Reduction in Mode-I SIF of an Edge Cracked C-Shaped Specimen Using Piezoelectric Actuator

. The focus of the current research is on the feasibility of an adhesively bound piezoelectric actuator in the active repair of an edge-cracked C-shaped specimen. First, only constant uniaxial tensile loading is used to determine the Mode-I stress intensity factor. The Mode-I stress intensity factor is then examined while taking into account the actuation effect provided by the piezoelectric patch when an external voltage is applied. For an edge-cracked C-shaped specimen, the total stress intensity factor is determined analytically using the concept of the superposition principle of the linear-elastic crack problem. The outcome demonstrates a considerable decrease in the Mode-I stress intensity factor following active repair using the piezoelectric actuator. Parametric analysis has been carried out to comprehend repair performance and choose the best-sized actuators for active repair.


Introduction
Active repair of a damaged structure utilizing piezoelectric (PZT) actuators has significantly reduced crack damage progression in structures due to its electro-mechanical impact during the past few decades.Similarly, there are many other forms of passive restoration of damaged structures employing composite materials, which has been quite popular in recent years.Transducers made of piezoelectric materials can function as a sensor or an actuator.Typically, the sensor will be used to determine the health state, while the actuator will be utilized to repair the structure.Piezoelectric actuators have demonstrated the ability to reduce and regulate the shear stress concentration and joint edge peel in adhesively bonded joint systems during the past 20 years.As a result, the research of piezoelectric actuator applications in damaged structures and adhesively bonded combination systems is based on three distinct repair inquiry methodologies: analytical, numerical, and experimental.A piezoelectric actuator can also be used for other investigations, such as the control of delamination in composite material beams.
Prior research found that piezoelectric materials had exceptional existing and sensing capabilities.The assessment and selection criteria for piezoelectric actuator configuration for the various applications of smart structures, with a focus on form and vibration control, were laid forth by Prasad et al. [1].According to such investigations, Lee et al. [2] showed that using piezoelectric actuators and sensors to repair a damaged structure while taking into account theoretical and analytical solutions was an efficient way to regulate delamination.Sohn et al. [3] created a unique way for signal processing techniques to forecast delamination in composite plates, highlighted another crucial result.This has emerged as a crucial discovery in delaminated composite materials when taking into account shell-type structures.Piezoelectric materials were employed by Wang et al. [4] to manage delamination and prevent fractures in composite beams.The authors came at the conclusion that adequate voltage is needed to control the delamination of the beam, which only depends on the delamination's position.Liu [5] conducted a similar investigation on the use of piezoelectric patches to repair a delamination beam.According to this study, lower voltage is practical for operations that are both cheap and safer.Additionally, the piezoelectric patch's design is significantly influenced by the patch's length, thickness, and layer.Piezoelectric patches may be used to regulate delamination in composite beams subjected to low-velocity impacts, according to research by Iannucci et al. [6].Because of their customizable mechanical characteristics, smart materials have been used in structural repair applications, according to Wang et al. [7] Aabid et al. [8][9] have undertaken several investigations on the prevention of delamination, vibration, and noise.Additionally, PZT was used for energy collecting, although there were few studies on how to repair aircraft structures.Therefore, the purpose of this effort is to develop concepts and fill up research gaps from earlier works.Abuzaid et al. [10][11][12] carried out the initial investigations in which they investigated the electromechanical response in active repair and the management of the damaged structure.They discovered that the piezoelectric actuator can regulate the joint edge peel in the adhesive-bonded joint system and repair the thin plates at lower shear stress concentrations.The scientists demonstrated that adhesive bonding had the effect of transmitting the stresses generated by the piezoelectric actuator patch to the fractured plate in order to decrease the stress intensity factor (SIF).Wang et al. [13] determined the link between the voltage and parameters and the interface of material n beam.The monomorph PZT plate was analyzed by Hudec [14].PZT materials have been employed in a wide variety of smart structures and devices because they can be used for both actuation and sensing.In several double-coated layers of thin films, Inoue et al. [15] examined the impact of increasing the thickness of the PZT bimorph structure.Microelectromechanical system (MEMS) applications benefit from the high voltage and enormous generating force enabled by the thick PZT films.
In the present work, the Mode-I SIF of a C-shaped edge cracked specimen is examined, and the Mode-I SIF is then lowered by employing the actuation effect of the piezoelectric material in order to extend the service life of a damaged structure.The analytical model is developed based on the concept of Linear Elastic Fracture Mechanics (LEFM) and weight function method to investigate effect of piezoelectric actuators on crack repair.The effect of specimen dimensions, crack length and piezoelectric patch dimensions are considered for this investigation.

Without Repair
Tada's theoretical formula [16] was used to calculate SIF.For the linear elastic fracture mechanics (LEFM) of a cracked rectangular plate of infinite size where mode-I is the only effective opening mode considered, the expression of Mode-I stress intensity factor is given by Eq. ( 1); (1) Figure 1: Edge Cracked C-specimen with piezoelectric actuator where σ 0 is the uniaxial tensile stress as given by Eq. ( 2), that is perpendicular to the crack length 'a'.For greater accuracy, a finite plate for an infinite crack length obtained from finite element analysis is used; the solution is fit to the appropriate polynomial expression.SIF is the characterization of the stress distribution located at the crack tip in a linear elastic material.Since the plate is finite, the boundary condition introduces higher stress at the crack tip.The mode-I SIF for the cracked plate is given by Eq. ( 3) [16] without any patches for an edge cracked c-shaped specimen as shown in Fig. 1 K where W is the width of the plate and F(ξ) and f(ζ, χ, ξ) are the dimensionless geometry functions as represented by the Eqs.( 4)- (5).These functions plays vital role to estimate SIF for any cracked plate as it largely depend on shape and size of the cracked specimen.For a cracked specimen, Mode-I SIF is first estimated.The reduction in SIF is then examined using an active repair process.

Use of Piezoelectric actuator as active repair:
Any bonded structure under in-plane loading develop 3-D stresses and out plane deformation and stress along thickness is neglected then we would be able to predict 3-D behaviour from a 2-D computation.The weight-function method depends not only on the stress distribution but rather on the structure geometry.Let us consider that the crack length 'a' for the cracked infinite plate as proposed by Rice [17,18], K I(piezo) can be expressed as given by Eq. ( 6), where ℎ(x i , a) is the weight function and Γ c is the perimeter of the body, and it can be illustrated as Eq.(7), where E ́ is Young's modulus in either plane strain or plane stress conditions and K I is the SIF for uniform tensile stress applied on the panel.For plane-strain conditions, in-plane crack opening displacement can be related to Westergaard stress function as given by Eq. ( 8) [19] and finally the Mode-I SIF due to piezoelectric stress can be calculated from Eqs. ( 6)-( 8), and represented by Eq. ( 9).
The piezoelectric patches are polarized in the Z-direction.The expression of the compressive force between a metal substrate and a piezoelectric patch with the assumption of a complete bonding between them, is represented by Eq. ( 10) [20], Where ψ = EWt E p W P t p ⁄ and Λ = d 31 V t p ⁄ (11) tp is the thickness of the piezoelectric patch, d31 is the piezoelectric charge coefficient, Ep is the equivalent Young's modulus of the piezoelectric patch, T is the width of distributed electrodes on the piezoelectric patch, and V is the applied voltage.Substituting σ piezo = F piezo /A where A = W P H P in the above equation with the assumptions that the piezoelectric stress σ piezo is having perpendicular distribution to the crack surface, the problem can be solved.It is important to consider the dimensionless geometry function F(ξ) and f(ζ, χ, ξ) as the crack length is affected by the geometry of the cracked body.Using empirical expressions for F(ξ) and f(ζ, χ, ξ) proposed by Tada et al. [16], thus the stress intensity factor can be expressed by Eq. (12).
, 01 Tensile force only (c) Actuator effect only The principle of superposition indicates that the total stress intensity factor K I(piezo) is given by Eq.( 13) and illustrated as shown in Fig. 2.
K I(total) = K I + K I(piezo) (13) According to the above equation, the piezoelectric actuator's stress intensity factor K I(piezo) is crucial to the integrated structure.When the piezoelectric actuator's applied stress is directed in the opposite direction of the external load, the integrated structure's total stress intensity factor K I(total) will drop as a result.In fact, applying the piezoelectric actuator stress in the opposite direction will result in compression stress on the crack surfaces that will counterbalance the external tensile stress applied to the plate.Hence, by subtracting K I(piezo) fromK I , help to reduce the K I(total) .

Result and Discussion
In this section, various study was conducted to observe the efficacy of the repair method.The test specimen considered for the present developed model is C-shaped edge cracked plate subjected to a P=30 N force.The various dimensions of the specimen are W=30 mm, R=60 mm, thickness t=1 mm, X=15 mm, as shown in figure.The applied tensile stress becomes 1 MPa.Parametric analysis is conducted to recognize the proper choice of the PZT patch geometry.Analytical results are obtained by considering linear elastic fracture mechanics (LEFM).Since the applied stress is very small, plastic zone near the crack tip is also very small as compare to the dimensions of the specimen and the crack length.The repair technique was adopted considering the actuation effect of piezoelectric patch.The PIC 151 was used for the actuation effect and this patch was perfectly bonded with the cracked specimen with any suitable adhesive material.The effect of adhesive was not considered for this analysis.The material properties of the specimen, PIC 151 and adhesive are shown in table 1.The goal of this study is to reduce the severity of the crack and total stress intensity factor KI(total) is calculated after repair.For the actuation purpose, the external electric field is to be applied in such a way that the stress produced by the actuator is in the opposite direction the applied stress.The actuator width is so kept that it covers the entire crack length.The positive strain in the actuator will lead to develop the compressive stress near the crack tip, which results in reduction in Mode-I SIF.

Comparison between cracked and repaired specimen
To validate the repair technique using piezoelectric actuators, a comparison study is performed considering cracked specimen and repaired specimen under the application of different external voltage.The analytical SIFs results are presented in Fig. 3.According to the results, it is found that there is significant drop in SIF under the application of external voltage as positive strain in piezoelectric materials causes negative SIF and then modified stress distribution near the crack tip decreases the total SIFs.For this case Piezoelectric Patch dimensions taken as tp=0.5mm,hp=12.5 mm, Wp=25 mm and the crack length ranging between 1 mm to 10 mm.It is found that approximately 18% reduction in SIF when the external voltage is 300 V as compared to 6% when the voltage is 100 V. Also it has been observed that for a particular crack length, with the increase in the application of electric voltage, the total SIF is reduced.So, in order to obtain maximum reduction in SIFs, greater amount of electric voltage is be applied.

Effect of crack length
The total analytical SIFs for various crack length under different repaired voltage is depicted in Fig. 4. The various piezoelectric actuator dimensions Wp=25 mm, hp= 15 mm, tp=0.5 mm and the crack length to width ratio, a/W=0.1,0.2, 0.3, 0.4 and 0.5 are taken for this analysis.The compressive stress produced by the actuator near the crack tip is proportional to the external applied electric field which results in total SIFs decreases linearly with the increase in external voltage.From the results, it can be seen that the total SIFs are increased with the crack length and greater electric voltage is required to reduce the SIFs to the same level.These results shows that lower a/W ratios are required to maintain for better repair performance.

Effect of piezoelectric actuator thickness
The variation of SIFs for different piezoelectric patch thickness tp for different repair voltage are represented in Fig. 5.The various piezoelectric actuator dimensions Wp=25 mm, hp= 12.5 mm, tp=0.5, 0.75 and 1 mm are taken for this analysis.The variation of SIFs are represented for different crack length a= 2, 4, 6 and 8 mm as shown in Fig. (5a)-(5d).Though the adhesion of the piezoelectric patch to the damaged structure increases the stiffness of the integrated structure still the passive effect of the patch is neglected for this analysis.From results it can be seen that when the applied voltage is zero there is no effect of piezoelectric patch on SIFs as the compressive stress produced by the patch is zero.For a particular applied repair voltage 200 V and crack length a=2 mm, it is observed that total SIFs increases by 8% when the patch thickness increases to 1 mm from 0.5 mm as shown in Fig. 5(a).The SIFs are linearly decreasing linearly with the increase in repair voltage.Under the application of 300 V, the reduction in SIFs are 18.5%, 10.5% and 8.9% for the patch thickness 0.5 mm, 0.75 mm and 1 mm respectively as shown in figure 5a.To fulfil the goal this study, it is good to select the thin piezoelectric actuators with high electric voltage.The possible reason could be described as the strain produced by the piezoelectric actuators is changes inversely with the patch thickness.

Effect of piezoelectric actuator size
For the reduction of SIFs, the effect of piezoelectric patch size is crucial as the compressive stress developed by the piezoelectric patch near the crack tip is greatly influenced by the piezoelectric patch geometry.Fig. ( 6)-( 7) illustrated the variation of SIFs for different patch size.For this investigation, the different piezoelectric actuator's height and width are used.
Second, several patch widths, Wp=15 mm, 22.5 mm, and 30 mm are taken into account while the patch's height, hp=15 mm remains unaltered and its thickness tp=0.5 mm.Wp/hp ratios are 1, 1.5, and 2 for the chosen piezoelectric patch width.Fig. (7a)-(7d) demonstrated the effects of patch width for various crack lengths, a=3, 6, 9 and 12 mm.For Wp/hp =2, 1.5, and 1 correspondingly, the reduction in SIFs under the application of 500 V are 23.53%,20.1%, and 15.6%.(Figure 7a.) The investigation's findings showed that the patch with a larger Wp/hp ratio had a significant drop in SIF.So, the choice of patch size plays a vital role for best possible repair performance.

Conclusion
This study developed an analytical model based on the linear elastic fracture mechanics (LEFM) and weight function approach for active repair of an edge cracked C-shaped specimen subjected to uniaxial tensile force.The cracked specimen was embedded with piezoelectric actuators on both sides.Piezoelectric actuators produced compressive stress when external voltage was applied, balancing the stress singularity at the crack front.The Mode-I Stress Intensity Factor (SIF) was taken into account to evaluate the efficacy of repair.Parametric analysis had been carried out to choose suitable dimensions of piezoelectric actuators.It is evident from the results of the analysis that piezoelectric actuators are extremely effective in decreasing Mode-I SIF.The thin actuators with high electric field are found to be most efficient.Also higher actuators width was selected to obtain maximum possible reduction in SIFs of the repaired integrated structures.

Figure 3 :
Figure 3: Comparison of analytical SIFs for cracked and repaired specimen

Figure 4 :
Figure 4: Effect of crack length on analytical SIFs

Figure 5 :
Figure 5: Variations of analytical SIFs for different piezoelectric actuators thickness tp under different repair voltage (a)a=2 mm (b)a=4 mm (c)a=6 mm (d)a=8 mm