Time Response of Delaminated Active Sensory Composite Beams Assuming Non-Linear Interfacial Effects

Abstract

A layerwise laminate FE model capable of predicting the dynamic response of delaminated composite beams with piezoelectric actuators and sensors encompassing local non-linear contact and sliding at the delamination interfaces was formulated. The kinematic assumptions of the layerwise model enabled the representation of opening and sliding of delamination interfaces as generalized strains, thereby allowing the introduction of interfacial contact and sliding effects through constitutive relations at the interface. This realistic FE model, assisted by representative experiments, was used to study the time response of delaminated active sensory composite beams with predefined delamination extents. The time response was measured and simulated for narrowband actuation signals at two distinct frequency levels using a surface-bonded piezoceramic actuator, while signal acquisition was performed with a piezopolymer sensor. Four different composite specimens, each containing a different delamination size, were used for this study. Experimental results were directly compared with model predictions to evaluate the performance of the proposed analytical approach. Damage signatures were identified in both the signal amplitude and the time of flight, and the sensitivity to delamination size was examined. Finally, the distributions of axial and interlaminar stresses at various time snapshots of the transient analysis are presented, along with contour plots across the structure’s thickness, which illustrate the delamination location and wave propagation patterns.

1. Introduction

The effect of existing delamination on the time response of realistic active sensory composite beams was thoroughly investigated by the authors, both analytically and experimentally. The work comprises two parts: (I) development of an analytical model and (II) experimental validation with delamination detection. An analytical model was developed using layerwise mechanics to simulate composite beams that include through-width delamination cracks and interactions with piezoelectric devices. The novelty of this model lies in the natural contact conditions at the delamination interfaces, which prevent penetration in a realistic manner. Additionally, an interlaminar shear stress field was applied during contact, controlling the free axial movement. This paper presents results that explore the effect of local non-linear interactions on the global structural response and validates the analytical model predictions against representative experimental measurements.

Various works investigated the effect of delamination on the structural response using experimental measurements or analytical simulations [1,2,3]. The majority of these studies investigated the effect of delamination on the frequency parameters [4,5] and presented changes in the natural frequencies of composite beams with delamination [6,7,8,9,10], or even the vibration mode shape [11,12]. The effect of delamination was additionally studied both analytically and experimentally on another critical dynamic parameter, like the modal damping, by Saravanos and Hopkins [13]. The introduction of piezoelectric devices operating either as actuators or sensors led to the development of the more widespread SHM setup for damage detection in composite structures, known as the “pitch and catch configuration”. The pioneers in this direction were Keiler and Chang [14], who presented an experimental study to identify the damage effect on the frequency response functions of delaminated composite beams. Kessler et al. [15] presented an analytical and experimental study revealing the effect of various damage types on the frequency response functions and mode shapes of damaged composite beams. Also, Chrysochoidis and Saravanos [16] in an experimental study investigated the sensitivity of the modal characteristics to the size of delamination as well as the efficiency of different testing setups.

Over the last three decades, many studies have been published on modeling delamination in composite structures. Early works focused on the effects of a single delamination on the natural frequencies of composite beams [4,17,18] and plates [19] using experimental results and simple analytical beam and plate models of the so-called “four-region” approach. Barbero and Reddy [20] reported a layerwise laminate theory to describe plates with multiple delamination cracks between the layers. A generalized composite beam model for predicting the effect of multiple delaminations on the modal damping and modal frequencies of composite laminates was reported by Saravanos and Hopkins [13], which treated the opening and sliding at the delamination interfaces as additional degrees of freedom. A higher-order theory to model matrix cracking and delamination in laminated composite structures was presented [21,22]. Ghoshal et al. [23] modeled delamination using a Fermi–Dirac distribution of the axial displacements at the crack interface to obtain smoother transitions in the displacement and strain fields, and Chrysochoidis et al. [24] used a coupled electromechanical model to simulate delaminated beams and the damage local modes, with piezoelectric actuators and sensors, neglecting contact at interfaces. A layered solid spectral element was used to simulate the wave propagation of delaminated composite structures with piezoelectric wafers [25], while Shen and Cesnic [26] proposed a hybrid local interaction approach that considers local delamination dynamics to simulate the response of delaminated structures. A high-fidelity computational tool was published [27] that utilized spectral finite elements for the rapid simulation of wave propagation characteristics in damaged composite plates. With the modelling focus on the debonding appearing between the composite laminate and the piezoelectric patch [28,29], an efficient layerwise electromechanical FE model was proposed using the recently developed facet shell element based on coupled third-order zig-zag theory.

Focusing on the simulation of interactions occurring locally at the delamination interface, limited research has been conducted. The first study [30] concentrated on the modeling at the zone of delamination, assuming negligible opening and a shear stress applied between the sliding interfaces, while Ghohal et al. [23] and Chattopadhyay et al. [31] presented a model simulating the contact between the delamination faces of a composite, as a function of the relative position of the two crack faces. In another high-order layerwise approach, a penalty constraint was introduced at the interface, preventing penetration [32]. However, none of the available studies realistically accounted for the relative local movements (contact and slip).

The investigation into the effect of delamination on the global transient dynamic response and wave propagation, conducted through both computational and experimental studies, has resulted in various publications. Ostachowitz et al. [33] developed a four-region type analytical model to simulate the transient response of delaminated composite beams. Chrysochoidis and Saravanos [34,35] presented the transient response of smart composite beams experimentally and analytically, revealing the effect of damage for various delamination sizes. Rekatsinas et al. [27] developed a cubic spline layerwise time domain spectral element to study the effect of delamination in guided wave mode conversion. Special attention should be paid to capturing experimentally the nonlinear interactions that occur at the delamination zone [36,37,38,39].

On the same path as the research above, Ramadas et al. also moved [40] by studying the effect of the primary anti-symmetric guided wave mode in the presence of delamination in composite strips, a study which was later expanded to plates by the same authors [41]. As a consequence of the presence of a delamination, many researchers were inspired to develop quantitative approaches in terms of Time of Flight (ToF) [42,43,44]. By measuring the change in ToF, the through-thickness position of a delamination crack could be revealed. Another approach to delamination detection was proposed by Shoja et al. [45], which utilizes low-frequency guided waves along with the Pearson correlation coefficient.

The current state of the art reveals a variety of methods and approaches to studying and understanding the presence and characteristics of delaminations; on the other hand, it also suggests that there is room for more efficient tools and practices to emerge. Thus, the current paper addresses interfacial contact and sliding at delamination faces directly through the layerwise theory without the introduction of external terms. The formulation leverages new terms introduced in interlaminar generalized strains, which are directly related to crack opening and sliding, and are derived from the kinematic assumptions. The developed, realistic, and fully electromechanically coupled FE model is capable of describing actuation via a piezoceramic actuator, wave propagation across the length of the beam, interaction with the delamination interfaces, as well as the reception of the propagated wave at the piezoceramic sensor. Simulations are correlated with representative experiments for various delamination sizes, validating the model performance. Additionally, delamination signatures in predicted time responses, such as amplitude and time of flight, are detected through both analytical and experimental results. At the same time, the continuous propagation of axial and shear waves is presented across delaminated active sensory beams, with contour plots provided to map the exact position of the damage.

2. Theoretical Formulation

An analytical model was developed for analyzing the transient response of active sensory delaminated composite beams, which assumed contact between the delamination faces and control of the relative horizontal slip [46].

2.1. Equations of Equilibrium

The variational statement of the equations of equilibrium for the piezoelectric structure is as follows:

$${\displaystyle \underset{\mathrm{A}}{\int }-\mathrm{\delta }\mathcal{H}\mathrm{d}\mathrm{A}}+{\displaystyle \underset{\mathrm{A}}{\int }\mathrm{\delta }\underset{˜}{\mathrm{u}}(-\mathrm{\rho }\underset{˜}{\ddot{\mathrm{u}}})\mathrm{d}\mathrm{A}}+{\displaystyle \underset{{\mathrm{\Gamma }}_{\mathrm{\tau }}}{\oint }\mathrm{\delta }{\overline{\underset{˜}{\mathrm{u}}}}^{\mathrm{T}}}\overline{\mathrm{\tau }}\mathrm{d}\mathrm{\Gamma }+{\displaystyle \underset{{\mathrm{\Gamma }}_{\mathrm{D}}}{\oint }\mathrm{\delta }\overline{\mathrm{\phi }}\overline{\mathrm{D}}\mathrm{d}\mathrm{\Gamma }}=0$$

(1)

where A is the x-z surface of the beam; $\mathcal{H}$ is in general the electric enthalpy of the material; $\overline{\mathrm{\tau }}$ and $\overline{\mathrm{D}}$ are, respectively, the surface tractions and charge, acting on the boundary surface ${\mathrm{\Gamma }}_{\mathrm{\tau }},{\mathrm{\Gamma }}_{\mathrm{D}}$; $\underset{~}{\mathrm{u}}=\left\{\begin{array}{c}\mathrm{u}\\ \mathrm{w}\end{array}\right\}$ is the displacement vector.

The electric enthalpy includes the components of elastic strain energy and electric energy. Thus, the electric enthalpy in the m-th layer is expressed in variational form as follows:

$${\displaystyle \underset{\mathrm{A}}{\int }-\mathrm{\delta }\mathcal{H}\mathrm{d}\mathrm{A}}+{\displaystyle \underset{\mathrm{A}}{\int }\mathrm{\delta }\underset{˜}{\mathrm{u}}(-\mathrm{\rho }\underset{˜}{\ddot{\mathrm{u}}})\mathrm{d}\mathrm{A}}+{\displaystyle \underset{{\mathrm{\Gamma }}_{\mathrm{\tau }}}{\oint }\mathrm{\delta }{\overline{\underset{˜}{\mathrm{u}}}}^{\mathrm{T}}}\overline{\mathrm{\tau }}\mathrm{d}\mathrm{\Gamma }+{\displaystyle \underset{{\mathrm{\Gamma }}_{\mathrm{D}}}{\oint }\mathrm{\delta }\overline{\mathrm{\phi }}\overline{\mathrm{D}}\mathrm{d}\mathrm{\Gamma }}=0$$

(2)

where z_m and z_m+1 indicate the bottom and top surface location of the layer, respectively.

2.2. Governing Materials Equations

Each ply or layer is assumed to exhibit linear piezoelectric behavior. In the current beam case, in-plane and interlaminar shear strains are considered in the elastic field. The piezoelectric materials are assumed equivalent to monoclinic class 2 crystals with the poling direction coincident with the z-axis. The constitutive equations have the following form:

$$\begin{gathered} \left\{\begin{array}{c}{\mathrm{\sigma }}_1\\ {\mathrm{\sigma }}_5\end{array}\right\}=\left[\begin{array}{cc}{\mathrm{C}}_{11}^{\mathrm{{\rm E}}}& 0\\ 0& {\mathrm{C}}_{55}^{\mathrm{{\rm E}}}\end{array}\right]\cdot \left\{\begin{array}{c}{\mathrm{S}}_1\\ {\mathrm{S}}_5\end{array}\right\}-\left[\begin{array}{cc}0& {\mathrm{e}}_{31}\\ {\mathrm{e}}_{15}& 0\end{array}\right]\cdot \left\{\begin{array}{c}{\mathrm{E}}_1\\ {\mathrm{E}}_3\end{array}\right\} \\ \left\{\begin{array}{c}{\mathrm{D}}_1\\ {\mathrm{D}}_3\end{array}\right\}=\left[\begin{array}{cc}0& {\mathrm{e}}_{15}\\ {\mathrm{e}}_{31}& 0\end{array}\right]\cdot \left\{\begin{array}{c}{\mathrm{S}}_1\\ {\mathrm{S}}_5\end{array}\right\}+\left[\begin{array}{cc}{\mathrm{\epsilon }}_{11}^{\mathrm{S}}& 0\\ 0& {\mathrm{\epsilon }}_{33}^{\mathrm{S}}\end{array}\right]\cdot \left\{\begin{array}{c}{\mathrm{{\rm E}}}_1\\ {\mathrm{{\rm E}}}_3\end{array}\right\} \end{gathered}$$

(3)

σ₁, σ₅ and S₁, and S₅ are the axial and shear mechanical stresses and strains respectively; E₁ and E₃ are the electric field vectors; D₁ and D₃ are the electric displacement vectors; C₁₁, C_33, and C₅₅ are the elastic stiffness coefficients; e₁₅ and e₃₁ are the piezoelectric coefficients; and ε₁₁, ε₃₃ are the electric permittivity coefficients of the material. Superscripts E and S indicate constant electric field and strain conditions, respectively. The above system of equations encompasses the behavior of a beam’s piezocomposite layer. The electric field vector is the negative gradient of the electric potential φ:

$$\begin{array}{cc}{\mathrm{E}}_{\mathrm{k}}=-{\displaystyle \frac{\partial \mathsf{\varphi }}{\partial {\mathrm{x}}_{\mathrm{k}}}}& \mathrm{k}=\mathrm{1,3}\end{array}$$

(4)

2.3. Kinematic Assumptions

Kinematic hypotheses of a coupled layerwise theory for piezoelectric laminates [47] are adopted as the basis for the present model, admitting piecewise linear (zig-zag) fields through the thickness. A typical laminate is assumed to be subdivided into N discrete layers, where each discrete layer may be defined to contain a single ply, a sub-laminate, or a sub-ply, depending on the desired detail of approximation. Linear fields are assumed in each discrete layer for both the in-plane and electric potential fields throughout the laminate thickness. In addition, multiple delamination cracks are also considered at the interfaces of adjacent plies. In such a case, additional terms are included in the displacement field to represent the interlaminar discontinuities in the displacement fields, due to delaminations. The resultant zig-zag field maintains continuity across the discrete layer boundaries, yet allows for different in-plane displacement slopes within each discrete layer, and admits sliding and opening across a delamination crack (see Figure 1). Considering a grading of N discrete layers and taking into account N_d delaminations, the assumed displacements and electric field take the following form:

$$\mathrm{u}(\mathrm{x},\mathrm{z})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{u}}^{\mathrm{i}}(\mathrm{x}){\mathrm{\psi }}^{\mathrm{i}}(\mathrm{z})\right)}+{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}\left({\tilde{\mathrm{u}}}^{\mathrm{k}}(\mathrm{x}){\mathrm{\psi }}^{\mathrm{k}}({\mathrm{z}}_{\mathrm{k}})\mathrm{H}(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})\right)}$$

(5)

$$\mathrm{w}={\mathrm{w}}^{\mathrm{o}}(\mathrm{x})+{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}{\tilde{\mathrm{w}}}^{\mathrm{k}}\mathrm{H}(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})}$$

(6)

$$\mathrm{\phi }(\mathrm{x},\mathrm{z})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{\Phi }}^{\mathrm{i}}(\mathrm{x}){\mathrm{\psi }}^{\mathrm{i}}(\mathrm{z})\right)}$$

(7)

where superscripts $\mathrm{i}=1,\dots ,\mathrm{N}$ indicate the $\mathrm{N}-1$ discrete layers, $\mathrm{k}=1,\dots ,{\mathrm{N}}_{\mathrm{d}}$ the number of the delaminations through the thickness, and “$\mathrm{o}$” the mid-plane. In (5), the term ${\mathrm{u}}^{\mathrm{i}}$ is the in-plane displacement at the interface of each discrete layer, effectively describing the extension and rotation of the layer. Also, ${\tilde{\mathrm{u}}}^{\mathrm{k}}$ and ${\tilde{\mathrm{w}}}^{\mathrm{k}}$ are new degrees of freedom describing the sliding and opening across the faces of the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination. $\mathrm{H}\left(\mathrm{z}\right)$ is the Heaviside’s step function and ${\mathrm{z}}_{\mathrm{k}}$ is the distance of the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination from the mid-plane. Finally, ${\mathrm{\psi }}^{\mathrm{i}}$ and ${\mathrm{\psi }}^{\mathrm{k}}$ are the linear interpolation functions through the laminate thickness for the ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$ layer and the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination, given by the following:

$${\mathrm{\psi }}^{\mathrm{i}}(\mathrm{z})=\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{z}-{\mathrm{z}}^{\mathrm{i}-1}}{{\mathrm{z}}^{\mathrm{i}}-{\mathrm{z}}^{\mathrm{i}-1}}},\mathrm{z}\le {\mathrm{z}}^{\mathrm{i}}\\ {\displaystyle \frac{{\mathrm{z}}^{\mathrm{i}+1}-\mathrm{z}}{{\mathrm{z}}^{\mathrm{i}+1}-{\mathrm{z}}^{\mathrm{i}}}},\mathrm{z}>{\mathrm{z}}^{\mathrm{i}}\end{array}\right.$$

(8)

$${\mathrm{\psi }}^{\mathrm{k}}(\mathrm{z})={\displaystyle \frac{{\mathrm{z}}^{\mathrm{k}+1}-\mathrm{z}}{{\mathrm{z}}^{\mathrm{k}+1}-{\mathrm{z}}^{\mathrm{k}}}}$$

(9)

2.4. Strain–Displacement Relations and Electrical Field

In the context of kinematic assumptions (5) and (6), the axial and shear strains ${\mathrm{S}}_1$ and ${\mathrm{S}}_5$ of a laminate consisting of $\mathrm{N}$ layers and ${\mathrm{N}}_{\mathrm{d}}$ delamination cracks take the following form:

$${\mathrm{S}}_1={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{S}}_1^{\mathrm{i}}{\mathrm{\psi }}^{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}{\tilde{\mathrm{S}}}_1^{\mathrm{k}}{\mathrm{\psi }}^{\mathrm{k}}\mathrm{H}\left({\mathrm{z}-\mathrm{z}}_{\mathrm{k}}\right)}$$

(10)

$${\mathrm{S}}_3={\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}{\tilde{\mathrm{w}}}^{\mathrm{k}}}\widehat{\mathrm{\delta }}\left(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}}\right)$$

(11)

$${\mathrm{S}}_5={\mathrm{w}}_{,\mathrm{x}}^{\mathrm{o}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{u}}^{\mathrm{i}}{\mathrm{\psi }}_{,\mathrm{z}}^{\mathrm{i}}+{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}\left[\left({\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}}+{\tilde{\mathrm{u}}}^{\mathrm{k}}{\mathrm{\psi }}_{,\mathrm{z}}^{\mathrm{k}}\right)\mathrm{H}(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})+{\tilde{\mathrm{u}}}^{\mathrm{k}}{\mathrm{\psi }}^{\mathrm{k}}\widehat{\mathrm{\delta }}(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})\right]}}$$

(12)

where $\widehat{\mathrm{\delta }}\left(\mathrm{z}\right)$ is the Dirac function. In the axial strain Equation (10), ${\mathrm{S}}_1^{\mathrm{i}}={\mathrm{u}}_{,\mathrm{x}}^{\mathrm{i}}$ represents generalized axial strains of a healthy laminate, whereas ${\tilde{\mathrm{S}}}_1^{\mathrm{k}}={\tilde{\mathrm{u}}}^{\mathrm{k}}{,}_{\mathrm{x}}$ are new strain components expressing the effect of the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination on the axial strain above the crack. We further focus attention on new generalized terms appearing in interlaminar strains. In the normal strain Equation (11), ${\tilde{\mathrm{w}}}^{\mathrm{k}}\mathrm{\delta }(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})$ is a new component expressing the relative transverse movement of the delamination crack faces. We define ${\tilde{\mathrm{S}}}_3^{\mathrm{k}}={\tilde{\mathrm{w}}}^{\mathrm{k}}$ and take advantage of this term in order to naturally describe the contact phenomena between the delamination interfaces as explained below. In the shear strain Equation (12), the generalized strain term is as follows:

$${\tilde{\mathrm{S}}}_5^{\mathrm{k}}={\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}}+{\tilde{\mathrm{u}}}^{\mathrm{k}}{\mathrm{\psi }}_{{,\mathrm{z}}_{\mathrm{k}}}^{\mathrm{k}}$$

(13)

This represents a constant shear strain added to all layers above the crack by the presence of the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination, and the last generalized strain term is as follows:

$${\widehat{\mathrm{S}}}_5^{\mathrm{k}}={\tilde{\mathrm{u}}}^{\mathrm{k}}$$

(14)

This is multiplied by the Dirac function, which represents the relative sliding between delamination faces, and has meaning only at the delamination interface.

Also, combining Equations (4) and (7), the transverse electric field in each discrete layer is derived as follows:

$${\mathrm{{\rm E}}}_3^{\mathrm{i}}=-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{\Phi }}^{\mathrm{i}}{\mathrm{\psi }}_{,\mathrm{z}}^{\mathrm{i}}\right)}$$

(15)

This is a constant approximation of the transverse electric field in each discrete layer. The previous strain and electrical field equations were included in the variational form of equilibrium equations, and the equivalent stiffness and mass matrices were derived.

2.5. Delamination Interface Interactions

Up to this point, the formulation described above does not account for interfacial effects and interactions between the crack faces, which allow for penetration between them. To remedy this weakness, and take advantage of strain terms ${\widehat{\mathrm{S}}}_3^{\mathrm{k}}$ and ${\widehat{\mathrm{S}}}_5^{\mathrm{k}}$ in Equations (11) and (12), offering a realistic model for delaminated beams, the generalized stress ${\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}$ and ${\widehat{\mathrm{\sigma }}}_5^{\mathrm{k}}$ are considered in the formulation for the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination crack, which is further related to the generalized strains ${\tilde{\mathrm{S}}}_3^{\mathrm{k}}$ and ${\tilde{\mathrm{S}}}_5^{\mathrm{k}}$, respectively, through the following constitutive relationships:

$$\begin{array}{cc}{\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}=\left\{\begin{array}{cc}{\widehat{\mathrm{a}}}_{33}^{\mathrm{k}}{\tilde{\mathrm{S}}}_3^{\mathrm{k}},& {\tilde{\mathrm{S}}}_3^{\mathrm{k}}<0\\ 0,& {\tilde{\mathrm{S}}}_3^{\mathrm{k}}>0\end{array}\right.& \hspace{1em}{\widehat{\mathrm{\sigma }}}_5^{\mathrm{k}}=\left\{\begin{array}{cc}{\widehat{\mathrm{a}}}_{55}^{\mathrm{k}}{\tilde{\mathrm{S}}}_5^{\mathrm{k}},& {\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}\ne 0\\ 0,& {\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}=0\end{array}\right.\end{array}$$

(16)

where ${\widehat{\mathrm{A}}}_{33}^{\mathrm{k}}$ and ${\widehat{\mathrm{D}}}_{55}^{\mathrm{k}}$ are coefficients to be determined, and their values are defined in the next section. These generalized constitutive relations effectively provide nonlinear contact and sliding stiffness, thereby controlling penetration and free axial slip when the two delamination faces are in contact. Due to the presence of the Dirac function, the stresses ${\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}$ and ${\widehat{\mathrm{\sigma }}}_5^{\mathrm{k}}$ only have meaning at the delamination interfaces and represent the contact stresses at the interface. The above system of generalized stresses works as follows: when the faces of the ${\mathrm{k}}^{\mathrm{t}\mathrm{h}}$ delamination is open, ${\tilde{\mathrm{S}}}_3^{\mathrm{k}}$ is positive, and both ${\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}$ and ${\widehat{\mathrm{\sigma }}}_5^{\mathrm{k}}$ vanish; hence, no stresses are applied at the interface. On the other hand, when the crack closes $\left({\tilde{\mathrm{S}}}_3^{\mathrm{k}}\le 0\right)$, ${\widehat{\mathrm{\sigma }}}_3^{\mathrm{k}}$ becomes nonzero and resists penetration; at the same time, the shear stress ${\widehat{\mathrm{\sigma }}}_5^{\mathrm{k}}$ acts along the axial direction of the crack faces to resist slipping. The described system of constitutive Equations (16) above is nonlinear and introduces nonlinearity into the formulation.

2.6. Laminate Electric Enthalpy and Laminate Matrices

Considering the assumptions of this work and combining Equations (4), (10)–(12), and (14), the electric enthalpy of the laminate in each discrete layer $i$ takes the following form:

$$\mathrm{\delta }{\mathcal{H}}^{\mathrm{i}}={\displaystyle \sum _{\mathrm{\lambda }=1}^{{\mathrm{{\rm N}}}_{\mathrm{p}}^{\mathrm{i}}}{\displaystyle \underset{{\mathrm{z}}_{\mathrm{\lambda }}}{\overset{{\mathrm{z}}_{\mathrm{\lambda }+1}}{\int }}\left(\mathrm{\delta }{\mathrm{S}}_{\mathrm{i}}^{\mathrm{\lambda }}{\mathrm{C}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\lambda }}{\mathrm{S}}_{\mathrm{j}}^{\mathrm{\lambda }}-\mathrm{\delta }{\mathrm{{\rm E}}}_3^{\mathrm{\lambda }}{\mathrm{e}}_{31}^{\mathrm{\lambda }}{\mathrm{S}}_1^{\mathrm{\lambda }}-\mathrm{\delta }{\mathrm{S}}_1^{\mathrm{\lambda }}{\mathrm{e}}_{31}^{\mathrm{\lambda }}{\mathrm{E}}_3^{\mathrm{\lambda }}-\mathrm{\delta }{\mathrm{E}}_3^{\mathrm{\lambda }}{\mathrm{\epsilon }}_{33}^{\mathrm{\lambda }}{\mathrm{{\rm E}}}_3^{\mathrm{\lambda }}\right)\mathrm{d}\mathrm{z}}}$$

(17)

where $\mathrm{i},\mathrm{j}=\mathrm{1,5}$ and $\mathrm{\lambda }$ denotes the ${\mathrm{\lambda }}^{\mathrm{t}\mathrm{h}}$ ply in the ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$ discrete layer; ${\mathrm{N}}_{\mathrm{p}}^{\mathrm{i}}$ denotes the plies of the ${\mathrm{i}}^{\mathrm{t}\mathrm{h}}$ discrete layer. Also, assuming that, in most cases, continuous electrodes exist on the surface of a piezoceramic layer or wafer, which imposes a uniform axial voltage distribution, the axial component of the electrical field in Equation (4) was ignored. Interfacial contact and friction between the delamination faces are introduced via the terms and constitutive Equations (16) described previously. Combining Equations (10)–(12), (14) and (17), integrating through the thickness of each discrete layer and collecting the common terms, the variation of laminate enthalpy is related to the resultant laminate stiffness, piezoelectric and permittivity matrices:

$$\begin{array}{l}\mathrm{\delta }{\mathcal{H}}_{\mathrm{L}}=\mathrm{\delta }{\mathrm{w}}_{,\mathrm{x}}{\mathrm{A}}_{55}{\mathrm{w}}_{,\mathrm{x}}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{N}}\left(\mathrm{\delta }{\mathrm{w}}_{,\mathrm{x}}{\mathrm{B}}_{55}^{\mathrm{n}}{\mathrm{u}}^{\mathrm{n}}+\mathrm{\delta }{\mathrm{u}}^{\mathrm{m}}{\mathrm{B}}_{55}^{\mathrm{m}}{\mathrm{w}}_{,\mathrm{x}}\right)}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{N}}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left(\mathrm{\delta }{\mathrm{u}}_{,\mathrm{x}}^{\mathrm{m}}{\mathrm{D}}_{11}^{\mathrm{m}\mathrm{n}}{\mathrm{u}}_{,\mathrm{x}}^{\mathrm{n}}+\mathrm{\delta }{\mathrm{u}}^{\mathrm{m}}{\mathrm{D}}_{55}^{\mathrm{m}\mathrm{n}}{\mathrm{u}}^{\mathrm{n}}\right)}}+\\ {\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}\left(\mathrm{\delta }{\mathrm{w}}_{,\mathrm{x}}{\overline{\mathrm{A}}}_{55}^{\mathrm{k}}{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}}+\mathrm{\delta }{\mathrm{w}}_{,\mathrm{x}}{\overline{\mathrm{B}}}_{55}^{\mathrm{k}}{\tilde{\mathrm{u}}}^{\mathrm{k}}+\mathrm{\delta }{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}}{\overline{\mathrm{A}}}_{55}^{\mathrm{k}}{\mathrm{w}}_{,\mathrm{x}}+\mathrm{\delta }{\tilde{\mathrm{u}}}^{\mathrm{k}}{\overline{\mathrm{B}}}_{55}^{\mathrm{k}}{\mathrm{w}}_{,\mathrm{x}}\right)}+\\ {\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{N}}{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}\left(\mathrm{\delta }{\mathrm{u}}_{,\mathrm{x}}^{\mathrm{m}}{\overline{\mathrm{D}}}_{11}^{\mathrm{m}\mathrm{k}}{\tilde{\mathrm{u}}}_{,\mathrm{x}}^{\mathrm{k}}+\mathrm{\delta }{\mathrm{u}}^{\mathrm{m}}{\overline{\mathrm{F}}}_{55}^{\mathrm{m}\mathrm{k}}{\mathrm{w}}_{,\mathrm{x}}^{\mathrm{k}}+\mathrm{\delta }{\mathrm{u}}^{\mathrm{m}}{\overline{\mathrm{D}}}_{55}^{\mathrm{m}\mathrm{k}}{\tilde{\mathrm{u}}}^{\mathrm{k}}+\mathrm{\delta }{\tilde{\mathrm{u}}}_{,\mathrm{x}}^{\mathrm{k}}{\overline{\mathrm{D}}}_{11}^{\mathrm{m}\mathrm{k}}{\mathrm{u}}_{,\mathrm{x}}^{\mathrm{m}}+\mathrm{\delta }{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}}{\overline{\mathrm{F}}}_{55}^{\mathrm{m}\mathrm{k}}{\mathrm{u}}^{\mathrm{m}}+\mathrm{\delta }{\tilde{\mathrm{u}}}^{\mathrm{k}}{\overline{\mathrm{D}}}_{55}^{\mathrm{m}\mathrm{k}}{\mathrm{u}}^{\mathrm{m}}\right)}+}\\ {\displaystyle \sum _{{\mathrm{k}}_1=1}^{{\mathrm{N}}_{\mathrm{d}}}{\displaystyle \sum _{{\mathrm{k}}_2=1}^{{\mathrm{N}}_{\mathrm{d}}}\left(\mathrm{\delta }{\tilde{\mathrm{u}}}_{,\mathrm{x}}^{{\mathrm{k}}_1}{\tilde{\mathrm{D}}}_{11}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{u}}}_{,\mathrm{x}}^{{\mathrm{k}}_2}+\mathrm{\delta }{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{{\mathrm{k}}_1}{\tilde{\mathrm{A}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{{\mathrm{k}}_2}+\mathrm{\delta }{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{{\mathrm{k}}_1}{\tilde{\mathrm{B}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{u}}}^{{\mathrm{k}}_2}+\mathrm{\delta }{\tilde{\mathrm{u}}}^{{\mathrm{k}}_1}{\tilde{\mathrm{B}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{{\mathrm{k}}_2}+\mathrm{\delta }{\tilde{\mathrm{u}}}^{{\mathrm{k}}_1}{\tilde{\mathrm{D}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{u}}}^{{\mathrm{k}}_2}\right)}}+\\ {\displaystyle \sum _{{\mathrm{k}}_1=1}^{{\mathrm{N}}_{\mathrm{d}}}{\displaystyle \sum _{{\mathrm{k}}_2=1}^{{\mathrm{N}}_{\mathrm{d}}}\left(\mathrm{\delta }{\tilde{\mathrm{u}}}^{{\mathrm{k}}_1}{\widehat{\mathrm{D}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{u}}}^{{\mathrm{k}}_2}+\mathrm{\delta }{\tilde{\mathrm{w}}}^{{\mathrm{k}}_1}{\widehat{\mathrm{A}}}_{33}^{{\mathrm{k}}_1{\mathrm{k}}_2}{\tilde{\mathrm{w}}}^{{\mathrm{k}}_2}\right)}+}\\ {\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{N}}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left(\mathrm{\delta }{\mathrm{\Phi }}^{\mathrm{m}}{\mathrm{P}}_{31}^{\mathrm{m}\mathrm{n}}{\mathrm{u}}_{,\mathrm{x}}^{\mathrm{n}}+\mathrm{\delta }{\mathrm{\Phi }}^{\mathrm{m}}{\mathrm{G}}_{33}^{\mathrm{m}\mathrm{n}}{\mathrm{\Phi }}^{\mathrm{n}}\right)}}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{N}}{\displaystyle \sum _{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{d}}}\left(\mathrm{\delta }{\mathrm{\Phi }}^{\mathrm{m}}{\overline{\mathrm{P}}}_{31}^{\mathrm{m}\mathrm{k}}{\tilde{\mathrm{u}}}_{,\mathrm{x}}^{\mathrm{k}}\right)}}\end{array}$$

(18)

In the previous equation, ${\mathrm{D}}_{11}^{\mathrm{m}\mathrm{n}}$ is the in-plane generalized laminate stiffness, and ${\mathrm{A}}_{55}^{\mathrm{m}\mathrm{n}},{\mathrm{B}}_{55}^{\mathrm{m}\mathrm{n}}$, and ${\mathrm{D}}_{55}^{\mathrm{m}\mathrm{n}}$ are the interlaminar generalized laminate stiffness matrices of the undamaged section. In the delaminated region, additional stiffness terms exist: ${\overline{\mathrm{D}}}_{11}^{\mathrm{m}\mathrm{k}}$ is the generalized in plane laminate stiffness, and ${\overline{\mathrm{A}}}_{55}^{\mathrm{k}},{\overline{\mathrm{B}}}_{55}^{\mathrm{k}},{\overline{\mathrm{F}}}_{55}^{\mathrm{m}\mathrm{k}}$, and ${\overline{\mathrm{D}}}_{55}^{\mathrm{m}\mathrm{k}}$ are the generalized shear stiffness matrices. Each one of the overbar terms is a new stiffness term in the damage section, representing coupling between pristine and delamination degrees of freedom. Additionally, ${\tilde{\mathrm{D}}}_{11}^{{\mathrm{k}}_1{\mathrm{k}}_2}$, ${\tilde{\mathrm{A}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2},{\tilde{\mathrm{B}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}$, and ${\tilde{\mathrm{D}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}$ are new laminate stiffness matrices introduced by the delamination degrees of freedom. Similarly, ${\mathrm{P}}_{31}^{\mathrm{m}\mathrm{n}}$ are the generalized piezoelectric coefficients of the pristine laminate, and ${\overline{\mathrm{P}}}_{31}^{\mathrm{m}\mathrm{k}}$ represents additional generalized effective piezoelectric coefficients of the laminate due to delamination. The above overbar terms are new and describe the effect of displacement disconnect on the effective direct and converse piezoelectric effect of the respective piezoelectric layer, and ${\mathrm{G}}_{33}^{\mathrm{m}\mathrm{n}}$ are the generalized permittivity laminate matrices. All the above matrices are provided in the Appendix A section of this paper. Special attention is required for the matrices ${\widehat{\mathrm{A}}}_{33}^{{\mathrm{k}}_1{\mathrm{k}}_2}$, and ${\widehat{\mathrm{D}}}_{55}^{{\mathrm{k}}_1{\mathrm{k}}_2}$, which are the new generalized stiffness terms due to interfacial contact and slip between the delamination surfaces as described in the previous section, and are given from the following:

$${\widehat{A}}_{33}^{k_1{k}_2}=\sum _{l=1}^{N_p}{\int }_{z_l}^{z_{l+1}}{C}_{33}^l\widehat{\delta }(z-z_{k_1})\widehat{\delta }(z-z_{k_2})dz$$

(19)

$${\widehat{D}}_{55}^{k_1{k}_2}=\sum _{l=1}^{N_p}{\int }_{z_l}^{z_{l+1}}{a}_5^l\widehat{\delta }(z-z_{k_1})\widehat{\delta }(z-z_{k_2})dz$$

(20)

In the equations above ${\mathrm{C}}_{33}^{\mathrm{l}}$ and ${\mathrm{a}}_5^{\mathrm{l}}$ are coefficients that need to be determined. The first coefficient, without loss of generality, is assumed equal to the normal elasticity modulus E₃₃. The second coefficient depends on the quality of the two interfaces that are in contact and is activated only when ${\tilde{\mathrm{S}}}_3^{\mathrm{k}}$ is negative. The authors have performed various computational studies, concluding that the value of the term ${\mathrm{a}}_5^{\mathrm{l}}$ should vary in the range 1 × 10⁸ to 1 × 10¹⁰, as for lower values the response remains insensitive to the presence of this term, and on the other hand, if the value of this term is over the range eliminates the axial movement.

2.7. Finite Element Formulation

A beam finite element was formulated based on the previous laminate mechanics, implementing ${\mathrm{C}}_1$ continuous Hermitian polynomials ${\mathrm{H}}^{\mathrm{i}}\left(\mathrm{x}\right)$ for the local approximation of the transverse displacements and ${\mathrm{C}}_{\mathrm{o}}$ shape functions ${\mathrm{N}}^{\mathrm{i}}\left(\mathrm{x}\right)$ for the remaining axial and electric DOFs. In this manner, the local approximations of the generalized state variables in the element take the following form:

$$\begin{array}{l}{\mathrm{u}}^{\mathrm{i}}(\mathrm{x},\mathrm{t})={\displaystyle \sum _{\mathrm{m}=1}^L{\mathrm{u}}^{\mathrm{i}\mathrm{m}}}(\mathrm{t}){\mathrm{N}}^{\mathrm{m}}(\mathrm{x})\\ {\tilde{\mathrm{u}}}^{\mathrm{k}}(\mathrm{x},\mathrm{t})={\displaystyle \sum _{\mathrm{m}=1}^L{\tilde{\mathrm{u}}}^{\mathrm{k}\mathrm{m}}}(\mathrm{t}){\mathrm{N}}^{\mathrm{m}}(\mathrm{x})\\ {\mathrm{w}}^{\mathrm{o}}(\mathrm{x},\mathrm{t})={\displaystyle \sum _{\mathrm{m}=1}^L\left({\mathrm{w}}^{\mathrm{o}\mathrm{m}}(\mathrm{t}){\mathrm{H}}_1^{\mathrm{m}}(\mathrm{x})+\frac{{\mathrm{L}}_{\mathrm{e}}}{2}{\mathrm{w}}_{,\mathrm{x}}^{\mathrm{o}\mathrm{m}}(\mathrm{t}){\mathrm{H}}_2^{\mathrm{m}}(\mathrm{x})\right)}\\ {\tilde{\mathrm{w}}}^{\mathrm{k}}(\mathrm{x},\mathrm{t})={\displaystyle \sum _{\mathrm{m}=1}^L\left({\tilde{\mathrm{w}}}^{\mathrm{k}\mathrm{m}}(\mathrm{t}){\mathrm{H}}_1^{\mathrm{m}}(\mathrm{x})+\frac{{\mathrm{L}}_{\mathrm{e}}}{2}{\tilde{\mathrm{w}}}_{,\mathrm{x}}^{\mathrm{k}\mathrm{m}}(\mathrm{t}){\mathrm{H}}_2^{\mathrm{m}}(\mathrm{x})\right)}\\ {\mathrm{\Phi }}^{\mathrm{k}}(\mathrm{x})={\displaystyle \sum _{\mathrm{i}=1}^L{\mathrm{\Phi }}^{\mathrm{k}\mathrm{i}}}{\mathrm{N}}^{\mathrm{i}}(\mathrm{x})\end{array}$$

(21)

where $\mathrm{i}=1,\dots ,\mathrm{N}+1$ and $\mathrm{k}=1\dots ,{\mathrm{N}}_{\mathrm{d}}$ indicate discrete layer and delamination; $\mathrm{m}=1,\dots ,\mathrm{L}$ is the node number; L_e denotes the element length. A 2-node (L = 2) beam element was further developed and encoded in a prototype research code employing linear interpolation functions ${\mathrm{N}}^{\mathrm{i}}$ and cubic Hermitian polynomials ${\mathrm{H}}_1^{\mathrm{i}}$,${\mathrm{H}}_2^{\mathrm{i}}$.

Substituting Equations (18) and (21) into the governing equations of equilibrium (1), then collecting the common coefficients, the coupled piezoelectric system was expressed in the standard discrete matrix form:

$$\begin{array}{l}\left[\begin{array}{cc}\left[{\mathrm{M}}_{\mathrm{u}\mathrm{u}}\right]& 0\\ 0& 0\end{array}\right]\left\{\begin{array}{c}\left\{\ddot{\mathrm{u}}\left(\mathrm{t}\right)\right\}\\ \left\{{\ddot{\mathrm{\phi }}}^{\mathrm{S}}\left(\mathrm{t}\right)\right\}\end{array}\right\}+\left[\begin{array}{cc}\left[{\mathrm{C}}_{\mathrm{u}\mathrm{u}}\right]& 0\\ 0& 0\end{array}\right]\left\{\begin{array}{c}\left\{\dot{\mathrm{u}}\left(\mathrm{t}\right)\right\}\\ \left\{{\dot{\mathrm{\phi }}}^{\mathrm{S}}\left(\mathrm{t}\right)\right\}\end{array}\right\}+\left[\begin{array}{cc}{\left[{\mathrm{K}}_{\mathrm{u}\mathrm{u}}\right]}^{\left(\mathrm{t}\right)}& \left[{\mathrm{K}}_{\mathrm{u}\mathrm{\phi }}^{\mathrm{F}\mathrm{F}}\right]\\ \left[{\mathrm{K}}_{\mathrm{\phi }\mathrm{u}}^{\mathrm{F}\mathrm{F}}\right]& \left[{\mathrm{K}}_{\mathrm{\phi }\mathrm{\phi }}^{\mathrm{F}\mathrm{F}}\right]\end{array}\right]\left\{\begin{array}{c}\left\{\overline{\mathrm{u}}\left(\mathrm{t}\right)\right\}\\ \left\{{\mathrm{\phi }}^{\mathrm{F}}\left(\mathrm{t}\right)\right\}\end{array}\right\}=\\ =\left\{\begin{array}{c}\left\{\mathrm{P}\left(\mathrm{t}\right)\right\}-\left[{\mathrm{K}}_{\mathrm{u}\mathrm{\phi }}^{\mathrm{F}\mathrm{A}}\right]\left\{{\mathrm{\phi }}^{\mathrm{A}}\left(\mathrm{t}\right)\right\}\\ \left\{{\mathrm{Q}}^{\mathrm{F}}\right\}-\left[{\mathrm{K}}_{\mathrm{\phi }\mathrm{\phi }}^{\mathrm{F}\mathrm{A}}\right]\left\{{\mathrm{\phi }}^{\mathrm{A}}\left(\mathrm{t}\right)\right\}\end{array}\right\}\end{array}$$

(22)

Submatrices $\left[{\mathrm{K}}_{\mathrm{u}\mathrm{u}}\right]$, $\left[{\mathrm{K}}_{\mathrm{u}\mathsf{\varphi }}\right]$, and $\left[{\mathrm{K}}_{\mathsf{\varphi }\mathsf{\varphi }}\right]$ indicate the elastic, piezoelectric, and permittivity matrices of the structure; $\left[{\mathrm{M}}_{\mathrm{u}\mathrm{u}}\right]$ indicates the mass matrix. It is worth noting that the previous system matrices implicitly incorporate the delamination parameters into the system response. A proportional damping matrix $\left[{\mathrm{C}}_{\mathrm{u}\mathrm{u}}\right]$ is also assumed to be calculated through mass and stiffness matrices for a specific frequency range. The piezoelectric and capacitance matrices have been partitioned into active and sensory parts. Superscripts F and A indicate, respectively, sensory (free) and active (applied) electric potential components; $\left\{\mathrm{P}\right\}$ is the applied mechanical forces vector, and $\left\{{\mathrm{Q}}^{\mathrm{F}}\right\}$ is the applied electric charge at the sensors.

Also, the above system is nonlinear, since the stiffness matrix $\left[{\mathrm{K}}_{\mathrm{u}\mathrm{u}}\right]$ depends on the local delamination displacements. For the solution of this non-linear problem, a modified Newmark implicit time integration method is used, combined with Newton–Raphson iterations [48], to calculate the model response when the delamination faces contact criteria are activated. The convergence of the Newton–Raphson solution scheme was satisfied in each time step.

3. Materials, Experimental Methods, and Simulation Parameters

Experiments were conducted in order to investigate the time response of delaminated composite beams. Composite beam specimens with a single delamination crack (Figure 2a) were tested using piezoceramic devices, as shown in Figure 2b. The beams had a [0/90/45/−45]_S laminate configuration, with T300/934 graphite epoxy plies of fiber volume ratios in the range of 0.57–0.63, a nominal thickness of 0.127 mm, a length of 280 mm, and a width of 25 mm. Four different sizes of delamination were artificially created at the mid-plane of the composite using Teflon tape during the pre-preg manufacturing procedure, covering the full width across the beam. The Teflon tape was later removed to avoid compromising the validity of the experimental measurements. Four specimens were tested: one undamaged and three with small, medium, and large delamination cracks covering 0%, 10.9%, 21.8%, and 43.6% of the total beam length, respectively.

These specimens were tested under a free-free supporting configuration in order to eliminate friction due to supports. Specimens were excited via piezoceramic devices having dimensions 40 × 10 × 1 mm. The piezoceramics were the PZ27 models of the Ferroperm-Piezo brand. Additionally, to complete the “pitch and catch” testing configuration, the authors used piezopolymer sensors attached to the specimen’s surface to measure the specimen’s vibration. Both the actuator and the sensor were bonded near the modal points of the 1st mode shape, as illustrated in Figure 2a. The piezopolymer dimensions were 30 mm × 12 mm × 70 µm, while the mechanical properties of the piezocomposite materials, as well as the mechanical and electrical properties of the piezoelectric devices, are incorporated in

Table 1. Properties of composite and piezoelectric materials.

Property	T300/934	Pz27 (Ferroperm)	PVDF (LDT1_028K)
E11 (GPa)	127	58.82	3
E33 (GPa)	7.9	43.1	6
v13	0.275	0.371	0.3
G13 (GPa)	3.4	22.98	1
d31 (m/V)	0	−170 × 10−12	−23 × 10−12
d33 (m/V)	0	425 × 10−12	33 × 10−12
d15 (m/V)	0	506 × 10−12
ε31 (farad/m)	0	15.94 × 10−9	106 × 10−12
ε11 (farad/m)	0	15.94 × 10−9	106 × 10−12
ρ (kg/m3)	1578	7700	1780

.

3.1. Testing Apparatus

In order to obtain the free supporting configuration, the specimens were supported with strings attached approximately at the modal points of the first bending mode. A virtual instrument was developed based on Labview^® to digitally generate tone-burst excitation signals having various central frequencies. The signal was converted to analog using a 16-bit DAQ card. The actuation signal was amplified through an appropriate piezodrive amplifier, and it was applied at the terminals of a piezoceramic actuator epoxied on the surface of the composite specimen. The response of the beam was acquired at the other free end of the specimen, measuring the voltage at the terminals of the PVDF piezopolymer film sensor. The measured voltage at the piezoelectric sensors was digitized through a high-speed DAQ board. The described testing configuration is illustrated in Figure 3.

3.2. Simulation Parameters

To simulate the response of the active-sensory composite beam with the delamination debonding, the in-house model used the following discretization:

(1)

Through the thickness, following the linear layerwise assumptions and the requirement for accurate displacement and stress distributions, 30 discrete layers were used; 24 layers (3 for each ply) evenly distributed for the composite laminate, 2 layers on top of the laminate (6.3 μm thick each) for the bonding adhesive of the piezelectrics and 4 layers for the piezoceramic actuator or 1layer for the piezopolymer sensor.

(2)

Across the length, for the FEM mesh, we used 30 beam elements distributed across the length. The length of each beam was selected appropriately to simulate the exact position of the actuator–sensor pair, as well as to provide a denser mesh in the delamination zone.

The time response was performed with a time step of Δt = 0.02 ms, and the actuation signal was applied from the beginning of the analysis.

4. Results and Discussion

This section evaluates the performance of the analytical model against the experimental measurements. The numerical predictions of both linear and non-linear systems are compared, and the derived outcome reveals the necessity of the interfacial contact proposed by the authors.

4.1. Analytical Model Efficiency

In this section, the transient response of smart delaminated composite beams with piezoelectric devices was measured. Four different sizes of delamination were tested, covering 0%, 10.9%, 21.8%, and 43.6% of the total specimen length. The derived actuation voltage was a narrowband 7-cycle tone burst signal with a central frequency of around 1 kHz. Based on the previous work of the authors performed on similar specimens [24], most of the dominant modal frequencies of these beams appear in the from range 80 to 500 Hz, so the excitation frequency level of l kHz was selected in order to avoid the excitation of dominant modal frequencies. The time signal, as well as its frequency content, is presented in Figure 4a and Figure 4b, respectively. Additionally, the transient response of the specimens mentioned above was simulated via the realistic analytical model.

Two different predictions were assumed. In the first case, the transient response of composite beams allowing crack faces to penetrate was simulated. In contrast, in the second scenario, a non-linear system was assumed through the introduction of interfacial contact and a shear control of the axial movement, simulating a type of friction. The values used for the applied shear force, as presented in Equation (20), were for the small delamination a₅₅ = 1 × 10¹⁰, and for the other two damage sizes a₅₅ = 1 × 10⁹. Additionally, proportional damping matrices were calculated for 1KHz.

Figure 5a–d presents both experimental measurements and simulation results for each of the four specimens. Initially, for the pristine specimen (Figure 5a), the two time signals are nearly identical, especially during the first six or seven cycles. After this initial period, the simulation appears to be more damped compared to the experiment. This is due to the damping, which was not analytically formulated but was incorporated as proportional damping through the stiffness and mass matrices. Continuing with the rest of the plots, there is an excellent agreement between the experiment and the model. The introduction of non-linear interactions at the local delamination area enhances the correlation, particularly for beams with small or medium-sized damage. The analytical model accurately captures key characteristics of the time response, such as oscillation frequency, the time of flight, and amplitude, especially for the first six or seven cycles of the transient analysis. The disagreement is evident in the free vibration amplitude due to the damping modeling, as discussed previously. These results demonstrate that the developed realistic model can efficiently simulate the transient response of delaminated active sensory composite beams.

Additional simulations and measurements were obtained for the same set of composite specimens by increasing the central frequency of the applied actuation signal to 5 kHz, as demonstrated in Figure 4c,d. This frequency was selected in order to excite the local oscillations at the delamination interface, as it was previously found [24] that the opening modes of the delamination exist in the range from 4 to 6 kHz, varying with the delamination length. The respective results are illustrated in Figure 6a–d. Each plot represents a different delamination length. Simulations in this case were only for the non-linear model using the same shear force parameters as previously. In this case, for the intact strip and the one with small delamination sizes, the simulations are satisfactorily correlated with the measurements. However, this good agreement disappears after the initial period, presenting differences in both the oscillation amplitude and frequency. For the case of large delamination size (43.6%), differences in comparison appear from the beginning, as each signal requires a different time to travel from the actuator to the sensor. Generally, it was observed for both excitation central frequencies selected for simulations and testing that significant disagreements were captured in the time responses for large delamination sizes. These disagreements were probably owed to the way friction was simulated at the delamination interface. The shear force applied while the interfaces were in contact always had the same magnitude in all cases. Probably, simulating the magnitude of this force as a function of the normal force at the damage interface, assuming a pre-sliding and sliding region, provides more accurate predictions. Nevertheless, the results suggest that improvements in the sliding model may be required, and this could be a topic for future work.

4.2. Delamination Detection

Following a different perspective on the same results, the authors attempted to elucidate the effect of delamination on the transient response. Delamination indices were investigated on various parameters of the time response, such as (i) the signal amplitude, (ii) the time of flight, i.e., the time required for the signal to travel from the actuator to the sensor, and (iii) the frequency of the voltage signal oscillation. From the two actuation signals, time responses were obtained and used only for the 1KHz actuation frequency.

Initially, for the specimen entailing a small delamination crack length, Figure 7a,b presents analytical predictions and experimental measurements, respectively. Each one of these figures illustrates two different curves representing the healthy and the delaminated beam. It is evident from the results that the introduction of damage causes a drastic reduction in the signal amplitude. Additionally, slight differences are observed in the signal frequency and the time of flight. Both the simulations and the measurements lead to the same conclusions. Similar predictions and simulations for the specimens with the medium and large damage sizes are depicted in Figure 8a,b and Figure 9a,b. Focusing on the investigation of the delamination effects on the time of flight, Figure 10a,b presents the time response for each of the four specimens for the initial 2.5 ms via analytical simulations or experimental measurements, respectively. To assist the observation, we consider the time of flight to be the period between the beginning of excitation and the reception of the first cycle at the sensor. It is evident that the ToF increases as the size of the delamination increases with $t_{PR}{<t}_S<t_M<t_L$. This conclusion is also validated by the experimental measurements shown in Figure 10b.

For medium and large damage sizes, discrepancies appear in the amplitude, period, and time of flight, which become more pronounced in the case of large delamination sizes. Correlations of time responses for strips with various damage sizes versus the healthy specimen results provide the following conclusions: (i) as the size of delamination increases changes at time of flight and period are also increased and (ii) signal amplitude cannot consist of an indicator of the damage size, as this parameter changes does not appear to be relative to the delamination length. The amplitude of the measured and simulated sensory voltage appears to be more sensitive to another system characteristic, such as the sensor position. However, in any case, the time response appears to be capable of revealing even small delamination sizes using the described pitch and catch configuration. This is very important, assuming the weak effect of small delamination size on the modal characteristics as presented previously [34].

4.3. Axial and Shear Wave Propagation

This section continues the investigation of delamination characteristics in the time response of delaminated strips. Taking into consideration the scope of the present section, it is assumed that only the axial variable field was considered from the analytical model; hence, a study on the propagation of axial and shear waves at a low frequency range is presented. The healthy and the beam entailing small delamination sizes were modeled as described previously, with the difference that the thickness of the piezoelectric actuators was simulated to be 0.1 mm. The through-the-thickness propagation of axial waves stresses through a section located at the midspan of the beam and at the middle of the delamination, as well (see Figure 11), is presented in Figure 12a–j. Simulations were performed for 1 kHz central frequency tone burst, as previously described. The axial stresses are simulated at the top and bottom faces of each discrete layer. The light horizontal lines illustrate the different ply orientation (eight plies) as each ply was simulated with three individual layers, and a vertical line was added to demonstrate the zero stress plane. Each one of the figures presents a different snapshot of the wave propagation. In any one of the distributions through the thickness, a discontinuity appears at the middle, indicating the position of the crack through the thickness. Also, as the transient response continues, the amplitude of predicted axial stresses increases. Additionally, wave propagation presents a signal initially driving towards the right, from the actuator to the sensor. After reaching the right boundary of the specimen, it reflects and drives towards the other end, the left direction, and this vibration continues till the burnout of the vibration. Discontinuity at the middle indicates that two separate axial waves are vibrating separately on the two sides of the delamination.

Similarly, Figure 13a–j illustrates the propagation of a shear wave through a section located at the integration point of a 2-node finite element at the left edge of delamination with both nodes without delamination DOFs (see Figure 11). Shear stresses are simulated at the middle of each layer. Similar to the axial stresses, the light horizontal lines illustrate the different ply orientation (eight plies) as each ply was simulated with three individual layers, and a vertical line was added to demonstrate the zero stress plane. Variation in the shear stresses indicates the position of damage, resulting in a different distribution compared to the healthy beam. Initially, due to the excitation period and the actuator position on the top beam’s surface, the signal is not symmetric for both the healthy and delaminated beams. Similarly to axial stresses, the magnitude of shear stresses increases during the period of transient analysis, and the signal propagates from left to right and vice versa. The most important conclusion of the shear wave propagation is that shear stresses are sensitive to revealing the existence of delamination using stress distribution, even if the measuring position is away from the damage. Additionally, it is crucial to investigate the magnitude of interlaminar stresses that appear at the crack edges during free vibration and determine the critical stress values for crack propagation.

4.4. Contour Plots of Axial and Shear Stresses

To complete this study of delamination signatures on the low-frequency time response of delaminated composite beams, contour plots were also created. For the same excitation as in the previous case, and two delamination sizes, the small (10.9%) and the medium (21.8%) axial and shear stresses were predicted in any position through the beam’s thickness. Figure 14a–f presents contour plots of the simulated axial stresses. Each plot illustrates the magnified x-z view of the composite beam as presented in Figure 11. Piezoceramic devices, including actuators and sensors, are similar to the previous case, and their positions are also indicated in each graph. In Figure 14, plots (a), (c), and (e) represent, for t = 0.5 ms of the transient analysis, the healthy beam and the beams with small and medium delamination sizes, respectively. Similarly, plots (b), (d), and (f) of Figure 14 are the axial stress contours for t = 1.6 ms. From the first three plots, representing the t = 0.5 ms snapshot since the beginning of the transient analysis, the exact position of delamination is mapped when the signal arrives at the crack. The extent of delamination crack, as well as its position through the thickness, is extensively indicated. Similar conclusions are drawn from the second set of plots, which present a snapshot of the system during free structural vibration after the relaxation of the transient phenomena.

As described in the previous case for axial stresses, Figure 15a–f presents contour plots of the interlaminar stresses for any point through the beam thickness at two different time moments of the transient analysis. Similarly, the x-z view is presented. The first snapshot, at t = 1.8 ms, shows the initial arrival of the shear wave at the delamination region during its propagation, providing a slight indication of the exact extent of the damage. At the second snapshot, at t = 8.6 ms, the contour shows a stationary shear wave propagating across the beam. When a delamination crack exists through the thickness, it appears as a disturbance in the distribution of shear stresses. The exact crack position and extent of damage are presented. Also, damage affects the magnitude of shear stresses in a region around the crack, indicating the appropriate positions to measure the shear stresses and reveal damage. Finally, comparing the contours of the axial and shear waves, it is clear that the axial wave signal propagates faster across the beam. It is a logical conclusion as a longitudinal wave propagates faster than a transverse wave.

5. Summary and Conclusions

This paper presents an analytical and experimental study of the time response of delaminated active-sensory composite beams. A previously developed analytical model capable of simulating delaminated composite beams with piezoelectric devices was extended to naturally include contact and sliding at the delamination interfaces. The resultant electromechanical system was non-linear due to the interface contact, and the stiffness matrix was updated in each time step of the transient analysis. The time response of this non-linear system was predicted using a modified Newmark time integration method, combined with Newton–Raphson iterations in each time step, until the least squares error was minimized. Representative experiments were further performed, and the time response of composite beams was measured using an experimental setup that provided actuation via a piezoceramic actuator and vibration acquisition using a piezopolymer sensor, both of which were attached to the beam surface. Four different specimens were tested: one healthy and three delaminated, each with a different damage length. Tests and simulations were conducted under free-supporting conditions to eliminate disturbances in the response from the boundaries.

An excellent agreement was observed between predictions and measurements, especially for the 1 kHz actuation. The non-linear analytical model, which assumes contact and control of axial movement at the delamination region, improved correlation with measurements. A good performance of the simulation in capturing the experiment was also presented for higher frequencies (5 kHz). However, for large delamination sizes, differences appeared between simulations and measurements. This was probably due to the way friction was modeled at the damage interface and to the assumed constant normal displacement field. Additionally, due to the model’s weakness in capturing the oscillation amplitude during the free vibration period, damping modeling would probably improve correlations.

Signatures of damage were detected in the time of flight, oscillation frequency, and amplitude of the sensory signal. Changes in these characteristics presented sensitivity even for small delamination sizes. Also, captured modifications on the time of flight and period were relative to the delamination length. These conclusions were duplicated from the experimental measurements.

Continuing the axial and shear wave propagation was predicted through sections along the damaged and healthy composite beams. Axial and shear waves exhibited sensitivity to the existence of cracks, and especially the shear stress distributions through the thickness were sensitive to revealing damage, even when the simulation position was away from the damage. Finally, contour plots of the axial and shear stresses were illustrated at various snapshots. These figures demonstrate the propagation of an axial or shear wave from the actuator towards the sensor, as well as the wave interactions with the delamination crack. These plots demonstrate interactions throughout the thickness and can provide valuable information about the optimal positions of actuators and sensors, facilitating the construction of an efficient health monitoring system.