Enhancing Aircraft Crack Repair Efficiency PDF

Heliyon 10 (2024) e32166 Contents lists available at ScienceDirect Heliyon journal homepage: www.cell.com/h...

Heliyon 10 (2024) e32166 Contents lists available at ScienceDirect Heliyon journal homepage: www.cell.com/heliyon Research article Enhancing aircraft crack repair efficiency through novel optimization of piezoelectric actuator parameters: A design of experiments and adaptive neuro-fuzzy inference system approach Abdul Aabid a, *, Meftah Hrairi b, **, Md Abdul Raheman c, Yasser E. Ibrahim a a Department of Engineering Management, College of Engineering, Prince Sultan University, PO BOX 66833, Riyadh, 11586, Saudi Arabia b Department of Mechanical and Aerospace Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50725, Kuala Lumpur, Malaysia c Department of Electrical and Electronics Engineering, NMAM Institute of Technology (NMAMIT), Nitte (Deemed to be University), Nitte, Karkala Taluk, Udupi, 574110, Karnataka, India A R T I C L E I N F O A B S T R A C T Keywords: This study addressed the critical problem of repairing cracks in aging aircraft structures, a safety Aircraft crack repair concern of paramount importance given the extended service life of modern fleets. Utilized a Stress intensity factor finite element (FE) method enhanced by the design of experiments (DOE) and adaptive neuro- Piezoelectric actuators fuzzy inference system (ANFIS) approaches to analyze the efficacy of piezoelectric actuators in Finite element method Experiment design mitigating stress intensity factors (SIF) at crack tips—a novel integration in structural repair Adaptive neuro-fuzzy inference system strategies. Through simulations, we examined the impact of various factors on the repair process, including the plate, actuator, and adhesive bond size and characteristics. In this work, initially, the SIF estimation used the FE approach at crack tips in aluminum 2024-T3 plate under the uniform uniaxial tensile load. Next, numerous simulations have been performed by changing the parameters and their levels to collect the data information for the analysis of the DOE and ANFIS approach. The FE simulation results have shown that changing the parameters and their levels will result in changing of SIF. Several DOE and ANFIS optimization cases have been performed for the depth analysis of parameters. The current results indicated that optimal placement, size, and voltage applied to the piezoelectric actuators are crucial for maximizing crack repair efficiency, with the ability to significantly reduce the SIF by a quantified percentage under specific condi­ tions. This research surpasses previous efforts by providing a comprehensive parameter optimi­ zation of piezoelectric actuator application, offering a methodologically advanced and practically relevant pathway to enhance aircraft structural integrity and maintenance practices. The study innovation lies in its methodological fusion, which holistically examines the parameters influ­ encing SIF reduction in aircraft crack repair, marking a significant leap in applying intelligent materials in aerospace engineering. * Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (A. Aabid), [email protected] (M. Hrairi). https://doi.org/10.1016/j.heliyon.2024.e32166 Received 12 January 2024; Received in revised form 26 May 2024; Accepted 29 May 2024 Available online 1 June 2024 2405-8440/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). A. Aabid et al. Heliyon 10 (2024) e32166 Abbreviations A Actuator cross-sectional area ANFIS Adaptive neuro-fuzzy inference system APDL ANSYS parametric design language CZM Cohesive zone material DF Degrees of freedom DOE Design of experiments FE Finite element NSIF Normalized stress intensity factor OA Orthogonal array PZT Lead zirconate titanate S Position of the actuator SIF Stress intensity factors t Actuator thickness ta Adhesive thickness V Voltage 1. Introduction Modern aircraft often exceed their intended lifespan, and the maintenance of aging structural components has become a global concern. Researchers in structural health monitoring are increasingly focused on this subject due to the growing demand for an efficient, cost-effective, and reliable monitoring system to ensure the functionality and safety of these structures. The cyclic stresses and corrosive operating environments that aircraft endure make them vulnerable to fractures over time. For instance, fatigue cracks may develop in corroded rivet holes, necessitating their detection and correction before they lead to catastrophic failures. Therefore, over the last two decades, some works have been reported to repair such cracks in aircraft structures. For the repair, a bonded composite patch was found in several studies [2–6], while bonded piezoelectric actuators became popular over the last decade for the repair of aircraft structures [7–9]. In initial investigations, researchers used mathematical models to examine damaged structures by assessing fracture parameters, including stress intensity and stress concentration factor and the FE method. This methodology was subsequently applied to the computation of SIF for bonded laminate structures [12,13]. The research expanded to encompass variations in these parameters after successfully determining fracture parameters , different damage propagation modes [15,16], and the effects of single-sided and double-sided composites. Bonded composite repair methods through experimental studies were found in 2007 by Hosseini-Toudeshky et al. , who investigated fatigue crack growth in mode I for repaired aluminum plates with a bonded com­ posite patch. Additionally, they obtained numerical results by varying the patch parameters to investigate optimum results and compared them with the experimental work to validate simulation results. Recent studies estimated the SIF of a crack tip and crack tip initiated from the circular hole in one case. In another case, without a hole , the results show variation in SIF, as a crack hole breaks the structure fast when the tensile load is applied. This was determined using a numerical method, and the stress concentration factor (SCF) was also determined using the same FE method by evaluating the stress values around the hole. In both studies, the authors used the FE method to investigate the effect of the bonded composite patch and adhesive bond parameters by changing the crack length and circular hole. An optimization technique was also helpful after investigating the bonded composite repair method via experimental, FE, and mathematical modeling. Most of the studies used The design of experiments to optimize the best possible outcomes with fewer experimentations, numerical simulations, and mathematical modeling work. Piezoelectric materials are intelligent materials, and lead zirconate titanate (PZT) is the most widely used piezoelectric ceramic material. The PZT materials have electromechanical effects that can be used for electrical and mechanical investigation. PZT can be sensed and actuated with force and voltage. The early investigation used PZT actuators as intelligent structure-like structures with highly distributed actuators and sensors. Piezoelectric materials have been found to investigate vibration control , noise control , and delamination control [23,24]. In previous research, the PZT method has been actively employed to repair damaged structures under dynamic loads by utilizing its electromechanical properties to induce localized moments. Researchers such as Wu and Wang utilized PZT actuator patches to numerically and analytically restore delaminated plates subjected to uniaxial tensile strain. Dawood et al. used the LS-DYNA explicit FE code to investigate delamination control in structures with PZT actuators through low-velocity impact tests. Abuzaid et al. [28,29] conducted experiments, FE simulations, and analytical work to repair center- and edge-cracked aluminum plates under uniaxial tensile loads using PZT actuator patches. Aabid et al. developed models for repairing center-holed aluminum plates with bonded PZT actuators, successfully addressing high-stress zones on circular holes by adjusting voltage within specific limits. In recent investigations, researchers have also explored using PZT materials adhesively bonded to structures in various applications such as vibration, noise, and actual control of columns, beams, thin plates, etc. For instance, composite patches have assessed vibration excitation and separation in laminates. At the same time, concrete beams have been used for damage evaluation in reinforced concrete structures with lap splices of tensional steel bars. Additionally, the effectiveness of PZT actuators has been examined 2 A. Aabid et al. Heliyon 10 (2024) e32166 when combined with viscoelastic bonding layers on elastic structures. In some cases, researchers have evaluated the performance of surface-bonded piezoelectric sensors and actuators on host structures, considering factors such as adhesive layer effects and non-uniform thickness, through FE simulations. However, it’s worth noting that there have not been any reported studies on reducing crack propagation in thin plates in the past four years, except for the work conducted by Aabid et al. and Abuzaid et al.. Therefore, the current focus is reducing the SIF factor bonded with PZT actuators. Another novelty of the current work is the adopted methodology to determine the results, the DOE and ANFIS approach. Also, DOE selected parameters based on the existing work of bonded composite patch and found that the chosen parameters impact the current work’s response, which has been suc­ cessfully determined. Current research is focused on improving aircraft structural components by adding bonded piezoelectric actuators, primarily made of PZT. The uniqueness of this approach lies in integrating PZT materials, known for their high efficiency in converting electrical energy into mechanical strain. This allows precise control over stress distribution around cracks, improving crack repair processes. One significant advantage of these structures is their ability to actively control and reduce SIF in real-time, dynamically responding to aircraft loads. This active control extends the lifespan of repairs and components, as PZT actuators can adaptively manage mechanical stress concentration caused by cyclic loading and operational conditions. Additionally, using PZT actuators enhances structural health monitoring capabilities significantly. These smart materials serve as both sensors and actuators, autonomously detecting and responding to cracks. This dual function enhances aircraft safety and reduces maintenance downtime. In designing repairs for cracks in aircraft structures, it is crucial to ensure that the patch absorbs a substantial portion of the load near the crack while maintaining structural integrity during service without debonding. Achieving this balance depends on several factors, including the intrinsic and geometrical properties of the actuator and adhesive. Prior researchers have explored changes in the SIF as a function of these factors, but their investigations were limited to altering one parameter at a time. This study aims to enhance this approach by investigating the phenomenon globally. Initially, we focus on critical parameters: actuator position, cross-sectional area, actuator thickness, and adhesive thickness. By employing a complete factorial design, we simulate various cases to explore the parameter space comprehensively. To achieve this, we utilize the DOE and ANFIS methods, which have shown relevance for this type of analysis. These advanced techniques enable us to identify the most influential aspects that significantly impact the final repair outcome. Integrating the DOE and ANFIS methods into the design of these structures is another innovative aspect. This work’s novel contribution lies in applying the DOE and ANFIS approaches to explore the effects of these characteristics on the repair process and optimize performance. This combination allows for more sophisticated analysis and optimization of the multiple parameters affecting the repair process, such as actuator position, cross-sectional area, thickness, and adhesive bond thickness. This is the first study to employ the DOE and ANFIS methods for bonded PZT actuators in damaged structures, providing valuable insights into the current problem. The novel use of these advanced analytical techniques is unprecedented in PZT actuator-based crack repair, positioning our research at the forefront of technological advancements in aerospace repair. Through this approach, we identified the most influential parameters for achieving our repair objectives and suggested adjustments to enhance the actuator’s quality. The primary goal of this work is to optimize the actuator’s performance by determining the values of its optimal parameters. By thoroughly investigating the interaction between different parameters, we aim to improve crack repair efficiency while ensuring the actuator’s effectiveness in absorbing the imposed load. The insights gained from this study will contribute to advancements in crack repair methodologies, Fig. 1. Center-cracked aluminum plate with piezoelectric actuators. 3 A. Aabid et al. Heliyon 10 (2024) e32166 benefiting the aerospace industry and promoting safer and more reliable aircraft structures. 2. Problem formulation The present study focuses on aerospace applications and utilizes a thin plate model to simulate a thin-walled structure. As depicted in Fig. 1, the investigation involves a damaged rectangular plate with an integrated piezoelectric actuator subjected to homogeneous uniaxial stress. The primary objective is to induce stress using an electric field and to achieve this, the piezoelectric actuator is completely bonded around the high-stress region of the fractured plate. Depending on whether the piezoelectric actuator is operating in extension or contraction mode, it is expected to either reduce or increase the SIF. The distance between the actuator and the fracture edge plays a crucial role in transmitting stress to the crack surfaces, directly influencing the effectiveness of the repair. It is expected that the edge of the piezoelectric actuator will be near the fracture point since the integrated construction receives intense stresses at that location. Additionally, it is assumed that tremendous tension at the actuator’s edge will persist for approximately twice the thickness of the actuator in distance. The study considers explicitly Mode-I fracture, and the plate material is assumed to be homogeneous, isotropic, and linearly elastic. By exploring the interaction between the piezoelectric actuator and the damaged plate under these conditions, this research con­ tributes essential insights into optimizing the actuator’s positioning and operation mode for effective crack repair. The proposed methodology holds significant importance as it addresses the critical aspects influencing the repair process in thin-walled aerospace structures, potentially leading to enhanced aircraft safety and maintenance practices. 3. Finite element method A commercial FE code, ANSYS 18.1, was employed using the FE approach. The displacement extrapolation technique employs nodal displacements near the fracture tip to explain this behavior, as the gradient of the stress and strain fields close to the crack front is rather substantial. As shown in Fig. 2, singular components are utilized to fairly represent the displacement of the fracture point Fig. 2 (a). The nodes used to determine the fracture tip displacements required by the equation are displayed in Fig. 2(b). As recommended by Ref. , Singular elements are employed to accurately depict the crack tip displacement, as illustrated in Fig. 2 (a). In the region surrounding the crack tip, a refined mesh was utilized to imprison the fast-changing stress and deformation √̅̅ fields. For linear elastic issues, the displacement at the crack tip or crack front varies as r where r is the distance from the crack tip, as illustrated in Fig. 2 (b). The stress and strain are singular at the fracture tip by adjusting a √1̅r. The KCALC command in ANSYS de­ termines the SIF at the crack for an LEFM analysis. The analysis considered the crack’s position and nodal displacement in the area around it. √̅̅̅̅̅̅ 2G |V| KI = 2π √̅̅ (1) 1+k r where, |V| is the motion of one crack face in relation to the motion of the opposite crack face. The last term of Eqn. (1) factor √ |V|̅ r which must be assessed in terms of nodal displacement and positions. G is the fracture toughness (material property indicating the material’s resistance to crack propagation) and k is the dimensionless geometric constant. Three points are accessible, as shown in Fig. 2(b). V is normalizing to where V at node I (r = 0) equals zero. Then A and B figured out how to make it such that (Eqn. (2)), |V| √̅̅ = A + Br (2) r at point J and K. Next, let r approaches 0, then it becomes, |V| lim√̅̅ = A (3) r→0 r after using Eqn. (3) then Eqn. (1) becomes, Fig. 2. Crack tip location (a) singular element and (b) nodes. 4 A. Aabid et al. Heliyon 10 (2024) e32166 √̅̅̅̅̅̅ 2μA KI = 2π (4) 1+k This study computed The SIF with reasonable precision using ANSYS FE analysis software. The SIF is calculated based on the singular element nodal displacement using Eqn. (4) after the displacement at the crack tip acquired using FE analysis. ANSYS APDL has done the SIF calculations with the use of the interaction integral method. In the current ANSYS software, the plane stress auxiliary fields are used only at the end nodes for an open crack front. In general, the change field slightly affects the results on the interior nodes. Therefore, a grid independence study will be used to get accurate results. 3.1. Geometry and modeling The dimensions of the integrated structure under investigation are illustrated in Fig. 1. The rectangular plate, measuring 1 mm in thickness, has specific dimensions: 2 W = 80 mm and H = 200 mm. A 20 mm-long fracture within the plate exists, exposed to an external load of 1 MPa. Meanwhile, the actuator, with a thickness of 0.5 mm, has dimensions Hp = 0.1H and 2Wp = 0.2H. Positioned at 1 mm from the crack tip, the piezoelectric actuator is identified as ‘S.’ Refer to Table 1 for details regarding the properties of the cracked plate, piezoelectric actuator, and adhesive bond. 3.2. Meshing and analysis The present study involves a coupled-field analysis exploring the interaction between structural and electric fields, specifically piezoelectric analysis. In this context, the piezoelectric actuator is represented using the coupled element SOLID226, a recommended approach by ANSYS , which consists of 20 nodes with up to 5 degrees of freedom each. Additionally, the SOLID186 element, a higher-order 20-noded element suitable for solid structure analysis, is utilized to model the cracked plate and adhesive layer, especially in linear applications. To accurately calculate the SIF, having a finely meshed region around the fractured front is imperative. Fig. 3 illustrates a typical mesh configuration with refinement in the crack tip area and the inclusion of the piezoelectric actuator. We employ the interaction integral method to compute the SIF, which uses virtual crack extension principles. This approach yields precise results while reducing the mesh requirements. Parametric analyses are conducted to explore various geometrical and electrical boundary conditions using the ANSYS parametric design language (APDL). Fig. 2(a) presents the fracture tip and face at node I, where stress and deformation fields exhibit steep gradients. We design the fracture tip mesh with specific properties to introduce stress and strain singularities, employing single quadratic elements with mid-side nodes situated at quarter points near the crack tip (or crack front). In Fig. 3, can see a comprehensive set of FE models, encompassing all aspects of the quarter model in line with specified boundary conditions as shown in Fig. 4. The crack front is represented using ten singular elements, while the PZT actuator, damaged plate, and adhesive bond are modeled using 4,999, 12,393, and 2499 high-order reduced integration solid elements, respectively. This study’s precise FE modeling approach ensures accurate and efficient analysis of the piezoelectric actuator’s interaction with the cracked plate. This research advances crack repair methodologies by employing sophisticated FE techniques, paving the way for more reliable and effective aircraft maintenance practices, which are crucial for ensuring enhanced aircraft safety and performance. To define the boundaries of the quarter model, homogeneous uniaxial stress of 1 MPa is applied perpendicular to the crack length on both sides of the plate surface through its thickness or on one side of the current quarter model. The model is fixed at the quarter lines, with displacement considered either in the x- and y-directions or fixed to achieve symmetry. A similar approach is adopted for the adhesive bond and piezoelectric actuator, ensuring they are also fixed on the quarter lines. Furthermore, the plate’s z-direction is kept constant with no moment assumed, and displacement occurs exclusively in the y-direction when a static load is applied to it (as depicted in Fig. 4). The ANSYS procedure was followed to determine the effect of the PZT used on SIF. After meshing the model, the first layer of PZT nodes needed to be selected for coupling. The “Select Entities” option in the select menu was utilized to achieve this, setting the location to zero and selecting the nodes accordingly. Subsequently, the nodes were coupled for voltage degree of freedom using the APDL coupling menu, with a reference number of 1. The next step involved applying the voltage to the first layer of the PZT actuator and inserting the voltage value into the first couple of nodes. These steps were then repeated for the other face of the PZT actuator. Table 1 Properties of the cracked plate, PIC151, and adhesive. Parameter Cracked plate PIC151 patch Adhesive Density 2715 kg/m3 7800 kg/m3 1160 kg/m3 Poisson’s Ratio 0.33 0.345 Young’s Modulus 68.95 GPa 5.1 GPa Shear Modulus 1.2 GPa Compliance Matrix S11 = 15.0 × 10− 12 m2/N S33 = 19.0 × 10− 12 m2/N Electric Permittivity Coefficient εT11 = 1977 εT33 = 2400 PZT strain coefficient d31 = - 2.10 × 10− 10 m/V d32 = - 2.10 × 10− 10 m/V 5 A. Aabid et al. Heliyon 10 (2024) e32166 Fig. 3. Typical FE mesh of the patched plate with integrated piezoelectric actuator and a singular element at the crack tip. Fig. 4. Applied Boundary conditions for a center-cracked plate. Fig. 5. Coupled elements of PZT actuator. 6 A. Aabid et al. Heliyon 10 (2024) e32166 Once the voltage was applied to the actuators, a static analysis of the plate was conducted, followed by solving the model. Fig. 5 il­ lustrates the PZT model with the coupled elements. 3.3. Cohesive zone model The adhesive layers are vital in transferring mechanical responses caused by piezoelectric patches to the cracked plate, so shear and adhesive shear coefficients are significant. Hence, this study included the cohesive zone model to determine the stress interface be­ tween them, and it is more interesting if this factor. The PZT actuator and cracked plate interface separation often cause the bonded actuator to fail, further leading to the damaged plate failing. As a result, the PZT actuator’s needed strength is supplied by the contact between the actuator and the plate. Therefore, the cohesive zone material (CZM) model is employed in the numerical investigation of this phenomenon. The traction – separation law, which connects the traction field (σ) to separation (u), represents the mechanical reaction of the cohesive interface. The numerical simulation of this study uses the bilinear traction – separation model in both normal and tangential axes. Eqns. (5) and (6) give the critical energies for the mode I (Gcn ) and mode II (Gct ) separation of the patch. 1 Gcn = σ max ucn (5) 2 1 Gct = σ max uct (6) 2 Where, σmax is the maximum normal contact stress, causing the debonding at the plate-actuator faces. Thus, the energy criteria based on the power law were used to specify the mixed mode criterion of failure, which included the separation of modes I and II (Eqn. (7)). ( )2 ( )2 Gn Gt + =1 (7) Gcn Gct 3.4. Grid independence test Three different mesh sizes, as outlined in Table 2, are used to investigate the impact of mesh refinement on computational out­ comes. Simulation parameters were as follows: a 10 mm crack length, a 20 mm square representing the adhesive bond and piezo­ electric actuator, a 0.03 mm adhesive bond thickness, and a 0.5 mm piezoelectric actuator thickness. Throughout the mesh study, a constant voltage of 150 V was applied. A grid-structured mesh with component sizes tailored to the mesh type is used to mesh the adhesive bond and piezoelectric actuator. Conversely, the damaged plate was divided into elements by selecting each line, resulting in an unstructured mesh. Table 2 illustrates that enhancing the mesh from medium to fine slightly improved the accuracy of the SIF solution, with a maximum relative difference of 14 percent. However, it’s worth noting that the medium mesh size provided adequate resolution and accuracy while reducing computation time by approximately half. Consequently, we concluded that the medium mesh size was suitable for subsequent simulations. Thoughtful consideration of mesh size and selecting an appropriate medium mesh delivered accurate and efficient computational results and reduced computation time without compromising the precision of our findings. This optimization enhances the reliability and practicality of our study, facilitating additional simulations and bolstering the significance and trustworthiness of our research contribution. 4. Optimization technique Optimization techniques, including DOE, machine learning, and fuzzy logic, have proven to be highly effective in solving complex problems in aeronautical, civil, and mechanical engineering. In the current work, DOE is utilized to optimize simulation data and identify the optimum solution for SIF reduction. DOE involves conducting experiments or simulations for two or more factors with multiple runs. A factorial design is particularly efficient for this type of simulation, as it allows the exploration of all possible com­ binations of factor values in a single full trial or simulation replication. When factors are arranged in a factorial design, they are typically crossed, meaning their effects are investigated as variations in response caused by changes in factor levels. These effects, known as main effects, are of significant interest in experiments or simulations and have wide applications in mechanical engineering, including materials science [37,40–42]. Running all conceivable complete accompaniment combinations ensures that all impacts, including primary and interactions, may Table 2 Mesh independence test. Mesh Type No. of Elements (Nodes) CPU Runtime (seconds) SIF (MPa) Coarse 11,832 (28,692) 309 0.113349 Medium 33,452 (73,633) 600 0.096678 Fine 68,920 (133,633) 1522 0.093678 7 A. Aabid et al. Heliyon 10 (2024) e32166 be assessed. Here, every element that appears in the whole model is as follows (Eqn. (8)): X = φ0 + φ1 Y1 + φ2 Y2 + φ3 Y3 + φ12 Y1 Y2 + φ13 Y1 Y3 + φ23 Y2 Y3 + φ123 Y1 Y2 Y3 + ε (8) A complete factorial design permits an approximation of all eight ‘Phi’ coefficients. {φ0 , …, φ123 }. Significant use of fractional factorial design is in screening simulations. Generally, screening simulations were performed in the initial step of the project, mainly when it is expected that numerous factors considered initially have slight or no effect on the response. A complete factorial experiment fraction or subset can only run the significant factors often identified by the fractional factorial designs. For the present investigation, Taguchi’s L27 orthogonal array containing the specified parameters and their levels for the active repair method was designed and structured. To tackle such issues, the Taguchi design is considered a highly fractionated factorial design and innovative statistical approaches. It is suggested that you use an L27 orthogonal array or a full factorial array – if cost is not a problem, in other words, if we have adequate time and the runs are not too long, or if the accuracy of the results is essential. The orthogonal array is typically employed in industrial applications to examine the impact of several control factors. Additionally, it is research in which the columns of the independent variables are orthogonal to one another. It is necessary to determine various levels and components to describe an orthogonal array. In the present work, the degrees of freedom for the three parameters in each group were calculated at different levels of one and a three-level L27 orthogonal array with twenty-seven simulation runs. The sim­ ulation’s overall degree of freedom is 27 - 1 = 26. Fig. 6 shows the procedure adopted to optimize the data. The use of DOE in this study is crucial as it offers a systematic and comprehensive approach to exploring the influence of multiple parameters on SIF reduction. By considering various combinations of factors, DOE enables the identification of critical parameters that significantly impact the desired outcome. This approach saves human efforts and computational resources and provides valuable insights that can lead to cost-effective and energy-efficient crack repair strategies for aeronautical structures. The DOE–based opti­ mization in this work contributes to the significance and applicability of the research, fostering advancements in aircraft maintenance practices and promoting safer and more reliable aerospace systems. 4.1. Plan of simulations The popularity of intelligent materials in engineering applications has grown significantly, especially for patching up flaws or cracks, leading to an increased utilization of PZT materials for active structural restorations. The PZT actuator layer plays a crucial role in generating local forces, and in mitigating stress singularity, a research-based approach is employed to prevent a loss of bending stiffness caused by a discontinuity in the cracked or damaged region. In previous studies, Yala et al. [19,44] and Fekih et al. have utilized the DOE technique to enhance bonded composite patch repairs in airplane structures. This demonstrates the effectiveness of DOE in optimizing repair strategies and improving structural integrity. The current work builds upon this knowledge and uses DOE to explore the potential of PZT actuators for repairing cracks in aerospace structures. By systematically examining various factors that influence the repair process, this research aims to identify the optimal configuration of the PZT actuator for achieving effective and reliable crack repair. The employment of DOE-based optimi­ zation in this study adds significance and value to the research, potentially advancing the field of active structural restorations in aerospace engineering. The findings may pave the way for more efficient and durable crack repair techniques, contributing to increased safety and longevity of aircraft components. As the demand for advanced repair methodologies grows, this investigation holds potential for real-world applications and is crucial in driving advancements in aerospace maintenance practices. A test was conducted on a 2024-T3 aluminum plate featuring a central fracture and integrated PZT actuators, as depicted in Fig. 1. A fully bonded PZT actuator effectively arrested crack propagation within the high-stress zone of the aluminum plate by inducing Fig. 6. The procedure of response surface analysis. 8 A. Aabid et al. Heliyon 10 (2024) e32166 stresses through an electric field, facilitating plate repair. To optimize parameters for this task, we employed DOE, a valuable strategy for optimizing variables within a known objective function. In this case, we aimed to minimize the SIF by operating four key factors: actuator position ‘S,’ actuator cross-sectional area ‘A,’ actuator thickness ‘t,’ and adhesive thickness ‘ta.’ This approach was chosen to handle a multi-variable function with controlled values, aligning with our goal of reducing SIF. The range evaluated was more in line with Yala et al. , who looked into similar types of performance concerning the current model. As a result, this range is used for active repair to determine the best values of these variables that can help reduce SIF. Table 3 lists the values of factors referred to as levels. The most suitable orthogonal array (OA) for testing is the L27 array, which has a total of 27 degrees of freedom (DF), and these values are selected in increasing order, as shown in Table 4. The DF for four parameters in the three groups was calculated at several levels of 1. A three-level L27 orthogonal array with 27 simulation runs was selected. The total DOF for the simulations is 27 − 1 = 26. Table 4 illustrates the response values that NSIF obtained from the FE simulations. 5. ANFIS method The ANFIS method was employed in the current work to enhance the optimization process for crack repair in aircraft structures using piezoelectric actuators. ANFIS is a robust and adaptive computational approach that combines the strengths of fuzzy logic and neural networks, making it well-suited for complex and nonlinear systems. ANFIS offers several advantages when repairing aircraft cracks with piezoelectric actuators. Firstly, ANFIS can effectively handle uncertainty and imprecise information inherent in real-world engineering problems, providing optimization efficiency. This mainly benefits aerospace applications, where environmental conditions and operational variability can introduce uncertainties. Secondly, ANFIS allows for data-driven learning and the creation of accurate models based on experimental data. By utilizing historical data from simulations or experiments, ANFIS can identify intricate relationships between input parameters and the desired output (SIF reduction, in this case). This learning capability enables ANFIS to continuously improve its predictions and adapt to changes in the system, making it ideal for dynamic aerospace environments. Furthermore, ANFIS offers a transparent and interpretable framework, facili­ tating a deeper understanding of the optimization results. Engineers can gain insights into the impact of various parameters on the repair process, helping them to refine strategies and make informed decisions. To generate output, ANFIS employs fuzzy logic reasoning and neural network techniques. Fig. 7 shows the architecture of the first- order Sugeno fuzzy model with two inputs, x and y, and four rules: A1 , A2 , B1 , and B2 , and one output f. Jang has described various ANFIS architectures and learning algorithms for the Sugeno fuzzy model. The four fuzzy if-then rules for a first-order Sugeno fuzzy model are: Rule 1 : if x is A1 and y is B1 , then f11 = p11 x + q11 y + r11 Rule 2 : if x is A1 and y is B2 , then f12 = p12 x + q12 y + r12 Rule 3 : if x is A2 and y is B1 , then f21 = p21 x + q21 y + r21 Rule 4 : if x is A2 and y is B2 , then f22 = p22 x + q22 y + r22 where A1 , and A2 are the membership functions of the input x, B1 , and B2 are the membership functions of the input y; these mem­ bership functions can be triangular, trapezoidal, or Gaussian functions, for the inputs x, y, fij (i, j = 1, 2) do the fuzzy rules specify the outputs inside the fuzzy region and pij , qij , and rij are the design parameters that are determined during the training process. Two inputs, one output, five layers, and the following operations are performed by each of the layers in Fig. 7: Layer 1. The layer is called the fuzzification layer consisting of adaptive nodes and the output O1i of ith a node in layer 1 is the membership grade of inputs x and y is given by: ⎧ ⎪ ⎨ O1 = μ (x), i Ai for i = 1, 2 ⎩ Oi = μB (y), ⎪ 1 i− 2 for i = 3, 4 where μAi (x) and μBi− 2 (y) are the membership functions for the inputs x, y, with fuzzy sets Ai and Bi− 2 respectively. These membership functions can be triangular, trapezoidal, sigmoidal, or Gaussian functions. For example, the Gaussian membership function (Eqn. (9)) μ Table 3 Selection process parameters and levels. Parameters/Levels Position of the actuator ‘S’ Actuator cross-sectional area ‘A’ Actuator thickness ‘t’ Adhesive thickness ‘ta’ L1 0.75 225 0.5 0.025 L2 1.0 400 0.75 0.03 L3 1.25 625 1.0 0.035 Units mm mm2 mm mm 9 A. Aabid et al. Heliyon 10 (2024) e32166 Table 4 Standard orthogonal array L27 (34) (experimental design matrix with results). Runs Input Parameters Response Output S A t ta NSIF 1 0.75 225 0.5 0.025 0.443706 2 0.75 225 0.75 0.03 0.519506 3 0.75 225 1.0 0.035 0.57479 4 0.75 400 0.5 0.03 0.422777 5 0.75 400 0.75 0.035 0.473577 6 0.75 400 1.0 0.025 0.504582 7 0.75 625 0.5 0.035 0.43676 8 0.75 625 0.75 0.025 0.465967 9 0.75 625 1.0 0.03 0.48866 10 1.0 225 0.5 0.025 0.479372 11 1.0 225 0.75 0.03 0.548186 12 1.0 225 1.0 0.035 0.598832 13 1.0 400 0.5 0.03 0.453232 14 1.0 400 0.75 0.035 0.498066 15 1.0 400 1.0 0.025 0.526072 16 1.0 625 0.5 0.035 0.464992 17 1.0 625 0.75 0.025 0.489278 18 1.0 625 1.0 0.03 0.508227 19 1.25 225 0.5 0.025 0.505432 20 1.25 225 0.75 0.03 0.570027 21 1.25 225 1.0 0.035 0.617714 22 1.25 400 0.5 0.03 0.474824 23 1.25 400 0.75 0.035 0.516393 24 1.25 400 1.0 0.025 0.542765 25 1.25 625 0.5 0.035 0.484435 26 1.25 625 0.75 0.025 0.50605 27 1.25 625 1.0 0.03 0.523015 Fig. 7. First order Sugeno fuzzy model. for the input x is given by: − (x− c)2 μ(x; σ , c) = e 2σ 2 (9) where σ and c are the standard deviations and mean for the gaussian function, respectively. ∏ Layer 2. This layer, known as a rule layer, comprises fixed nodes with the label " " and serves as a multiplier for the node’s incoming signals. The outcome from each node is supplied and indicates the firing strength of the rule and is given by (Eqn. (10)): O2ij = wij = μAi (x) ⋅ μBj (y), i, j = 1, 2 (10) 10 A. Aabid et al. Heliyon 10 (2024) e32166 Layer 3. The normalization layer comprises fixed nodes denoted by the letter N and performs normalization to the firing strengths. The output of each node is supplied and indicates the normalized firing strength and is given in Eqn. (11): wij O3ij = wij = 2 , i, j = 1, 2 (11) ∑ wij i,j=1 Layer 4. This layer is called the defuzzification layer and consists of adaptive nodes. Each node’s output is the result of multiplying the normalized firing strength by a first-order polynomial and is given in Eqn. (12): ( ) O4ij = wij f ij = wij pij x + qij y + rij , i, j = 1, 2 (12) where pij , qij , and rij are called the consequent parameters. ∑ Layer 5. The layer has a single constant node with the label " " that computes the total output, which is the tally of all the incoming signals, and is represented in Eqn. (13): 2 ∑ ∑ 2 O51 = wij f ij , i, j = 1, 2 (13) i=1 j=1 Layer 1 and Layer 4 are the two layers consisting of adaptive nodes. Layer 1 with parameters Ai and Bi , and Layer 4 with parameters pij , qij , and rij. In ANFIS, the adjusting algorithm adjusts the modifying parameters such that the output matches the training data. A hybrid learning approach is used to change these adjustable parameters and entails two processes. The premise parameters are fixed during the forward pass of the hybrid learning process, and node output proceeds until Layer 4 and later parameters are recognized using the least square approach. The error signals go backward, the consequent parameters are held constant, and the premise parameters are changed using the gradient descent technique during the reverse pass. Further details about the hybrid learning algorithm can be found in Ref.. By integrating ANFIS with the DOE approach, the current work leverages the strengths of both methodologies, resulting in a more efficient and accurate optimization process. This combination allows for a systematic exploration of the parameter space and identifies the optimal configuration of piezoelectric actuators for crack repair. Ultimately, the ANFIS-DOE approach leads to improved crack repair performance, reduced repair time, and enhanced cost-effectiveness in aircraft maintenance, contributing to safer and more reliable aircraft structures. 6. Results and discussion 6.1. Validation of the numerical model 6.1.1. Without actuators Before proceeding to repair validation, the fracture mechanics theoretical solution was also used to verify the FE model of a healthy plate without a bonded piezoelectric actuator. According to Eqn. (14), there is an acceptable pact between the findings obtained Fig. 8. Validation of numerical simulation results without actuators. 11 A. Aabid et al. Heliyon 10 (2024) e32166 from theoretical results and the current FE results, as indicated in Fig. 8. The results were obtained for a different crack length of 5 mm–15 mm, showing that the crack length of 10 mm perfectly matched both results. Hence, the default crack length of 10 mm was used in further parametric studies for repaired and unrepaired plates. () ( )3 √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ πa 0.752 + 2.02 + 0.37 1 − sin 2b a πa √̅̅̅̅̅̅ 2b b KI = σ πa tan (14) πa 2b cos 2b πa where ‘a’ represents the crack length, and ‘b’ represents the width of the damaged plate of Eqn. (14). 6.1.2. With actuators To confirm the performance of the actuator, compared the displacements obtained through FE analysis with those calculated using an analytical method for a PZT actuator (PIC151) manufactured by PI Ceramic, operating in extension mode (d31), under an electric voltage of 100 V. The constitutive equations for PZT materials are given by S1 = sE11 T1 + d31 E3 15(a) D3 = d31 T1 + εT33 E3 15(b) The strain of the free actuator increases when the external mechanical loading (T1 = 0) is ignored. S1 = d31 E3 16(a) δo = S1 Ip 16(b) V E3 = 16(c) tp where, S1 = mechanical strain, T1 = stress D3 = electrical displacement E3 = electric field sE11 = mechanical compliance at zero fields εT33 = dielectric constant at zero stress d31 = PZT-coefficient δo = displacement Ip = PZT length tp = PZT thickness The actuator has thickness and length parameters of tp = 0.3 and lp = 20 mm, respectively. The displacement result produced using the FE technique was δoF = − 1.378 μm, which was found to be in excellent agreement with the analytical results δo = − 1.40 μm. Additionally, by contrasting the simulated results from the current study with the experimental data from Abuzaid et al. , the PZT Fig. 9. Present FE findings are validated with experimental data from Ref.. 12 A. Aabid et al. Heliyon 10 (2024) e32166 effect on SIF reduction was confirmed. Fig. 9 demonstrates a substantial agreement between the two outcomes, exhibiting a maximum relative error of approximately 10 %. The main reason for the variation between these results can be attributed to challenges related to accurately positioning and aligning the PZT actuator and strain gauge on the host plate during the experiment. Additionally, an external notch outside the computational model initiates the crack. The consistent linear fluctuation of the SIF with the applied electric field adheres to estab­ lished principles despite any potential factors that might explain this allowable variation. 6.2. Finite element results 6.2.1. Reduction of stress intensity factor Fig. 10 demonstrates the percentage reduction of SIF in the patched plate compared to the unpatched plate. The plot displays the maximum reduction achieved with an applied negative voltage of 150 V. This effect was previously studied by Abuzaid et al. , where the impact of applied negative voltage ranging from 0 to 150 V was explored. The results reveal a linear decrease in SIF with increasing crack lengths. Moreover, it was observed that as the crack length decreases, the SIF increases while the percentage reduction diminishes. This phenomenon occurs because reducing the crack length necessitates a larger negative electric field to achieve the same level of SIF reduction, owing to the higher stress gradient in the crack tip region. For instance, applying a voltage of 150 V yields approximately 52.3 % reduction in SIF for a crack length of 5 mm, whereas for a crack length of 12.5 mm, the reduction is only 45.8 %. These findings provide valuable insights into the relationship between crack length, applied voltage, and the resulting decrease in SIF. Understanding this correlation is crucial in devising efficient crack repair strategies and enhancing aerospace components’ structural integrity and safety. It was highlighted that aircraft systems have certain limitations on the amount of voltage that can be safely and practically applied. Our current work adopted a maximum voltage of 150 V for the crack repair process to address this constraint. The reason for selecting this specific voltage level is to strike a balance between the desired reduction of SIF and the safety considerations associated with higher voltages. It is well-known that higher voltages have the potential to lead to a more significant decrease in SIF; however, there is a threshold beyond which the benefits may diminish, and adverse effects may arise. By adhering to the 150 V limitation, we aimed to ensure the crack repair process’s reliability and avoid any potential risks asso­ ciated with excessive voltages. The chosen voltage level was within the practical operating range for aircraft systems, considering such applications’ specific constraints and safety requirements. Moreover, it is crucial to emphasize that the PZT actuator’s influence on the crack repair performance is inherently tied to the presence of voltage. The PZT actuator relies on the application of voltage to induce its electromechanical effect, which is essential for reducing the SIF and strengthening the repaired structure. Without the application of voltage, the PZT actuator could not generate the required mechanical response to repair the crack effectively. In essence, the voltage is the driving force that enables the PZT actuator to contribute to the crack repair process. We recognize and respect the voltage limi­ tations in aircraft systems, and the selection of 150 V as the maximum voltage in our current work was a careful consideration to ensure both efficacy and safety in the crack repair process. This approach highlights the importance of working within practical constraints while harnessing the capabilities of PZT actuators to enhance the structural integrity and safety of aircraft components. Therefore, this voltage value is for all simulations of the current work. 6.2.2. Effect of piezoelectric actuator voltage The normalized SIF of the applied negative voltage from 0 to 150 V is shown in Fig. 11. The findings demonstrate that when the voltage rises, the NSIF drops linearly. This is primarily caused by the compression stress of the piezoelectric actuator, which is inversely Fig. 10. Reduction of SIF with bonded piezoelectric actuators. 13 A. Aabid et al. Heliyon 10 (2024) e32166 Fig. 11. Effect of PZT voltage. proportional to the applied voltage. Indeed, this has been explained by Abuzaid et al. [28,29]. Fig. 12 (a & b) shows the distributions of nodal stress and strain close to the crack tip for applied voltages of 0, 80, and 150 V using a piezoelectric actuator. As observed, as the applied voltage is raised, the node stress (Fig. 12(a)) and strain (Fig. 12(b)) for the cracked plate decrease. In the case of 150 V, the concentration of stress and strain can be reduced to its lowest point following repair. Since the applied electric fields place additional stress on the host structure due to the piezoelectric actuator, they are more effective in reducing SIF than low-applied electric fields, which only slightly reduce SIF. 6.2.3. Effect of piezoelectric actuator position The stress transfers to the crack surfaces, and consequently, the effectiveness of the repair is influenced by the actuator’s distance from the crack edge. The edge of the piezoelectric actuator is anticipated to be close to the fracture tip because intense stresses generated in the integrated structure emerge there. The piezoelectric effect refers to the ability of certain materials to generate an electric charge in response to applied mechanical stress or, conversely, to produce a mechanical strain when an electric field is applied. When a voltage is applied to a piezoelectric actuator positioned near a crack in a material, it induces a strain in the actuator. This strain is transferred to the material as stress, which, depending on the actuator’s position relative to the crack, can influence the opening or closing of the crack. The SIF quantifies the stress state near the tip of a crack and is used to predict the growth of the crack under stress. The closer the actuator is to the crack tip, the more direct and concentrated the induced stress on the crack surfaces, which more effectively influences the SIF. Ideally, placing the actuator close to the crack tip would maximize this effect. Fig. 13 shows that 0.75 mm is more successful in decreasing SIF. However, practical considerations, such as the difficulty of positioning the actuator so close to the crack or potential damage to the actuator from the crack’s stress field, limit the minimum feasible distance. The literature cited, works by Jin and Wang and Aabid et al. , provide empirical evidence or theoretical backing that the tension remains high within a zone that is approximately twice the thickness of the actuator. This suggests that while the optimal position for the actuator is as close to the crack tip as possible, there is a range of distances within which the actuator can effectively reduce the SIF. In the case of the study, the actuator is bonded at a 1 mm distance from the crack face, a practical compromise that balances the effectiveness of stress transfer and the constraints of actuator placement. The graph in Fig. 13 presumably demonstrates the variation of SIF concerning the distance “S" from the crack face at a 150 V applied voltage. To further clarify the mechanism, you could discuss how the piezoelectric actuator’s stress field interacts with the crack tip’s stress field and influences the SIF, possibly with FE modeling or analytical solutions from fracture mechanics. 6.2.4. Effect of actuator cross-sectional area When a piezoelectric actuator is applied to a cracked plate, it generates a strain field in response to the applied voltage, creating a stress field in the adjacent material. The geometry of the actuator, including its cross-sectional area, dictates the distribution and intensity of the induced stress field in the material. A larger cross-sectional area of the actuator means that the induced stress is distributed over a larger volume of the material. As the actuator size increases, the distributed stress becomes more effective at closing the crack surfaces, thereby reducing the SIF. This is because the induced stress can effectively counter the stress concentration at the crack tip. However, the results depicted in Fig. 14 indicate a turning point where increasing the actuator size no longer leads to a reduction but rather an increase in the SIF. The concept of stress concentration and adhesive bond mechanics may explain this counterintuitive phenomenon. As the actuator size becomes disproportionately large relative to the crack size, the uniformity of the induced stress field is disrupted. Instead of applying uniform compressive stress to close the crack, the larger actuator might induce bending or twisting stresses due to its interaction with the adhesive bond and the surrounding material. Moreover, the larger size of the actuator increases the likelihood of an imperfect bond between the actuator and the plate. Any 14 A. Aabid et al. Heliyon 10 (2024) e32166 Fig. 12. Effect of PZT actuator on damaged plate. 15 A. Aabid et al. Heliyon 10 (2024) e32166 Fig. 13. Effect of piezoelectric actuator position ‘S’. Fig. 14. Effect of actuator cross-sectional area ‘A’. Fig. 15. Effect of actuator thickness ‘t’. 16 A. Aabid et al. Heliyon 10 (2024) e32166 imperfections in the adhesive bond, such as voids or weak spots, are more likely to occur and propagate as the bonded area increases. The separation in the adhesive bond leads to local stress concentrations that can exacerbate the crack, thus increasing the SIF. The study by Kim and Lee also observed a similar pattern, which further supports the notion that actuator size must be optimized for the crack length. An oversized actuator may result in inefficient stress transfer and could lead to secondary issues such as debonding, further complicating the repair. The detailed mechanism behind the results shown in Fig. 14 involves the balance between effective stress distribution and the potential for adhesive bond failure. The initial reduction in SIF with increasing actuator size is due to more effective stress transfer. However, beyond a certain point, the mechanism is dominated by the adverse effects of large actuator sizes, such as non-uniform stress fields and adhesive bond failures. This explanation will require a sophisticated understanding of fracture mechanics, material science, and the piezoelectric behavior of the actuators used. 6.2.5. Effect of actuator thickness Fig. 15 illustrates the PZT actuator thicknesses, and the obtained results exhibit a satisfactory agreement between analytical and FE results. The findings demonstrate that the normalized SIFs slightly increase as the PZT actuator thickness increases for a lower passive voltage. However, it is crucial to note that the impact of increasing the PZT actuator thickness is not absolute, and its effectiveness may diminish as the applied voltage increases. Remarkably, at high applied voltages, the effect becomes contrary to what is typically associated with the PZT actuator. During this scenario, the PZT generates strain that inversely varies with the actuator thickness. As a result, it is preferable to employ a thin actuator with a relatively high voltage to achieve the most significant reduction in SIF. These findings reveal the importance of carefully considering the trade-off between PZT actuator thickness and applied voltage in crack repair applications. The selection of an appropriate actuator thickness and an optimal voltage level is critical to achieving efficient and effective crack repair with minimal SIF. This insight contributes to developing advanced crack repair strategies in aerospace engi­ neering, fostering safer and more reliable aircraft structures. 6.2.6. Impact of the adhesive bond thickness In response to the query regarding Fig. 16, it is crucial to understand the role of adhesive bond thickness in the context of piezoelectric actuator-enhanced crack repair. When activated by an electric field (in this case, at 150 V), the piezoelectric actuator attempts to change shape, creating a strain that is transferred to the crack faces through the adhesive layer. The effectiveness of this strain transfer—and consequently, the reduction of SIF—is significantly affected by the adhesive layer’s thickness. A thinner adhesive layer provides better transmission of the piezoelectric actuator’s strain to the crack faces due to the reduced compliance between the actuator and the substrate. This creates a more efficient stress transfer mechanism capable of closing the crack faces more effectively, thus reducing the SIF. This phenomenon aligns with the findings from composite patch repair studies [2,4,6], which similarly report that thinner bonding layers tend to result in more effective crack repair due to improved load transfer characteristics. Conversely, as the adhesive layer’s thickness increases, the mechanical compliance between the actuator and the substrate also increases. This compliance dampens the strain energy transfer, which, in turn, reduces the efficiency of the actuator’s ability to close or “clamp” the crack. As a result, the normalized SIF increases with adhesive thickness, suggesting that crack face displacement is not mitigated as effectively. The choice of a 0.03 mm bond thickness for the finite element (FE) analysis is consistent with the mean of recommended values in the literature. It is within the range suggested by Abuzaid et al. [28,29] for optimal crack repair effectiveness. The selected bond thickness represents a balance between sufficient adhesive strength to transfer the piezoelectrically induced stress and the need to minimize the mechanical compliance that can reduce the repair’s effectiveness. Therefore, the mechanism by which the SIF increases with adhesive thickness can be attributed to the increased compliance that a thicker adhesive layer introduces, hindering the stress Fig. 16. Effect of adhesive bond thickness. 17 A. Aabid et al. Heliyon 10 (2024) e32166 transfer from the piezoelectric actuator to the crack faces. The findings in Fig. 16 emphasize the importance of optimizing adhesive thickness in piezoelectric actuator applications to maximize repair effectiveness and enhance the overall integrity of the material being repaired. 6.2.7. Analysis of bonded interface via cohesive model The numerical simulations in this study utilized the cohesive zone model in ANSYS to analyze the specimen configuration described in Fig. 1. The separated region at the interface was monitored to assess its state. Stresses in the actuator and aluminum plate, along with interfacial normal and shear stresses, were measured through the simulations. To test the condition specified in Eqn. (7), far-field stress of 1 MPa was applied, and stresses and actuator separation at nodes in the plate-adhesive interface for PZT actuator interfaces were evaluated (Fig. 17). The maximum interfacial normal and shear stresses at these nodes were 1.5 MPa and 0.23 MPa, respectively, with corresponding normal and tangential displacements of 7.93 × 10− 4 and 1.20 × 10− 5 mm. By using Eqns. (5) and (6), the fracture energies were computed as Gcn = 0.12 J/m2 in mode I and Gct = 0.02 J/m2 in mode II. As a result, the right-hand side of Eqn. (7) yielded 4.65 × 10− 8, less than 1. It was observed that the nodes remained attached to the skin since the failure criterion was not met. This outcome was anticipated as the applied stress of 1 MPa was insufficient compared to the skin’s yield stress, aligning with the findings of Shinde et al.. In their study, patch separation occurred when the applied stress was slightly higher than the skin’s yield stress. These results affirm the stability of the interface and indicate that the actuator and plate were effectively bonded without experiencing separation. The agreement with prior research underscores the validity and reliability of the cohesive zone model and numerical simulations in assessing the interface’s behavior, which is critical in developing robust crack repair strategies for aerospace applications. 6.3. Design of experiments results 6.3.1. Analysis of variance The ANOVA table provides insight into the interaction impact of each component with all potential factor combinations. The ANOVA table for the Normalized Stress Intensity Factor (NSIF) was generated using Minitab 18 software. From Table 5, it is evident that the thickness of the piezoelectric actuator, with an F-value of 88.01, has the most significant one-way interaction effect. This is attributed to the production of compressive stress from the actuator to the crack area, and higher compressive loads generally result in a more significant reduction of NSIF. Another crucial finding is that the actuator’s distance must be kept close to the fracture surface to achieve effectiveness. For two-way interactions, the actuator’s cross-sectional area and thickness combination yields a much larger F-value of 14.85, making it a suitable combination for the present study. All four factors and six interactions exhaust the 26 degrees of freedom (DOF), leaving no room for further enumeration. Consequently, insignificant interactions and components were merged to address this limitation and provide a feasible solution. The Model F-value of 29.94 indicates that the model is significant, as revealed by the ANOVA study. Such a large F-value is only likely due to noise in 0.01 percent of cases. Significant model terms have P-values below 0.0500, with Model A, B, C, BC, and CD identified as essential terms in this context. On the other hand, model terms with values exceeding 0.1000 are deemed insignificant. To Fig. 17. Boundary conditions and nodes in the numerical analysis that monitored interfacial stresses. 18 A. Aabid et al. Heliyon 10 (2024) e32166 Table 5 Analysis of variance. Source Sum of Squares DF Mean Square F-value p-value Model 0.0565 10 0.0057 29.94