## Abstract

The understanding of strength recovery behavior under a dynamic loading environment provides a guidance for optimizing the design of composite structures for in-service applications. Although established for metals, the quantification of strength recovery in carbon fiber-reinforced viscoelastic composites is still an area under active research. This study aims to understand the effects of fatigue loading rates on the damage behaviors of stress-relaxed carbon fiber-based composites. Hence, the time-dependent strength recovery in woven composites is quantified experimentally using two mutually exclusive approaches under identical fatigue loading environments. In the first approach, the strength recovery is quantified by the dissipated non-linearity in Lamb wave propagation due to the damage state of the composite materials. This is quantified and shown coupled with second- and third-order non-linear parameters. In the second approach, ultrasonic acoustic pressure waves are utilized to quantify the fatigue-induced internal stress and the damage accumulation. A comparison of these two approaches leads to the assessment of strength reduction which is experimentally validated with the remaining strength of the specimens.

## Introduction

Carbon fiber-reinforced polymer (CFRP) composites are widely used as structural components in a variety of industrial applications including, but not limited to, aerospace, automotive, civil, and defense infrastructures owing to their high specific stiffness and strength. Since composite structures are considered to endure higher fatigue cycles with a considerable loading rate, extensive research has been performed on several damaged mechanisms related to the issues involving fiber, matrix, and their interfaces [15], despite careful design and manufacturing of CFRP composite in a high-tech manufacturing environment, in-service degradation of mechanical properties due to the micro-damage occurs. Hence, predicting the life of a composite structure is one of the key issues in effective design of the complete final product [6,7]. With the profound development of composite manufacturing techniques throughout the last decades and the significant demand for the structural mass optimization with an aim to diminish power consumption in industrial applications, the in-service composite structures are exposed to different loadings progressively closer to their static strength [2,8]. Since CFRP composites are more prone to stress relaxation and creep, the study on the time-dependent behavior of the matrix is inevitable [9]. This fact enhances a rapidly growing area of interest for the researchers to deal with the challenges of new designs and fabrication of composites for long-term applications. Therefore, a better comprehensive view of viscoelastic properties constitutes a directive for optimization and prediction of long-term usage of such composite structures.

Various non-destructive evaluation (NDE) techniques have been developed for prognostic and diagnostic assessment of a structural element [1921]. In the recent years, the use of non-linear ultrasonic Lamb waves has proved its reliability for material state awareness and structural health monitoring due to the multiple advantages of guided waves over bulk waves [2227]. In this study, the physics of non-linear higher harmonic interactions of Lamb waves with anisotropic materials are utilized to assess the macro-damage state of CFRP-woven composites. In addition, time-dependent strength reduction is monitored and quantified experimentally over an 8-h period. To validate these results, traditional ultrasonic bulk waves generated in a scanning acoustic microscope (SAM) are simultaneously utilized to quantify the strength reduction of the composite specimens. Finally, a quantitative comparison of strength reduction is presented by determining the remaining strengths of the specimens undergoing these two approaches. A theoretical background on higher order non-linearity technique has been highlighted in Sec. 1 of this article. Based on this technique, the experimental design has been described in Sec. 2 that discusses the sample preparation, fatigue loading design, experiment setup with pitch-catch and pulse-echo techniques. The last section discusses the results found in the experimental process, and a comparison of these two techniques have been presented.

## Theoretical Background of Acoustic Non-Linearity Parameter

One dimensional wave equation in the x-direction can be written as
$ρ∂2u∂t2=∂σxx∂x$
(1)
Constitutive equation for non-linear materials in 1D can be written as [28]
$σxx=Exxεxx(1+βεxx+γ(εxx)2+⋯)$
(2)
where σxx, ɛxx, and Exx are stress in the x-direction, strain in the x-direction, and Young's modulus. β and γ are second- and third-order non-linearity parameters, respectively.
By substituting Eq. (2) into Eq. (1), we can write the equation of motion including the second- and third-order non-linearity as [28]
$ρ∂2u∂t2=Exx∂2u∂x2+βExx∂u∂x∂2u∂x2+γExx∂2u∂x2[∂u∂x]2$
(3)
The classical solution of displacement parameter u can be obtained by applying perturbation theory and hence u can be written as
$u=A1cos(kx−ωt)−A2sin2(kx−ωt)$
(4)
Absolute second- and third-order non-linearity parameter can be expressed as [29]
$β=8k2x.A2A12.f(ω)$
(5)
$γ=32k4x2.A3A13.f(ω)$
(6)
In Eqs. (5) and (6), the amplitude of the fundamental, second, and third harmonics are denoted by A1, A2, and A3, respectively. The propagation distance, wavenumber, and a frequency function are defined by x, k, and f(ω), respectively. The normalized second- and third-order non-linearities, $β~$ and $γ~$, can be expressed as [30] given below:
$β~=A2A12∝βx$
(7)
$γ~=A3A13∝γx2$
(8)

## Experimental Design

### Sample Preparation.

In this research, the specimens are prepared from commercially available four-layered carbon fiber-reinforced woven composite plates. These plates are manufactured with carbon fabric substrates combined with an epoxy resin system. Each layer of the fabric is weaved with 3k carbon fiber tows in [0,90] direction. The density of the prepared composite plate for the specimens is reported as 1605 kg/m3 by the vendor. According to the experimental design, 21 specimens are required and are fabricated based on ASTM D 3039 standard. The final dimensions of the samples are 250 mm (L), 25 mm (W), and 1.5 mm (T). In the next step, the constitutive matrix expressed below is used to determine the dispersion curve as found in Refs. [31,32]:
$[81.6427.7427.7427.7476.9815.5127.7415.5176.98000000000000000000500050005]GPa$

Following the experimental design, nine specimens out of 21 specimens, are reserved for performing the PC approach experiment and nine specimens are reserved for performing the SAM approach experiment. The remaining three specimens are utilized to verify the stress–strain properties of the prepared composite plate samples and the average stress–strain curve is shown in Fig. 1(c). In the PC approach, two piezoelectric wafer active sensors (PWAS) are attached to each composite specimen surface using Epoxy 9340. A three-day time period is devoted to each epoxied PWAS to cure it sufficiently at room temperature. The thickness of the adhesives is measured to be ∼150 µm at the time of attachment. This measurement is controlled by holding a micrometer between the bottom side of the specimen and the top surface of the PWAS, and the standard deviation is calculated as 0.31 µm. Commercially available PZT-5H sensors (purchased from STEMiNC, Devenport, FL) have consistent piezoelectric material properties which are not influenced by minor lab room temperature changes (usually, the lab room temperature is fixed at 25 ± 2 °C) [33]. Hence, PZT-5H sensors are utilized as our PWAS during the pitch-catch experiment. The material properties of PZT-5H are reported as given in Ref. [33] (Table 1).

The capacitance of the sensors is measured at the pristine stage, and at each relaxation stage, this is found as ∼1.2 nF with a standard deviation of 0.013 nF. The distance between the two epoxied sensors on each specimen is 90 mm (center to center), and each sensor was equidistant from the centerline of the specimen. In the SAM approach experiment, no PWAS is attached to the nine specimens; however, during the relaxation stage, specimens are placed in water at room temperate in the SAM machine under a 25 MHz transducer. SAM is a machine which is generally employed in failure analysis and non-destructive evaluation supplied by PVA TePla Analytical Systems GmbH [26]. It utilizes concentrating acoustic waves to inspect, measure, or image a specimen which is called scanning acoustic tomography. Scanning acoustic microscopy runs by directing focused acoustic waves from a transducer at a small point on each fabricated specimen. Using this technique, it is possible to detect the scattered pulses traveling in a particular direction. The time of flight (TOF) of the detected pulses is defined as the time taken for it to be emitted by a transducer, scattered by an object, and received back again by the transducer. The time of flight can be used to determine the distance of the inhomogeneity from the source when the speed of the propagating wave through the specimen is known. Based on the TOF measurement, a value is assigned to the scanned position. This process is repeated several times by moving and focusing the transducer in a preselected area until the entire region of interest on the specimen is scanned.

### Time-Dependent Stress Relaxation Tests

#### Pitch-Catch Approach.

Toward the end of each 75,000 fatigue cycle completion, in the PC approach experiment, a standard five-count tone burst signal with a central frequency of 320 kHz is utilized to produce Lamb waves, which was determined by performing a tuning experiment as shown in Fig. 1(d). The peak-to-peak amplitude of the signals is assigned to be 20 V. Sensor signals are collected averaging 500 sample signals to improve the signal-to-noise ratio. The capacitance of the actuating and receiving sensors was monitored throughout the experiments to ensure their appropriate functioning. The PC approach is illustrated by Route 1 as shown in Fig. 2.

#### Scanning Acoustic Microscope Approach.

Similar to the pitch-catch approach, toward the end of each 75,000 fatigue cycles completion, in the SAM approach, the specimens are placed under 50 mm of distilled water at room temperature. The scanning procedure for time-dependent stress relaxation is started by exciting the ultrasonic p-wave. This is done by means of a 25 MHz transducer which is mounted ∼35 mm above the test specimen. The generated signals pass through the water and form a focal region on the top surface of the specimen. Once the acoustic wave interacts with the top surface, some part of the wave energy reflects back to the transducer and the rest of the energy goes through the specimen. Afterward, the wave energy interacts with the bottom surface of the specimen and reflects back to the transducer. The signals reflected from the top surface (water–composite interphase) and the bottom surface (composite–water interphase) are received by the actuating transducer. The time difference between these two reflections is the time of flight which is displayed by the SAM [34]. The signals received from a point on the composite specimen are averaged from 10,000 samples to minimize the noise effect. An area at the mid-section of the specimen is chosen, and the signals from 1500 points distributed in the xy plane are collected at each trial. This process is repeated at intervals of 15 min for 8 h. The SAM approach is illustrated by Route 2 in Fig. 2. The major differences between the SAM and the PC approaches are the excitation method and the direction. In the PC approach, PWAS is employed to excite in the longitudinal direction of the specimen (L direction) using Lamb wave, whereas in the SAM approach, a 25 MHz transducer is employed to excite p-wave in the thickness direction (T direction).

## Result and Discussion

### Quantification of Stress Relaxation Using Scanning Acoustic Microscope Approach.

In this approach, fatigue-induced stress behavior during unloaded specimens is characterized using a SAM. As mentioned earlier, specimens of the SAM approach go through exactly the same loading conditions as described in the PC approach. The major difference in these two techniques is the method of data acquisition. In the SAM approach, the specimens are kept inside distilled water and a pulse-echo experiment is performed. The source of acoustic excitation is a 25 MHz ultrasonic transducer. After actuation from the transducer, the ultrasonic time-domain signal is received from nearly 1500 points of each specimen as shown in Fig. 5. These points are chosen from the mid-section of each specimen which has homogeneous loading effect. The signals are acquired right after the specimens are unloaded from the MTS and are recorded every 15 min for 8 h.

The determination of the ultimate remaining strength as the specimens go through finite fatigue cycles is the direct evidence of the claim presented in this study. As such, the remaining ultimate tensile strength of the specimens is determined right after the relaxation period of 225,000 fatigue cycles. To determine the remaining tensile strengths, the stress–strain profiles of the damaged samples are determined right after the third-stage relaxation period. The remaining ultimate tensile strengths are compared with the ultimate strengths determined at the pristine state, and the reduction in tensile strengths is presented in Fig. 7. Clearly, as determined by the PC and SAM approaches, specimens which went through 5 Hz fatigue loading rate have encountered the lowest remaining ultimate strength. In this experiment with 5 Hz loading cycle, 11% reduction in ultimate tensile strength is found for the specimens that went through the pitch-catch approach while 11.2% reduction in the ultimate tensile strength is found for the specimens that went through the SAM approach. This proves that the results obtained by the two approaches are convincingly close. Therefore, it can be summarized that time-dependent stress relaxation is an inherent part of viscoelastic composite materials.

## Acknowledgment

The research is funded by NASA Langley Research Center (LaRC; Funder ID: 10.13039/100000104), United States; Contract No. NNL15AA16C.

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