwhile, the phase between additional frequencies with the driving frequency. plots between the intact and debonded model in the outside The comparison of phase portrait with driving fre- debonding region have similar sizes, so the diagnostics of quency ½ is depicted in Figure 11. The phase portrait of small debonding using phase plot measured at the points both intact and 5% debonded models measured in the in- outside the debonding area (N3 and N4) is not sensitive. side debonding region (N1 & N2) shows a larger size than that in the outside debonding region. The difference of 34 | Tuswan et al. phase portrait between the intact and debonded model in vere modulated waveform than that in the previous driving the inside debonding region is more visible while the re- frequency, as one can see that the modulated waveform of sponse in the outside debonding region has a similar size. velocity response is more severe than that of displacement The debonding detection using a phase plot with driving response. frequency ½ measured at points outside the debonding Further, the comparison of phase portrait with driving area is not recommended. frequency 5250 rad/s is depicted in Figure 13. The compari- son of phase plots in the inside debonding shows a larger trajectory size than that in the outside debonding region. Moreover, the comparison of phase plots between intact and debonded models in the outside debonding region is more visible than the previous excitation frequency. Figure 10: Dynamic response of frequency ratio η=1/2 between intact and debonded models a) displacement b) velocity. Figure 12: Dynamic response of frequency ratio η=3/4 between intact and debonded models a) displacement b) velocity. Figure 11: Comparison of phase portrait calculated at different measurement points at frequency ratio η=1/2. The time responses between the intact and debonded models are further analysed for the frequency ratio η of Figure 13: Comparison of phase portrait calculated at various mea- ¾ with the same excitation amplitude. A comparison of surement points at frequency ratio η=3/4. time responses between intact and debonded models at the centre of debonding region (N1) is presented in Figure 12. It follows from these plots that the amplitudes of the The investigation is further conducted when the fre- debonded plate’s responses are larger than those in the quency ratio is increased up to η= 2, corresponding to driv- time signals of the intact plate. The bigger the debonding ing frequency 14.000 rad/s, which falls between mode 4 ratio, the larger the amplitude of time signals. Moreover, and mode 5. The dynamic response of debonded models the time responses of the intact model are steady-state mo- is changed noticeably from the previous result. Compar- tion, but the displacement and transverse velocity response ing the transverse displacements at N1 between intact and of debonded models is a periodic function with a more se- debonded models, the response of debonded models varies A numerical evaluation on nonlinear dynamic response of sandwich plates | 35 as a quasi-periodic function with large amplitudes, and a signal is severely distorted by the existence of the higher driving frequency, see Figure 14a. However, the velocity response of this point compared to the intact model is char- acterized by the uncommonly large amplitude, distorted waveform, and loss of periodicity. It happens due to irreg- ular contact interactions that occur between the surface in the damaged area [38]. To prove this phenomenon, the comparison of phase portrait at four various measurement points is drawn in Figure 15. The phase trajectory in the debonded models at all measurement points is distorted from the elliptical form (see Figure 15). Thus, the phase tra- jectory in the inside debonded region (N1 & N2) is irregular and includes frequencies excited by contact surfaces. More- over, the dynamic behaviour of the point outside debonding Figure 15: Comparison of phase portrait calculated at different region (N3 & N4) is assumed a quasi-periodic motion. measurement points at frequency ratio η=2. Regarding debonding detection, a comparison of phase portraits at the different debonding ratios with frequency and tangential (CSHEAR1 and CSHEAR2) contact stress at ratio η=2 is presented in Figure 15. It can be found from the time similar with Figure 16a are illustrated in Figure the result that the higher the debonding ratio, the differ- 16c-d. It can be shown when fully or partially closed con- ence of phase portrait of the debonded model compared tacts, the distribution of normal and shear contact forces with intact one is more visible. It signifies that the higher appear at the debonded region to fulfilling stress continuity the debonding ratio, the larger the amplitude of displace- requirements. Otherwise, when the contact is fully open, ment and velocity. It is caused by higher stiffness loss in no normal and shear contact forces appear in the debonded the debonding model with a high ratio. region and stress-free contact conditions are realised [33]. As a consequence of transition behaviour between the open and closed forms during oscillations, various nor- mal and shear contact forces distributions during analysis time exist in the debonding region. The comparison of nor- mal contact stress and contact shear stress evolution in debonded models is shown in Figure 17. As shown, the debonded models have higher normal and shear contact forces than the intact model. It can be analysed that with increasing debonding ratio, the magnitude of the contact forces increases. The evolution of the contact forces be- Figure 14: Dynamic response of frequency ratio η=2 between intact and debonded models a) displacement b) velocity. tween intact and debonded models change with analysis time so that contact behaviour changes towards debond- ing ratio variation. Additionally, the magnitudes of normal As the debonded models are excited by the concen- contact forces are larger than the shear forces, as seen in trated harmonic force, the models begin to oscillate, so the Figure 17. It occurs due to contact surfaces in the debond- debonded contact behaviour must be investigated in de- ing region interact with each other in the normal direction. tail. Several deformation contours of vibrating models at However, sliding also occurs due to the rotation and lat- different moments of time corresponding to its ’breathing’ eral movement of the faceplate in the debonded surface. are presented in Figure 16a, i.e., fully closed at time 0.003 Thus, the contact forces arising in the debonding region s and 0.017 s, partially open at 0.0377 s, and fully open are influenced by several factors, including inertial forces, at 0.0452 s. As shown during oscillations, the contact of the local deformation of the debonded region, and global debonded surfaces oscillates from close to open status and deformation of the model caused by external excitation or vice versa during time analysis. To certainly understand a combination of all factors [33]. the contact behaviour of 10% debonding ratio between the debonded surfaces, the contours of both normal (CPRESS) 36 | Tuswan et al. Figure 16: a) Deformed shapes of the debonded zone b) CPRESS c) CSHEAR1 d) CSHEAR2 at instants of time a) 0.0003 s, b) 0.017 s, c) 0.0377 s, d) 0.0452 s. problems due to concentrated impulse load was performed with ABAQUS/Explicit. First, time history responses corre- sponding to transverse displacements, velocities, accelera- tions, and displacement trendline at point N1 (see Figure 2b) were compared. Figure 18 indicates the comparison of transient responses computed at the centre of debonding in the upper faceplate of both intact and debonded models. The intermittent contact causes a significant difference in transient responses in both waveforms and signal proper- Figure 17: Comparison of a) normal contact force b) contact shear ties between intact and debonded models. As illustrated in force of debonded models. Figure 18a, the waveform of the debonded model is more disturbed than that of the intact model because the dynam- 4.4 Result of transient dynamic analysis ics of the debonded surfaces are modelled by intermittent contact. Further, the existence of debonding increases the To gain a deeper investigation of debonding diagnostic, amplitude of the transient responses. The bigger the ratio the comparison of the dynamic behaviour of debonding A numerical evaluation on nonlinear dynamic response of sandwich plates | 37 of debonding, the higher the amplitude of the transient responses, as clearly seen in the displacement trendlines in Figure 18d. It occurs due to the result of superimposing between vibrational waves caused by external loads and those generated by contact interactions [25]. So, the effect of debonding due to transient loading of the sandwich models is significant. The evolution of contact forces due to impulse load- ing in the time interval 0 - 0.002 s is illustrated in Fig- ure 19, which describes the comparison of the normal and shear contact forces with various debonding ratios. As seen, debonding causes an increase both in the normal and shear contact forces. The magnitude of the normal and shear con- tact forces increases with the increase of debonding ratio. Moreover, the magnitude of the normal contact forces is higher than the shear forces due to the interaction of con- Figure 18: Comparison of transient response between intact and tacting surfaces in the debonded region occurs in the nor- debonded models a) displacement b) velocity c) acceleration d) mal direction. displacement trendline. To present a better visualisation of both normal and shear contact force distribution due to impulse load, the visualisation of contact at instants of time of 10% debonded model is presented in Figure 20. The contact forces arising between the faceplate and the core si passing from closing to opening and vice versa. Figure 20a shows the visualisa- tion of normal and shear contact forces when the model is fully closed at time 0.00045 s. Moreover, Figures 21b-c shows normal and shear contact force distributions when the debonded model is partially closed. As shown, the nor- Figure 19: Transient response of debonded models a) normal con- mal contact traction due to the contact–impact motion of tact force b) contact shear force. the debonded surfaces changes the sliding mechanism de- scribed by the shear contact stress distributions. debonded region along with the global mode changes in the mode shapes of the debonded model. Further, the numerical result of dynamic responses, 5 Conclusions both harmonic and impulse loading, shows that the debonding can be identified by comparing time responses The damage detection of the hybrid sandwich using free between intact and debonded models. The dynamic re- and forced vibration was studied in this work. The influence sponses analysed in the four different driving frequencies of dynamic behaviour on the debonding ratio using FE soft- of the debonded models due to harmonic excitation de- ware ABAQUS can be drawn. The free vibration using the pend on the intermittent contact. A variety of motions such Lanczos method was used to extract eigenfrequencies and as periodic, quasi-periodic, and irregular on the time re- eigenmodes. To obtain an understanding of the dynamic sponses and phase portraits in the debonded region are behaviour of debonded models, both the transient dynamic found. The dynamic responses of the debonded models are analysis and the forced dynamic analysis were analysed higher than that of the intact model. In terms of transient using ABAQUS/Explicit. dynamic analysis, numerical results show that the presence The result shows that debonding causes natural fre- of debonding affects the short-time response of debonded quency reduction and alters the mode shapes as well. It models. Using transient time responses, the magnitude of can be determined that the natural frequency decreases the time responses of the debonded models is higher than with the increase of the debonding ratio. The higher modes the intact model. To sum up, using the change of responses are found to be more sensitive to debonding existence. It resulted from free and forced vibration analyses of both un- can be found that the presence of local oscillations in the 38 | Tuswan et al. [5] Sandeep SH, Srinivasa CV. Hybrid Sandwich Panels: A Review. Int J Appl Mech Eng. 2020;25(3):64-85. [6] Chen Y, Hou S, Fu K, Han X, Ye L. Low-velocity impact response of composite sandwich structures: modelling and experiment. Compos Struct. 2017;168:322-34. [7] Fatt MSH, Sirivolu D. Marine composite sandwich plates under air and water blasts. Mar Struct. 2017;56:163-85. [8] Huang SJ. An analytical method for calculating the stress and strain in adhesive layers in sandwich beams. Compos Struct. 2003;60(1):105-14. [9] Tsai SN, Taylor AC. Vibration behaviours of single/multi- debonded curved composite sandwich structures. Compos Struct. 2019;226:1-13. [10] Sahoo S. Free vibration behaviour of laminated composite stiff- ened elliptic parabolic shell panel with cutout. Curved and Layer Struct. 2015;2:162-82. [11] Tuswan, Zubaydi A, Piscesa B, Ismail A, Ilham MF. Free vibration analysis of interfacial debonded sandwich of ferry ro-ro’s stern ramp door. Proc Struct Int. 2020;27C:22-9. Figure 20: Distributions of normal contact (CPRESS) and shear [12] Zhao B, Xu Z, Kan X, Zhong J, Guo T. Structural damage detection contact (CSHEAR1, and CSHEAR2) of debonded model at instants of by using single natural frequency and the corresponding mode time a) t= 0.00045 s b) t= 0.00655 s c) t= 0.0358 s. shape. Shock Vib. 2016;2016:1-8. [13] Ismail A, Zubaydi A, Piscesa B, Ariesta RC, Tuswan. Vibration- based damage identification for ship sandwich plate using finite damaged and damaged models, the debonding diagnostics element method. Open Eng. 2020;10:744-52. can be implemented. [14] Kaveh A, Zolghadr A. An improved CSS for damage detection of truss structures using changes in natural frequencies and mode Funding information: The research has received finan- shapes. Adv Eng Softw. 2015;80:93-100. [15] Prabowo AR, Tuswan T, Ridwan R. Advanced Development of cial support from The Ministry of Education, Culture, Re- Sensors’ Roles in Maritime-Based Industry and Research: From search, and Technology of The Republic of Indonesia under Field Monitoring to High-Risk Phenomenon Measurement. Appl "Penelitian Dasar Unggulan Perguruan Tinggi" research Sci. 2021;11(9):3954. scheme with contract number 3/E1/KP.PTNBH/2021 and [16] Burlayenko VN, Sadowski T. Nonlinear dynamic analysis of har- 895/PKS/ITS/2021. monically excited debonded sandwich plates using finite ele- ment modelling. Compos Struct. 2014;108:354-66. [17] Tuswan, Zubaydi A, Piscesa B, Ismail A. Dynamic characteris- Author contributions: All authors have accepted responsi- tic of partially debonded sandwich of ferry ro-ro’s car deck: a bility for the entire content of this manuscript and approved numerical modelling. Open Eng. 2020;10:424-33. its submission. [18] Burlayenko VN, Sadowski T. 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A numerical evaluation on nonlinear dynamic response of sandwich plates with partially rectangular skin/core debonding

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Abstract

Curved and Layer. Struct. 2022; 9:25–39 Research Article Tuswan Tuswan, Achmad Zubaydi*, Bambang Piscesa, Abdi Ismail, Rizky Chandra Ariesta, and Aditya Rio Prabowo A numerical evaluation on nonlinear dynamic response of sandwich plates with partially rectangular skin/core debonding https://doi.org/10.1515/cls-2022-0003 1 Introduction Received May 20, 2021; accepted Aug 15, 2021 Abstract: As one of the most dangerous defects in the sand- As one of the unique thin-walled structures, sandwich ma- wich panel, debonding could significantly degrade load car- terial is broadly used in shipbuilding industries due to com- rying capacity and affect dynamic behaviour. The present bining high performance with a lightweight design. Sand- work dealt with debonding detection of the rectangular wich panels are profitable during construction and repa- clamped hybrid sandwich plate by using ABAQUS software. ration because they can reduce the weight by eliminating The influence of various damage ratios on the linear and the stiffeners compared to the stiffened plate structures. nonlinear dynamic responses has been studied. The finite However, weight reduction often leads to an increase in vi- element model was initially validated by comparing the bration problems occurring under manufacture and service modal response with the experimental test. Rectangular conditions. Increased vibration can cause a variety of struc- debonding was detected by comparing dynamic responses tural damage. Face sheet/core debonding is the serious of free and forced vibrations between intact and debonded damage that can reduce the stiffness [ 1, 2, 3] and threaten models. A wide range of driving frequency excitation cor- the safety and service life [4]. responding to transient and harmonic concentrated loads The difficulty in controlling the appropriate bonding was implemented to highlight nonlinear behaviour in the during the manufacturing stage is the main problem [5, 6], intermittent contact in the debonded models. The results in which the debonding location is often invisible in the showed that debonding existence contributed to the nat- interface layer between two adjacent surfaces [7]. Debond- ural frequency reduction and modes shape change. The ing has resulted from manufacturing failure when a small numerical results revealed that debonding affected both region’s adhesive layers have not been sufficiently bonded. the steady-state and impulse responses of the debonded Moreover, this early-stage damage may propagate to create models. Using the obtained responses, it was detected that larger debonding, altering the vibration characteristic and the contact in the debonded region altered the dynamic behaviour. The debonding can lead to frequency reduction, global response of the debonded models. The finding pro- leading to structural failure in the lower mode [8, 9]. Since vided the potential debonding diagnostic in ship structure bonding quality during manufacture determines the per- using vibration-based structural health monitoring. formance and integrity of the structure, an effective and efficient debonding detection in the early stage is necessary Keywords: debonding detection, hybrid sandwich, finite el- to assess structural health and performance. ement analysis, dynamic contact, forced vibration, marine. Tuswan Tuswan: Department of Naval Architecture, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Abdi Ismail: Department of Naval Architecture, Institut Teknologi 60111, Indonesia; Department of Naval Architecture, Faculty of Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya 60111, In- Engineering, Universitas Diponegoro, Semarang 50275, Indonesia donesia; Ship Machinery Study Programme, Faculty of Vocational *Corresponding Author: Achmad Zubaydi: Department of Naval Studies, Indonesia Defense University, Belu, Indonesia Architecture, Institut Teknologi Sepuluh Nopember, Kampus ITS Rizky Chandra Ariesta: Department of Naval Architecture, Institut Sukolilo, Surabaya 60111, Indonesia, E-mail: zubaydi@na.its.ac.id Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Bambang Piscesa: Department of Civil Engineering, Institut 60111, Indonesia Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya Aditya Rio Prabowo: Department of Mechanical Engineering, 60111, Indonesia Universitas Sebelas Maret, Surakarta 57126, Indonesia Open Access. © 2022 Tuswan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 26 | Tuswan et al. Assessment of the structure using vibration-based To address this issue, the numerical damage assess- structural health monitoring (VSHM) is widely used by ex- ment of a hybrid sandwich containing a steel faceplate and amining the changes in measured vibration response [10]. polymer core was developed in this study. The essential The basic principle is comparing the change of dynamic contribution of this work is to study the nonlinear effect of characteristics between intact and debonded structures debonding contact between the faceplate and core due to such as natural frequencies [11, 12, 13], mode shapes [14], free and forced vibrations. An initial study was performed frequency response function (FRF) curvatures [15], modal by comparing the numerical model using ABAQUS software damping, and time or frequency domain data [16, 17]. Re- [27] with the experimental test to check the correctness of cently reported VSHM studies based on linear and non- debonding modelling. The three-dimensional model of the linear vibration analysis in the composite sandwich were clamped sandwich plate with rectangular shape debonded comprehensively presented. Burlayenko and Sadowski in- zone was firstly conducted free vibration analysis to extract vestigated the influence of debonding ratio, location, and natural frequency. The general dynamic based on explicit type on free vibration response [18]. It showed that the ra- time-stepping procedure due to transient and harmonic tio, location, and type of debonding influence the vibration load was then undertaken to check the effect of dynamic behaviour of the composite sandwich. Moreover, the pre- response due to contact dynamic under various debonding vious studies found that debonding is easily detected in ratios. The structure of this paper is constructed in sections higher mode than lower mode [19, 20, 21], but it is chal- as follows. Section 1 presents the introduction and objec- lenging to get higher modes in the practical experiment tive of the study. The theoretical backgrounds of model dis- due to restriction of excitation magnitude. Consequently, cretization, contact modelling, and finite element analysis nonlinear vibration utilising time-domain data of structural are reported in Section 2. Section 3 offers the Experimental responses of the composite sandwich is more popular. Com- test preparation and procedure, and Section 4 reports the pared with modal analysis, investigating the damage based result of the free and forced vibration test. In the last, a few on time-domain dynamic responses is easier [22], making concluding remark and summary is provided in Section 5. the result more reliable [23]. Several studies have been re- viewed in detail. Burlayenko and Sadowski have created an explicit 3D finite element model (FEM) to examine the 2 Numerical modelling of sandwich transient dynamic response of an impacted sandwich panel with debonding [24]. Moreover, damage detection using an panels implicit dynamic has been reviewed in [25] to investigate the responses of the debonded sandwich subjected to har- 2.1 Aspects of model discretization monic forces. A recent development study of debonding identification based on the different types of analysis in In the aspect of model discretization, there are several the composite sandwich is comprehensively reviewed by schemes available for modelling sandwich and lami- Burlayenko and Sadowski [26]. nate structure. They are classified into equivalent single Although a majority of previous investigations have layer (ESL), layerwise (LW), and continuum-based three- been reviewed to explore the dynamic behaviour of the dimensional (3-D) elasticity theories. ESL theory is simply composite sandwich, there is limited study in the com- efficient for general analysis with computationally efficient, prehensive assessment of dynamic behaviour and nonlin- but interlaminar stress and transverse compressibility are ear characteristic of hybrid sandwich panels containing ignored [26]. In order to alleviate the shear-locking phe- metal/polymer combinations as part of ship structure due nomenon in shell/plate, mixed interpolation tensorial com- to debonding problems. The issue is crucial since the me- ponent technique (MITC) technique with Carrera Unified chanical properties of two adjacent layers of hybrid sand- Formulation (CUF) framework is proposed by Kumar et al. wich have higher dissimilar properties, leading to initiate [28] to perform modal analysis of delaminated composite a debonding problem more severe. There is an important shell structures. Using this strategy, the shell element has issue investigating the response characteristic of debond- 9-nodes and allows displacement distribution along with ing problems due to the free and forced vibration of this the thickness of the multilayered shell. The implementation type of sandwich under different loading scenarios. More- of refined and advanced delamination model of CUF shell over, there is a limited study concerning the validity and model using various theories and different ordered theories correctness of proposed debonding modelling of hybrid was studied. Moreover, using 2D MITC finite element based sandwiches with the response obtained from the experi- on the CUF framework, Legendre-like polynomial expan- mental test. A numerical evaluation on nonlinear dynamic response of sandwich plates | 27 Table 1: Sandwich configuration and debonding geometry. sions are introduced by Pagani et al. [29] to implement ESL, LW, and variable kinematics theories. Models h −h −h D = D T (mm) t c 1 2 A nonlinear pattern in the height of the core during b (mm) (mm) deformation/ core transverse compressibility should be accounted for modelling modern sandwich panels. This Intact (0%) 4-20-4 0 0 makes it an obstacle to handle this modelling using ESL 2D 5% debonding ratio 4-20-4 67.08 0.00004 elements. In this regard, quasi three-dimensional model 10% debonding ratio 4-20-4 94.90 0.00004 or layerwise (LW) scheme is preferable due to accommo- 20% debonding ratio 4-20-4 134.2 0.00004 dating core compressibility with effective computational 30% debonding ratio 4-20-4 164.3 0.00004 cost. Solid-brick elements for the core and continuum shell elements for faceplates were used in this case. Table 2: Material constants of faceplates and core sandwich mate- rial. 2.2 Sandwich configuration and debonding Parameters Faceplate Core material geometry −3 ρ (kg m ) 7850 1465 11 9 E (Pa) 2.1 x 10 4.4 x 10 Ferry Ro-Ro 6300 Gross Tonnage (GT) stern ramp door was 10 9 G (Pa) 8.07 x 10 1.69 x 10 used as a reference model, as previously described in [21]. σ (MPa) 275 24.8 The sandwich panel geometry of the stern ramp door is σ (MPa) 292 24.8 illustrated in detail in Figure 1. The steel faceplate and υ 0.3 0.3 resin/clamshell core material with dimensions 300 mm in length and width are modelled. The configuration of the sandwich panel is 4 mm in the upper and lower faceplate trated in Figure 1. The rectangular debonded region at the thickness (h = h ) and 20 mm in core material thickness middle span was presented by a small artificial gap ( t = 0.1% t b (h ). The sandwich thickness configuration and debonding c x h ) between the upper faceplate and the core. No treat- geometry are presented in Table 1. The material constants ment of either geometrical entities or material constants is of the sandwich panel used for debonding detection are applied in the debonded area to guarantee a physically real shown in Table 2 and previously reported in [11, 21]. case condition. The ratio of debonding was represented by a damage parameter (D = A /A x 100) expressing the D T percentage of the region of debonding (A ) to the whole region of the interface layer (A = D x D ). In rectangular T 2 debonding, the length and width of the debonding have similar values (D = D ). The model was modelled by a solid/shell layerwise (LW) approach in the model discreti- sation, as previously recommended [30, 31]. In terms of the element type, the faceplate was modelled using a con- tinuum shell element (SC8R), and the core material was modelled using a solid element (3CD8R). Meanwhile, the clamped-free-clamped-free (CFCF) boundary condition, as illustrated in Figure 1, was applied. 2.3 Theoretical basis of modal analysis Free vibration analysis was used to investigate structural dynamic behaviour. In free vibration, there was no external Figure 1: Sandwich geometry with debonding [21]. load causing the motion. The motion has resulted from initial conditions, such as an initial displacement from an A total of v fi e models was created, as presented in Table equilibrium position. The equation of motion for forced 1. Debonding was modelled using the FEM by single rectan- vibration can be assumed in the form [32]. gular damage at the centre of the top adhesive layer, as illus- 28 | Tuswan et al. Then, the accelerations, velocities, and displacements in ¨ ˙ ¨ ˙ [M][U ] + [C][U ] + [K][U] = [F] (1) this time increment are expressed by U , U , U , re- i+1 i+1 i+1 spectively. Therefore, Eq. (5) at t = t + ∆t is expressed i+1 i i+1 where M, C, and K are mass, damping, and stiffness in the form: ˙ ¨ matrices. U is nodal points displacements, and U and U are their time derivatives referring to nodal velocities and ext cont ¨ ˙ MU + CU + KU = F − F (6) i+1 i+1 i+1 i+1 i+1 acceleration. For free vibration analysis, the damping of the structure can be neglected. Thus, the equation can be U = U and U = V are the assumption given at the ini- 0 0 written in Eq. (2). tial boundary conditions. The explicit algorithms determine Eq. (6) using cen- [M][U ] + [K][U] (2) tral difference formulation without iterations and tangent stiffness matrix by explicitly advancing the kinematic state where Eq. (1) shortens to an eigenvalue problem. The known from a previous increment to the next one [34]. Ini- governing equation for free vibration analysis can be writ- tially, the accelerations at the beginning of each increment ten in Eq. (3). ∆t are calculated. Then, Eq. (6) is rewritten: i+1 (︁ )︁ [K] − λ [M] ϕ = 0 (3) [ ] i i −1 ext cont U = M F − F (7) i i where λ is eigenvalue and ϕ is eigenvector correspond- i i ext where F is the vector of the external nodal forces at ing to the eigenvalues. The eigenvalues associated with the int time t and F is the sum of nodal internal F =KU , damp- natural frequency can be written in Eq. (4), as follows: , i i i damp cont ing F =CU and contact force F which are updated √︀ during the previous time increment ∆t . f = λ /2π (4) i i Then, the accelerations computed at t are used to ad- The analysis was conducted to present an insight into vance the velocity solution to t + ∆ t and the displace- i i+1 the oscillation response between intact and face sheet/core ment solution to t +∆ t as follows: i i+1 debonded models. The free vibration analysis in the first six modes using ABAQUS/Standard was carried using the Lanc- ∆t + ∆t i+1 i ˙ ˙ ¨ U = U 1 + U (8) i+1 i i− zos method to extract eigenvalues. The modal responses were used as a damage index to detect the debonding prob- ˙ ˙ U = U + ∆t U 1 (9) lem. i+1 i i+1 i+ The initial half-step lagging velocity U is calculated -1/2 from the initial velocity assuming the initial acceleration. 2.4 Descriptions of explicit time-stepping The computational time in the explicit analysis linearly procedure increases with problem capacity, and the explicit procedure is only stable if the time increment in Eq. (6) is smaller The explicit procedure accomplishes a large number of time than the stability limit of the central difference operator increments. A set of a nonlinear differential equation of [34]. An approximation to the stability limit (∆t ) can be crit motion with the assumption of linear elasticity and small formulated as follows: deformation is written as [33]: ∆t ≈ (10) crit ext cont ¨ ˙ MU (t) + CU (t) + KU(t) = F (U(t)) − F (U(t)) (5) Where L is the smallest element dimension in the mesh and c is dilatational wave speed. where Ü(t), (t) and U(t) is the global vectors of unknown To gain a deeper investigation of debonding diagnostic, accelerations, velocities, and displacements, respectively, ext the comparison of dynamic behaviour due to debonding at each instant of time. F is the vector of the external cont problem was investigated using general dynamic analysis. force, and F is the vector of contact forces. Then, M, C, First, after analysing the modal analysis of the debonded and K are the global mass, damping, and stiffness matri- model for evaluating its natural frequencies, the FE mod- ces serially. Eq. (5) is discretised in a time domain. The els were subjected to the concentrated harmonic load at time interval [0, T] is divided into subintervals in the form L−1 the upper faceplate. The investigations were developed in [0, T ] = ∪ t , t , where t < t , and t = 0, t = 0. [ ] i i+1 i i+1 0 L i=0 ABAQUS/Explicit that incorporated the explicit integration So, Eq. (5) can only be found in a finite number of time steps. For example, the time increment is ∆t = t − t . i+1 i+1 i A numerical evaluation on nonlinear dynamic response of sandwich plates | 29 algorithm for a general dynamic analysis with contact de- 2.5 Descriptions of contact formulation scribed in Section 2.5. The models with rectangular debond- ing were loaded with a concentrated harmonic force F(t) = The fundamental features of the contact formulation em- F0 sin Ω t subjected at the central point of the upper face- ployed in the current study to perform dynamic FEA were cont plate. The dynamic response of the model was investigated described briefly. The vector of F are presented by nor- for fixed excitation amplitude F = 100 N by varying the mal t and tangential t components of a contact traction 0 N T excitation frequency (Ω ). vector. The normal g and tangential g gap functions, N T In this respect, four different driving frequencies ( η=1/3, which describe the relative motions of contacting surfaces η=1/2, η=3/4, and η=2) were analysed so that the applied in the normal and tangential directions, can be formulated load can highlight dynamic phenomena that occur in the in Eqs. (14) and (15) [35, 36]: model. The driving frequency (η) is the ratio between the (︀ )︀ − + + excitation frequency (Ω ) and the first natural frequency g = x − x · n ¯ (14) on the intact model panel (f ), i.e., η = Ω / f . Comparing 0 0 (︀ )︀ dynamic responses between intact and debonded models +α − + + g = g a , g = x − x · a (15) T Tα Tα α were investigated in the different four different measure- 1 2 ment points at the upper faceplate. The dynamic responses where x is slave surface point and x (ξ , ξ ) is an corresponding to transverse displacement, velocity, and orthogonal projection on the master surface parameterized phase portrait were measured in both the inside debonding by x (α = 1, 2) and n is the unit vector normal to the master (N1 and N2) and outside debonding area (N3 and N4), as surface and a (α = 1, 2) is the tangent base vectors at the illustrated in Figure 2b. point x . The rate of the tangential gap function at this Further, the effect of debonding ratio due to transient point may be calculated in the geometrically linear case as: load will be analysed using ABAQUS/Explicit. To check the responses of debonded model, the clamped sandwich (︀ )︀ ˙ α + +α − + + ˙ β ¯ ¯ g ˙ = ξ a = g ˙ a with g ˙ = x ˙ − x ˙ · a = α ξ (16) models on both sides were subjected to the concentrated α α T T T αβ α α force at the centre point on the upper faceplate. The period + + + where a = a · a is the metric tensor at x . The αβ α of the force applied was set in such a way that it was shorter impenetrability criteria known as the Karush–Kuhn–Tucker than the analysis time (1/10 of the total analysis time). inequalities can therefore be formulated as follows [37]: {︃ F , 0 ≤ t ≤ t* F(t) = (11) t ≤ 0, g ≥ 0, and t g = 0 (17) N N N N 0, t > t* where t is the scalar quantity of the normal contact The amplitude of F = 50 kN and the time t = 0.05 0 * pressure, i.e., t = t n . The friction conditions that arise N N s. The dynamic responses corresponding to time histories in tangential directions can be expressed in Eq. (18). of displacement, velocity, acceleration, and displacement trendline within the upper faceplate of the model were compared between intact and debonded models. ‖t ‖ ≤ τ , ‖g ‖ ≥ 0, ‖t ‖ − τ ‖g ‖ = 0 (18) ( ) T crit T T crit T In this case, the damping ratio was estimated at 1% of the critical damping. Therefore, the coefficients of the where τ is a threshold of tangential contact traction crit Rayleigh damping matrix C were determined for each de- when a tangential slip occurs. The Coulomb friction law sired frequency range. The damping material in the system defines τ = µt where µ is the coefficient of friction. crit N was defined by the C matrix represented by Rayleigh damp- Using an analogy between plasticity and friction leads to ing with the following formula: the Eq. (18) along with loading–unloading conditions can be expressed in Eqs. (19) and (20). C = αM + βK (12) ∂ϕ (t ) t slip T T g ˙ = 𝛾 ˙ = 𝛾 ˙ (19) Where α dan β according to damping ratio can be cal- T ∂t ‖t ‖ T T culated as follows: ϕ ≤ 0, 𝛾 ˙ ≥ 0, and ϕ𝛾 ˙ = 0 (20) α βω ξ = + (13) 2ω 2 where 𝛾 ˙ is the slip rate parameter, and the potential function is expressed as ϕ(t ) = ‖t ‖ − µt . T T N 30 | Tuswan et al. In this case, between the surface in part 1 and part core material has been previously developed and tested [41] 2 within the debonded zone (Figure 2a), a contact for- based on the Det Norske Veritas-Lloyd’s Register (DNV-GL) mula was developed. The surface-to-surface contact ap- standard [42]. proach in terms of master and slave formulation in the The casting process manufactured the sandwich ma- ABAQUS/Explicit was utilised for modelling the debond- terial. The first process was cleaning and drying the cavity ing interface. The formulation of pure master-slave contact of faceplate surfaces and releasing them from surface rust. pair was applied due to high dissimilar mechanical con- The minimum surface roughness of 60 microns was cre- stants. Further, in this case, small-sliding displacement ated on the bonding surfaces before injection. The second kinematics to describe small oscillations of the interacting process was a mechanical blending of the core, and the mix- surfaces was assumed. The computational cost of using ture was injected into the steel mould. In the final stage, the a small sliding kinematic is less expensive than a finite blending core was cured, preferably up to 24 hours. After sliding formulation [38]. removal from the mould, the core was visually examined The constitutive behaviour of the contact in the normal from surface defects. For debonded models, the debond- direction between two adjacent surfaces in the debonded ing region was created by inserting the Teflon tape on the region was governed by the hard contact model. It is as- specimen mould. The debonding geometry and thickness sumed that the surfaces transmitted no contact pressure of four debonded models are created based on Table 1. The unless the nodes of the master surface contacted the slave assembly of the debonded model is presented in Figure 3. surface, and penetration was not allowed between them. In the case of tangential direction, the contact was devel- oped by the isotropic Coulomb friction model. The penalty parameter was automatically computed to provide some portion of reversible tangential motion specified by a user. Figure 3: Assembly of the debonded specimen. 3.2 Experimental setup and procedure Figure 2: a) Details of intermittent contact modelling in debonded region b) four different measurement points (N) in the upper face- EMA was conducted on the intact and 30% debonded spec- plate. imens to obtain natural frequencies experimentally. The time-domain response was obtained from the impact load input with the instrumentation setup illustrated in Figure 4. The clamped model of both sides was excited by the im- pact hammer at the centre of the plate. Simultaneously, 3 Experimental modal analysis the natural frequency was measured from the accelerom- (EMA) eter located at the centre of the upper faceplate in three different measurement points, see Figure 5. The range of 3.1 Material and specimen manufacturing accelerometer frequency was 1 – 2000 Hz. The accelerome- ters are affixed with the help of adhesive on the surface of The Sandwich panel is consisted of relatively thin but has the upper faceplate to fix the position [ 43]. Before taking high stiffness faceplates and a thick core with relatively measurements, it is necessary to measure the response of low-density soft material [39]. The compressive and ten- the accelerometer and impact hammer by calibrating it to sile stresses are mainly carried by the faceplates, while the determine the sensitivity. The time-domain response was core carries transverse shear stresses [40]. In this study, transformed using a Fast Fourier Transform (FFT) into the the steel faceplate and UPR/ clamshell core were used. The frequency domain. The peaks on the frequency domain are A numerical evaluation on nonlinear dynamic response of sandwich plates | 31 identified as the natural frequency of the models. The re- The accuracy and efficiency of the proposed numerical sult of the experimental natural frequency of intact and modelling are crucial to study and compare with experimen- debonded specimens will be compared with that of numer- tal data [46]. The results of natural frequencies between ical results. numerical methods and experimental vibration tests on the intact model and the 30% debonding model are pre- sented in Table 3. As seen in the result, it shows that the largest error is 11.22% in mode 1 in the intact model. The comparison results show a small error rate (< 15%), so that it is assumed that the proposed numerical modelling of the sandwich plate has good accuracy. Figure 4: Experimental vibration test setup. 4.2 Result of free vibration analysis To explore the issue of debonding problems on the modal characteristics, the undamaged and damaged models containing debonding were investigated in the first six 4 Numerical results and discussion modes. The model discretisation was similar to the pre- vious convergence study in Section 4.1. A total of v fi e mod- 4.1 Convergence study and validation test els containing different debonding ratios were analysed by ABAQUS/Standard. The natural frequency of the first To be confident in the result, mesh convergence was ini- six modes between intact and debonded models is illus- tially studied to get an accurate solution at the minimum trated in Figure 6a. As presented in Figure 6a, debonding element sizes required to obtain optimum CPU time [44, 45]. reduces the frequencies of the debonded model compared In this analysis, the general mesh size between 0.02 m and to the intact plate, and the frequency reduction is different 0.006 m was evaluated. The convergence study was per- for each mode. The result demonstrates that debonding formed by comparing the first three natural frequencies causes a significant frequency, especially 20% and 30% and mesh sizes of the intact model. All the geometries, ma- debonding ratio. However, the reduction of frequency in terial properties, and the applied boundary condition of the small debonding ratio (5%) is practically no change. the intact model are similar to Section 2.2. Figure 5 shows Figure 6b presents the normalised frequency (ω/ω ) the convergence study of the intact model in the first three as a function of the debonding ratio. The natural frequency modes analysed in the mesh sizes between 0.02 – 0.006 m. of damaged models (ω) has been normalised with respect It can be concluded that mesh element size 0.008 m with to the natural frequency of the intact model (ω ). It can be 7220 elements ensures sufficient accuracy with moderate mentioned that frequency changes are more rapid with the computational efficiency. increase of mode number. However, the natural frequency changes do not exhibit a defined trend as the mode number increases. Moreover, one can also be analysed that the rela- tively small debonding (D < 10%) does not almost change the frequency in the lower modes and only decreases the frequencies in the higher modes. It may be violated by the effect of the local thickening phenomenon [ 47], where the frequency of debonded model is higher than those of the intact model, for instance, in mode 3. It is also noted that mode 6 has the highest frequency reduction of all corre- sponding modes. Hence, its result can be summarised that the damage can influence natural frequencies and depend on the mode number. The frequency reduction increase due to a loss in stiffness and strength of the model is caused by Figure 5: Mesh convergence of the intact model in the first three initial discontinuity [11, 13, 17, 26]. modes. In general, it can be analysed that the natural frequen- cies and associated mode shapes of the debonded models 32 | Tuswan et al. Table 3: The natural frequency between experimental test (EMA) and numerical analysis (FEM). Intact model Debonded model (30%) Mode number FEM EMA Error (%) FEM EMA Error (%) Mode 1 986.05 875.4 11.22 875.79 784.2 10.46 Mode 2 1214.19 1095.2 9.80 1009.06 922.5 8.58 Mode 3 1862.67 1748.6 6.12 1104.06 1045.1 5.34 Figure 6: Comparison of a) natural frequency b) normalised natural frequency between intact and debonded model. shift from the initial model. The eigenmode is introduced responses. The amplitudes of displacement and velocity re- to understand the deformation of models. Figure 7 shows sponses of the debonded models are higher than that of the the eigenmodes between intact and 30% debonded models. intact model. The larger the ratio of debonding, the larger It can be concluded that local oscillations in the debonded the amplitude responses. It is caused by the stiffness reduc- region and the global modes of the entire sandwich lead to tion due to the existence of debonding, where the larger the changes in the mode shapes. debonding ratio, the larger the stiffness reduction. More- over, the displacement and velocity-time responses of the intact sandwich are periodic motions with a pure sinusoidal 4.3 Result of forced dynamic analysis that corresponds to its steady-state motion (see Figure 8a). In contrast, the response of debonded models has periodic Forced vibrations in terms of external harmonic excitation displacement motions with the sinusoidal waveform with a are of great importance as it raises in the practical field, modulated signal (see Figure 9b). The presence of contact which may cause severe damage [48]. The comparison of dy- motion in the area of debonding caused the modulated re- namic behaviour between the intact and debonded models sponse. It can be found that the higher the debonding ratio, was investigated using explicit dynamic analysis. Time his- the bigger the modulated signal of the responses. tory responses corresponding to transverse displacements, Analyzing in more detailed result, the comparison of velocities, and phase portrait are firstly compared at four phase portrait between intact and 5% debonding at dif- measurement points (see Figure 2b) at the upper faceplate ferent measurement points of the upper faceplates is pre- with a different driving frequency η=1/3, η=1/2, η=3/4 dan sented in Figure 10. Analysing these plots, one can see η=2. that the phase plots in the point inside debonding region The debonding detection with frequency ratio (η) of (N1 & N2) and the outside (N3 & N4) of the intact model 1/3 leading to the driving frequency of about 2333 rad/s is are elliptical forms. However, the phase plots of the 5% firstly analysed due to concentrated transverse force load. debonded model measured in the point outside debond- Fig 8a and b present displacement and velocity-time histo- ing (N3 & N4) are close to the elliptical form. In contrast, ries of the models with and without debonding at N1. As the phase plots in the inside debonding (N1 & N2) are sig- shown, the existence of the interfacial debonding in the nificantly disturbed from the elliptical form. Hence, it can model significantly influences all evaluated time history be assumed that the responses calculated in the outside A numerical evaluation on nonlinear dynamic response of sandwich plates | 33 Figure 8: Dynamic response of frequency ratio η=1/3 between intact and debonded models a) displacement b) velocity. Figure 9: Comparison of phase portrait calculated at different mea- surement points at frequency ratio η=1/3. Figure 7: Eigenmodes between intact and debonded model (30%). Figure 10 shows the comparison of displacement and velocity responses at frequency ratio½ (3500 rad/s) at N1. As can be seen, the same phenomenon occurs where the debonding region oscillate periodically with the driving presence of damage significantly influences time responses. frequency, but at the point in the debonding region, it ex- The displacement and velocity-time responses of the dam- periences a general periodic motion which is resulted by aged models are higher than the responses of the intact superposition between driving frequency and motion of ones. Moreover, the signal of displacement and velocity periodic contacts [38]. of the intact model is steady-state motion with excitation Further, the debonding diagnostics using phase por- frequency. Still, the displacement and transverse velocity trait between intact and small debonding with driving fre- response of debonded models is a periodic motion with quency 1/3 measured in the various points are depicted in a modulated signal that is much more complicated than Figure 9. From the result, it can be analysed that the dif- that in the previous driving frequency. As shown, the phase ference of phase plot between intact and debonded model plots of the debonded model are significantly disturbed by measured in the inside debonding area (N1 and N2) is more the elliptical form. The surfaces in the debonded zone run significant. The phase plot in the 5% debonding model into an aperiodic contact manner due to such interactions has a larger size than the intact one. Meanwhile, the phase between additional frequencies with the driving frequency. plots between the intact and debonded model in the outside The comparison of phase portrait with driving fre- debonding region have similar sizes, so the diagnostics of quency ½ is depicted in Figure 11. The phase portrait of small debonding using phase plot measured at the points both intact and 5% debonded models measured in the in- outside the debonding area (N3 and N4) is not sensitive. side debonding region (N1 & N2) shows a larger size than that in the outside debonding region. The difference of 34 | Tuswan et al. phase portrait between the intact and debonded model in vere modulated waveform than that in the previous driving the inside debonding region is more visible while the re- frequency, as one can see that the modulated waveform of sponse in the outside debonding region has a similar size. velocity response is more severe than that of displacement The debonding detection using a phase plot with driving response. frequency ½ measured at points outside the debonding Further, the comparison of phase portrait with driving area is not recommended. frequency 5250 rad/s is depicted in Figure 13. The compari- son of phase plots in the inside debonding shows a larger trajectory size than that in the outside debonding region. Moreover, the comparison of phase plots between intact and debonded models in the outside debonding region is more visible than the previous excitation frequency. Figure 10: Dynamic response of frequency ratio η=1/2 between intact and debonded models a) displacement b) velocity. Figure 12: Dynamic response of frequency ratio η=3/4 between intact and debonded models a) displacement b) velocity. Figure 11: Comparison of phase portrait calculated at different measurement points at frequency ratio η=1/2. The time responses between the intact and debonded models are further analysed for the frequency ratio η of Figure 13: Comparison of phase portrait calculated at various mea- ¾ with the same excitation amplitude. A comparison of surement points at frequency ratio η=3/4. time responses between intact and debonded models at the centre of debonding region (N1) is presented in Figure 12. It follows from these plots that the amplitudes of the The investigation is further conducted when the fre- debonded plate’s responses are larger than those in the quency ratio is increased up to η= 2, corresponding to driv- time signals of the intact plate. The bigger the debonding ing frequency 14.000 rad/s, which falls between mode 4 ratio, the larger the amplitude of time signals. Moreover, and mode 5. The dynamic response of debonded models the time responses of the intact model are steady-state mo- is changed noticeably from the previous result. Compar- tion, but the displacement and transverse velocity response ing the transverse displacements at N1 between intact and of debonded models is a periodic function with a more se- debonded models, the response of debonded models varies A numerical evaluation on nonlinear dynamic response of sandwich plates | 35 as a quasi-periodic function with large amplitudes, and a signal is severely distorted by the existence of the higher driving frequency, see Figure 14a. However, the velocity response of this point compared to the intact model is char- acterized by the uncommonly large amplitude, distorted waveform, and loss of periodicity. It happens due to irreg- ular contact interactions that occur between the surface in the damaged area [38]. To prove this phenomenon, the comparison of phase portrait at four various measurement points is drawn in Figure 15. The phase trajectory in the debonded models at all measurement points is distorted from the elliptical form (see Figure 15). Thus, the phase tra- jectory in the inside debonded region (N1 & N2) is irregular and includes frequencies excited by contact surfaces. More- over, the dynamic behaviour of the point outside debonding Figure 15: Comparison of phase portrait calculated at different region (N3 & N4) is assumed a quasi-periodic motion. measurement points at frequency ratio η=2. Regarding debonding detection, a comparison of phase portraits at the different debonding ratios with frequency and tangential (CSHEAR1 and CSHEAR2) contact stress at ratio η=2 is presented in Figure 15. It can be found from the time similar with Figure 16a are illustrated in Figure the result that the higher the debonding ratio, the differ- 16c-d. It can be shown when fully or partially closed con- ence of phase portrait of the debonded model compared tacts, the distribution of normal and shear contact forces with intact one is more visible. It signifies that the higher appear at the debonded region to fulfilling stress continuity the debonding ratio, the larger the amplitude of displace- requirements. Otherwise, when the contact is fully open, ment and velocity. It is caused by higher stiffness loss in no normal and shear contact forces appear in the debonded the debonding model with a high ratio. region and stress-free contact conditions are realised [33]. As a consequence of transition behaviour between the open and closed forms during oscillations, various nor- mal and shear contact forces distributions during analysis time exist in the debonding region. The comparison of nor- mal contact stress and contact shear stress evolution in debonded models is shown in Figure 17. As shown, the debonded models have higher normal and shear contact forces than the intact model. It can be analysed that with increasing debonding ratio, the magnitude of the contact forces increases. The evolution of the contact forces be- Figure 14: Dynamic response of frequency ratio η=2 between intact and debonded models a) displacement b) velocity. tween intact and debonded models change with analysis time so that contact behaviour changes towards debond- ing ratio variation. Additionally, the magnitudes of normal As the debonded models are excited by the concen- contact forces are larger than the shear forces, as seen in trated harmonic force, the models begin to oscillate, so the Figure 17. It occurs due to contact surfaces in the debond- debonded contact behaviour must be investigated in de- ing region interact with each other in the normal direction. tail. Several deformation contours of vibrating models at However, sliding also occurs due to the rotation and lat- different moments of time corresponding to its ’breathing’ eral movement of the faceplate in the debonded surface. are presented in Figure 16a, i.e., fully closed at time 0.003 Thus, the contact forces arising in the debonding region s and 0.017 s, partially open at 0.0377 s, and fully open are influenced by several factors, including inertial forces, at 0.0452 s. As shown during oscillations, the contact of the local deformation of the debonded region, and global debonded surfaces oscillates from close to open status and deformation of the model caused by external excitation or vice versa during time analysis. To certainly understand a combination of all factors [33]. the contact behaviour of 10% debonding ratio between the debonded surfaces, the contours of both normal (CPRESS) 36 | Tuswan et al. Figure 16: a) Deformed shapes of the debonded zone b) CPRESS c) CSHEAR1 d) CSHEAR2 at instants of time a) 0.0003 s, b) 0.017 s, c) 0.0377 s, d) 0.0452 s. problems due to concentrated impulse load was performed with ABAQUS/Explicit. First, time history responses corre- sponding to transverse displacements, velocities, accelera- tions, and displacement trendline at point N1 (see Figure 2b) were compared. Figure 18 indicates the comparison of transient responses computed at the centre of debonding in the upper faceplate of both intact and debonded models. The intermittent contact causes a significant difference in transient responses in both waveforms and signal proper- Figure 17: Comparison of a) normal contact force b) contact shear ties between intact and debonded models. As illustrated in force of debonded models. Figure 18a, the waveform of the debonded model is more disturbed than that of the intact model because the dynam- 4.4 Result of transient dynamic analysis ics of the debonded surfaces are modelled by intermittent contact. Further, the existence of debonding increases the To gain a deeper investigation of debonding diagnostic, amplitude of the transient responses. The bigger the ratio the comparison of the dynamic behaviour of debonding A numerical evaluation on nonlinear dynamic response of sandwich plates | 37 of debonding, the higher the amplitude of the transient responses, as clearly seen in the displacement trendlines in Figure 18d. It occurs due to the result of superimposing between vibrational waves caused by external loads and those generated by contact interactions [25]. So, the effect of debonding due to transient loading of the sandwich models is significant. The evolution of contact forces due to impulse load- ing in the time interval 0 - 0.002 s is illustrated in Fig- ure 19, which describes the comparison of the normal and shear contact forces with various debonding ratios. As seen, debonding causes an increase both in the normal and shear contact forces. The magnitude of the normal and shear con- tact forces increases with the increase of debonding ratio. Moreover, the magnitude of the normal contact forces is higher than the shear forces due to the interaction of con- Figure 18: Comparison of transient response between intact and tacting surfaces in the debonded region occurs in the nor- debonded models a) displacement b) velocity c) acceleration d) mal direction. displacement trendline. To present a better visualisation of both normal and shear contact force distribution due to impulse load, the visualisation of contact at instants of time of 10% debonded model is presented in Figure 20. The contact forces arising between the faceplate and the core si passing from closing to opening and vice versa. Figure 20a shows the visualisa- tion of normal and shear contact forces when the model is fully closed at time 0.00045 s. Moreover, Figures 21b-c shows normal and shear contact force distributions when the debonded model is partially closed. As shown, the nor- Figure 19: Transient response of debonded models a) normal con- mal contact traction due to the contact–impact motion of tact force b) contact shear force. the debonded surfaces changes the sliding mechanism de- scribed by the shear contact stress distributions. debonded region along with the global mode changes in the mode shapes of the debonded model. Further, the numerical result of dynamic responses, 5 Conclusions both harmonic and impulse loading, shows that the debonding can be identified by comparing time responses The damage detection of the hybrid sandwich using free between intact and debonded models. The dynamic re- and forced vibration was studied in this work. The influence sponses analysed in the four different driving frequencies of dynamic behaviour on the debonding ratio using FE soft- of the debonded models due to harmonic excitation de- ware ABAQUS can be drawn. The free vibration using the pend on the intermittent contact. A variety of motions such Lanczos method was used to extract eigenfrequencies and as periodic, quasi-periodic, and irregular on the time re- eigenmodes. To obtain an understanding of the dynamic sponses and phase portraits in the debonded region are behaviour of debonded models, both the transient dynamic found. The dynamic responses of the debonded models are analysis and the forced dynamic analysis were analysed higher than that of the intact model. In terms of transient using ABAQUS/Explicit. dynamic analysis, numerical results show that the presence The result shows that debonding causes natural fre- of debonding affects the short-time response of debonded quency reduction and alters the mode shapes as well. It models. Using transient time responses, the magnitude of can be determined that the natural frequency decreases the time responses of the debonded models is higher than with the increase of the debonding ratio. The higher modes the intact model. To sum up, using the change of responses are found to be more sensitive to debonding existence. It resulted from free and forced vibration analyses of both un- can be found that the presence of local oscillations in the 38 | Tuswan et al. [5] Sandeep SH, Srinivasa CV. Hybrid Sandwich Panels: A Review. Int J Appl Mech Eng. 2020;25(3):64-85. [6] Chen Y, Hou S, Fu K, Han X, Ye L. Low-velocity impact response of composite sandwich structures: modelling and experiment. 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Structural damage detection contact (CSHEAR1, and CSHEAR2) of debonded model at instants of by using single natural frequency and the corresponding mode time a) t= 0.00045 s b) t= 0.00655 s c) t= 0.0358 s. shape. Shock Vib. 2016;2016:1-8. [13] Ismail A, Zubaydi A, Piscesa B, Ariesta RC, Tuswan. Vibration- based damage identification for ship sandwich plate using finite damaged and damaged models, the debonding diagnostics element method. Open Eng. 2020;10:744-52. can be implemented. [14] Kaveh A, Zolghadr A. An improved CSS for damage detection of truss structures using changes in natural frequencies and mode Funding information: The research has received finan- shapes. Adv Eng Softw. 2015;80:93-100. [15] Prabowo AR, Tuswan T, Ridwan R. 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Journal

Curved and Layered Structuresde Gruyter

Published: Jan 1, 2022

Keywords: debonding detection; hybrid sandwich; finite element analysis; dynamic contact; forced vibration; marine

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