Shock and Vibration

Volume 2018 (2018), Article ID 5715863, 26 pages

https://doi.org/10.1155/2018/5715863

## Linear and Nonlinear Dynamic Analyses of Sandwich Panels with Face Sheet-to-Core Debonding

^{1}Department of Solid Mechanics, Lublin University of Technology, 40 Nadbystrzycka Str., Lublin, Poland^{2}Department of Applied Mathematics, National Technical University “KhPI”, 2 Kyrpychova Str., Kharkiv, Ukraine

Correspondence should be addressed to Vyacheslav N. Burlayenko

Received 5 September 2017; Accepted 31 December 2017; Published 14 February 2018

Academic Editor: Giuseppe Piccardo

Copyright © 2018 Vyacheslav N. Burlayenko and Tomasz Sadowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A survey of recent developments in the dynamic analysis of sandwich panels with face sheet-to-core debonding is presented. The finite element method within the ABAQUS™ code is utilized. The emphasis is directed to the procedures used to elaborate linear and nonlinear models and to predict dynamic response of the sandwich panels. Recently developed models are presented, which can be applied for structural health monitoring algorithms of real-scale sandwich panels. First, various popular theories of intact sandwich panels are briefly mentioned and a model is proposed to effectively analyse the modal dynamics of debonded and damaged (due to impact) sandwich panels. The influences of debonding size, form, and location and number of such damage incidents on the modal characteristics of sandwich panels are shown. For nonlinear analysis, models based on implicit and explicit time integration schemes are presented and dynamic responses gained with those models are discussed. Finally, questions related to debonding progression at the face sheet-core interface when dynamic loading continues with time are briefly highlighted.

#### 1. Introduction

Sandwich panels have long been recognized as one of the efficient structural elements. Due to their intrinsic properties like a high bending stiffness at minimum mass, the capability to be tailored for specific uses, high damping properties, and a great potential for energy absorption, they have found applications in diverse kinds of modern industries and structures. Along with advantages that suggest the sandwich concept, the sandwich panels suffer the consequences of their constructive features. The large differences between thicknesses and elastic moduli of the constitutive layers make the sandwich panels susceptible to debonding of the face sheet from the core at their interface. The sources of debonding usually are a result of imperfections in the manufacturing process, the degassing of the foam core under direct sunlight, the ability for water absorption of cellular types of the core followed by repeated cycles of freezing and thawing at the face sheet-to-core interface, low-velocity impacts, stress concentrations due to localized loading, and so forth. The presence of debonding is often invisible, but it affects the dynamics and the strength of sandwich structures as well as it may become a reason of premature failure below the level of design loads [1]. Therefore, a correct use of sandwich panels in different engineering applications requires a better knowledge of their mechanical behaviour, in particularly, their dynamic response. This knowledge can also provide a basis for nondestructive health monitoring techniques.

In modern sandwich panels, the core layer is often made of a soft and flexible material. This leads to the core compressibility and, as a result, to the change in the height of the core during deformation [2]. Thus, to provide a high-fidelity analysis of sandwich panels, the nonlinear deformation patterns in the core should be accounted for in prediction models. This makes the dynamic analysis of the sandwich panels more challenging than that of the laminates. Furthermore, high physical and geometrical mismatching is also an obstacle to handle sandwich panels using the equivalent single layer (ESL) two-dimensional (2D) finite element [3, 4] or dynamic stiffness element [5] models. In this regard, models adopting mixed theories or the layer-wise (LW) approach are preferable for sandwich structures [6]. The latter has given rise to a variety of high-order sandwich panels theories referred to as HSAPTs [7–9], in which the high-order effects in the sandwich panel do not result from some prior assumptions for the displacement field. These theories invite any plate/shell theory formulations for the face sheets and a 3D elasticity theory or equivalent one for the core [10–12].

One of the other aspects that relates to the analysis issues of sandwich panels is a microscopically discrete structure of the core, for example, truss cores, honeycomb cores, and corrugated or folded cores of various shapes. The traditional analysis scheme of such sandwich panels is that, first, the core is simplified as an equivalent generally anisotropic homogeneous material and then modelled by using any theory and solution method. Thus, the analysis accuracy strongly depends on correct estimations of equivalent rigidities. Some examples of finite element analyses of sandwich panels with discrete cores can be found, for example, in [13–18].

Keeping in mind the challenges in modelling sandwich structures, many efforts have been made to study their vibration responses using both linear and nonlinear models. In doing so, some studies have used the assumptions on geometrical nonlinearity in the dynamics of sandwich panels, for example, [19, 20], while the other ones have examined the influence of material nonlinearity. The effect of a viscoelastic core on damped forced oscillations has been considered, for example, in [21, 22]. The problem of the interlaminar slip between constitutive layers in the nonlinear free vibration has also been discussed, for example, in [23, 24]. Apart from these nonlinear problems, the dynamics of sandwich panels with debonding are another computationally challenging task, even when small displacements and a material linearity are assumed. The difficulties are concerned with modelling physical phenomena arising from debonding. First, the detached surfaces are free of shear and normal stresses and as a result the overall stiffness of the structure is reduced. Second, during loading or oscillations these surfaces may slip longitudinally one with respect to another and/or undergo normal compressive stress if contact between them exists. Such local changes within the debonded region give rise to alterations in the global dynamics of sandwich panels and, also, the stress fields accompanying this dynamic behaviour may result in fracture in the face sheet-to-core interface.

To simplify the nature of the problem, the earliest solutions on free vibration were based on the split beam approach under assumptions that the decoupled layers either freely overlap each other or are constrained to move together [25–30]. Later, improved nonlinear models excluding penetration between the layers coming into contact were developed. In [31], to prevent overlapping, the dynamics of a delaminated beam was studied using a piecewise linear virtual spring model, whereas the kinematic contact conditions were established using the node-to-node contact model in [32]. To date, a large volume of the literature on this subject is available [33]. Some recent advanced studies point at the need to use models accounting for coupling between normal and flexural actions in vibrations [34]. As found, it is a reason of parametrically induced vibration of delaminated beams and plates [35–37]. Other studies confirmed the efficiency of the LW-based finite element models compared to those using the ESL approach [38, 39] and showed that such models provide a stress recovery for localized effects [40].

Yet, the dynamic finite element analysis (FEA) contributes to developments of vibration control methods of sandwich structures [41, 42]. Alternatively, the results of the dynamic FEA of sandwich panels can be used for increasing the efficiency of the structural health monitoring (SHM). In the latter, natural frequencies, mode shapes, frequency response functions, and time or frequency domain data can be extracted from either linear or nonlinear dynamic FEA. For instance, in [43] natural frequencies extracted from the eigenvalue analysis have been used to detect and locate the saw cut within a composite plate, whereas mode shapes collected from a similar linear analysis have successfully been applied to the delamination location in a composite beam in [44]. Other authors have developed a strain-based damage index based on a linear model for the prediction of the delamination location in composite plates [45]. The curvatures of mode shapes have been utilized for quantifying the damage magnitude in a honeycomb beam in [46]. Also, a linear dynamic analysis is used in FRF-based damage detection techniques [47]. In [48] such approach has successfully been implemented for detecting debonding in a honeycomb beam. For the sake of debonding detection, the linear dynamic FEA has been carried out in [49] to highlight the relative changes of dynamic response between a healthy sandwich plate and a debonded one. More advanced techniques using modal dataset to detect debonding in sandwich structures can be found in some recent works, for example, [50–52].

Motivated by the idea to examine the influence of debonding on the modal dynamics of sandwich panels in detail, a number of numerical and experimental studies have been performed during the last two decades. For instance, in [53–55] natural frequencies and corresponding vibration modes have been calculated for flexible debonded sandwich beams. The changes of the modal characteristics in sandwich plates caused by debonding depending on the debonding size and form and of boundary conditions and sandwich core properties have been presented in [56]. Similar researches using linear models to study the dynamic behaviour of sandwich plates that have undergone a postimpact damage or containing a multidebonding have been carried out in [57–59] and [60–62], respectively. A linear model has been also adopted to examine a dynamic stability of a delaminated beam under harmonic longitudinal loading in [63]. An experimental study on the modal dynamics of a delaminated plate has been done in [64]. Changes in the peaks and valleys of the FRFs due to delamination in a honeycomb sandwich beam have been proven by experimental tests in [65]. Effects of the debonding length on fatigue and vibration of sandwich composites have been tested in [66].

Because, in general, the dynamics of composite structures are rather nonlinear, the numerical simulations could throw light on the nature of various nonlinear mechanisms, in particular, due to debonding and in turn this knowledge can make the SHM methodologies more reliable [67, 68]. Insight into the real-life dynamics of debonded sandwich panels can be gained by accounting for the “real contact” conditions between the debonded parts. Hence, nonlinear models have been developed to handle the nonlinearities caused by either of or both of the factors, namely, opening and closing of the debonding (breathing) and contact/impact between the debonded layers during vibration. In [69] the contact problem in delaminated surfaces has been modelled in terms of fictitious linear springs to study the dynamic transient behaviour of a delaminated plate. An analytical model of a 1D sandwich beam accounting for the real contact has been created in [70] to examine the influence of the contact phenomenon on the transient behaviour of the beam. In [71–73] an explicit 3D finite element contact model has been developed with ABAQUS for analysing the transient dynamics of an impacted sandwich panel. This model has also been used in [74, 75] to study the effect of the debonding size on the transient dynamic response and the stress state in sandwich plates. Other contact models addressed to discovering the role of interlaminar contact on the nonlinear dynamics of sandwich and laminated panels accounting for geometrical nonlinearity have been proposed in [76–78]. A nonlinear contact model within an implicit approach has been elaborated in [79, 80] to explore the dynamics of sandwich panels with debonding, which are subjected to harmonic forces. The research reported a complex dynamic behaviour of such panels highly dependent on the driving frequency that is inherent in structures with conditions of friction and contact [81]. A potential of such nonlinear models to identify damages in composite structures has been demonstrated in [82].

The discussion above clearly demonstrates that high-fidelity dynamic finite element analysis methods of sandwich panels based on linear and nonlinear models are still needed for many engineering problems. It seems that research on the modelling aspects of the dynamics of sandwich panels with debonding is scattered across different models and diverse research areas. Often such studies exploit only one either linear or nonlinear model, and then it is not so easy to juxtapose them within the same problem being considered. Besides, many of the studies are focused on sandwich beams or unidirectional panels. This spatial reduction simplifies the nature of the physical phenomena, because it dictates a specific structural behaviour. To the best of our knowledge, there is a deficiency of 3D models for studying the dynamic behaviour of debonded sandwich plates. Some of them have been developed in our previous papers. The present work can be viewed as a comprehensive survey summarizing the recent developments in 3D finite element modelling of the dynamics of sandwich plates with debonding. Our previous findings in this research area are considered and reexamined under the light of our goal to characterize the dynamic behaviour of sandwich panels with debonding ranging from free vibration to nonlinear oscillations by both linear and nonlinear 3D finite element models. So, the present work takes a step towards the classification of computational models for quantitatively estimating the dynamics of sandwich panels with debonding and understanding the responses of such structures based on the numerical examples.

#### 2. Outlines of Dynamic FEAs

In this section, basic aspects of the finite element procedures used in the current work to perform dynamic finite element analyses (FEAs) are briefly presented.

##### 2.1. Equations of Motion

The finite element analysis of the system of elements provides the solution of the equations of motion corresponding to the nodal degrees of freedom, which, in general, can be written as follows [83]:where is a certain moment of time and , , , and stand for inertia, damping, and internally and externally applied forces, respectively. In the case of a linear problem, (1) reads aswhere , , and are, respectively, mass, damping, and stiffness matrices of the element assemblage and are nodal points displacements, and and are their time derivatives referring to nodal velocities and acceleration, respectively. The matrices of the physical properties of the finite elements are related to the force vectors as , , and . Under assumptions that initial conditions and boundary conditions applied to the system remain unchanged during loading, the solution of (2) completely describes a linear response of the system.

In presence of contact between elements of the system, the system response is linear, if other nonlinearities except contact do not exist, only prior to the contact conditions met. Otherwise changes in boundary conditions at a certain load level occur. Then, (2) is to be supplemented by contact constraints imposed at a set of all prospective contact pairs over the loading time as follows:and the constraints appropriate for the prescription of a Coulomb friction law, for example, as [84]In (3) and (4), and stand for normal and tangential “gap” functions defined for all active contact pairs, and are normal and tangential contact traction acting at those pairs, is a Lie derivative (a convective derivative that defines a frame invariant measure of rate of in the case of slipping), is the coefficient of friction, and is a nonnegative scalar. For conciseness, we skip details and refer to works in this area, for example, [84]. The contact traction contributes to equilibrium of the system as contact force vector acting on some nodal points at a certain moment of time* t;* that is, (2) takes the form

##### 2.2. Eigenvalue Problem

Dynamic characteristics of a linear structural system (2) are governed by the natural frequencies and the corresponding mode shapes. The determination of them requires the solution of an eigenvalue problem:Herein* ω* is an undamped circular frequency and is the corresponding vibration mode. Then,

*n*uncoupled solutions of (6) can be composed in the form convenient for frequency extraction:where in the matrix the columns are orthogonalized mass-normalized eigenvectors and is a diagonal matrix containing squared eigenfrequencies.

##### 2.3. Mode-Based Dynamic Analysis

To provide predictions of the steady-state dynamic response of a linear system (2) due to harmonic excitation at a given driving frequency , a modal superposition method can be used. Then, in the modal subspace (2) takes the formHere the nodal displacements are expressed in terms of the vector of generalized displacements and the normal modes being found from the eigenvalue analysis; the matrices , , and are the projections of , , and onto the modal subspace; that is, and and and . In the case of linear analysis is the identity matrix, and in case of no damping coupling , where are modal damping rations. Hence, the equations in system (8) with respect to the generalized displacements are decoupled and for the* r*th mode have the formOnce all are evaluated, the physical response of the original system in terms of nodal displacements, velocities, acceleration, and stresses can be recovered. The analysis is done as a frequency sweep by applying the loading at a series of different frequencies and recording the system response [85].

##### 2.4. Direct Integration Analyses

The equations in system (5) cannot be solved independently; a direct integration over time is required. One would normally resort to numerical approximation procedures to do it. The explicit central difference time integrator used in ABAQUS/Explicit [85] presents (5) at a typical time increment Δ*t* in the formwhere is a lumped mass matrix obtained by transform of the consistent mass matrix and the velocities and the displacements are calculated at each time increment according to the expressionsThis explicit time integration scheme is conditionally stable only. An approximation to its stability limit can be achieved by evaluating the highest modal natural frequency in the FE mesh as [83].

The implicit Hilber-Hughes-Taylor (HHT) temporal integrator is used by ABAQUS/Standard [85]. This integration scheme applied to (5) yieldswhereThe unconditional stability occurs for , , and [83]. The linearized problem in (12) within the* j*th Newton-Raphson iteration has the formwith updating unknowns as .

##### 2.5. Modelling Workflow

In Figure 1 the modelling workflow followed in this work is summarized. The finite element analyses carried out with ABAQUS are presented in the order of their computational cost and modelling complexity, that is, the computational procedures mentioned above and used in those analyses and some approaches and features applied to the development of linear and nonlinear models. Thereby, for sandwich panels, first, the frequency eigenvalue analysis (6) and the modal-based harmonic analysis (8) are performed using linear models and, then, the general nonlinear dynamics under an impulse load and harmonic loading are worked out using explicit (9) and implicit (12) time integration schemes. Both a debonded sandwich panel and the same healthy one are simulated in the calculations. By tracking the differences between their dynamic responses, the effects of debonding on the linear and nonlinear dynamic behaviour of the sandwich panel are evaluated. Moreover, using both linear and nonlinear models for the same sandwich panel, the impact for each of them for highlighting dynamic effects can be clearly seen.