## 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 [1–5], 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.

Time-dependent strength reduction is one of the straightforward ways of characterizing the damaged state of polymer composites under fatigue loading [10]. Unlike isotropic and homogenous materials such as metals and alloys, the damage state modeling of composite structures is complex due to their heterogeneous arrangement of material constituents, especially under fatigue-induced loading cycles. Fatigue-induced residual stresses are those that exist along the cross section of a structure after the removal of the applied load. Kim et al. analyzed residual stress relaxation under low- and high-cycle fatigue behavior of shot-peened medium-carbon steel [11]. A detailed review on residual stresses resulting from four major classes of manufacturing operations such as shot peening and related surface treatments, cold expansion of holes, welding, and machining can be found at [12]. A good discussion on stress relaxation behavior of polymers can be found in Ref. [13]. Unlike metals, fatigue damage in fiber-reinforced composites can occur in the form of matrix cracking, fiber/matrix debonding, delamination and localized fiber breakage, or various complex combinations of these [14–16]. The damage mechanism of carbon fiber-reinforced composites has been popularly described in terms of microstructural damage states or by classical laminate theory [17]. It has been widely believed that the microstructural damage leads to the failure of complete composite structures. This necessitates the development of structural health monitoring systems. Thus, the global effects on stress relaxation under the fatigue loading environment play a vital role in designing structural loading specifications. Furthermore, a conclusive aspect to envisage the residual lifetime of such structures is a methodology for incessant monitoring of the fatigue-induced micro-damage state of CFRP composites [18]. Since limited works have been reported to quantify fatigue-induced stress reduction in composites, this requires a better understanding of non-destructive prognostic points of view.

Various non-destructive evaluation (NDE) techniques have been developed for prognostic and diagnostic assessment of a structural element [19–21]. 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 [22–27]. 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

*x*-direction can be written as

*σ*

_{xx},

*ɛ*

_{xx}, and

*E*

_{xx}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.

*u*can be written as

*A*

_{1},

*A*

_{2}, and

*A*

_{3}, 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, $\beta ~$ and $\gamma ~$, can be expressed as [30] given below:

## Experimental Design

### Overview of Experimental Design.

A series of tensile–tensile fatigue loading cycles were considered to induce localized stresses and distributed damage in the prepared composite specimens with the purpose of investigating the repercussions of different applied fatigue loading rates on the viscoelastic properties of the composite material. A comparison of different design approaches is the adopted methodology in this study with the intention of verification and quantification of time-dependent strength reduction in composite materials. The first approach, named as pitch-catch (PC) approach, is explained by the Fig. 1(a), where the specimens go through 225,000 fatigue loading cycles at three stages. At each stage, each specimen undergoes 75,000 cycles of the MTS machine in order to experience the assigned fatigue loading cycles. Then, after removing the specimen from the MTS Machine, each specimen remains unloaded at room temperature for 8 h of relaxation. During this 8-h relaxation period, pitch-catch experiment is performed at 15-min intervals by means of generating guided wave signal. The second adopted experimental approach, called the SAM approach, is explained in Fig. 1(b), where the fatigue loading conditions and unloaded relaxations are the same as those of the PC approach while *p*-waves are utilized to estimate the damage conditions at 15-min intervals. Details of these experiments are described in the following sections.

### Sample Preparation.

^{3}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]:

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 *µ*m 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 *µ*m. 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).

Symbol | Quantity | Unit |
---|---|---|

d33 | 650 × 10^{−12} | C/N |

d31 | −320 × 10^{−12} | C/N |

g33 | 19.0 × 10^{−3} | (V/m)/(N/m^{2}) |

g31 | −9.5 × 10^{−3} | (V/m)/(N/m^{2}) |

Coupling co-eff | 0.75 | – |

Modulus of elasticity | 5.0 × 10^{10} | Pa |

Symbol | Quantity | Unit |
---|---|---|

d33 | 650 × 10^{−12} | C/N |

d31 | −320 × 10^{−12} | C/N |

g33 | 19.0 × 10^{−3} | (V/m)/(N/m^{2}) |

g31 | −9.5 × 10^{−3} | (V/m)/(N/m^{2}) |

Coupling co-eff | 0.75 | – |

Modulus of elasticity | 5.0 × 10^{10} | Pa |

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.

### Fatigue Testing Parameters.

To investigate the material responses, three loading rates are employed which are as follows: low frequency (2 Hz), moderate frequency (5 Hz), and high frequency (10 Hz). For each loading rate, three specimens are reserved for the pitch-catch approach experiments and three specimens are utilized for the SAM approach experiments. As mentioned earlier, the MTS 820 machine is applied to perform tensile–tensile fatigue testing. By previously setting up the fatigue loading ratio, three specimens are tested under incremental tensile load to establish the stress–strain behavior of the woven composites. The average ultimate strength is found as ∼8400 lbf. In this study, the fatigue loading ratio *σ*_{min}/*σ*_{max} is mentioned as 0.01 where *σ*_{max} is 60% of the *σ*_{ult}. After the completion of each 75 K tensile–tensile fatigue cycles at a specified loading rate, each specimen is set aside in an unloaded situation to undergo stress relaxation within ∼5 min at room temperature.

### 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 *x*–*y* 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 the Pitch-Catch Approach.

Time-domain signals collected during the stress relaxation process are analyzed using non-linearity parameters and presented in Fig. 3. Before performing the non-linearity analysis, instrument non-linearity is taken into consideration. Pristine state signals collected at 15-min intervals for 8 h from the randomly chosen specimens were analyzed for the existence of non-linearity. The obtained acoustic signals were compared with each other at all intervals. The result shows that no significant change in acoustic non-linearity is present at the pristine state [35]. In addition, this ensures the absence of instrument non-linearity during the fatigue-induced stress relaxation process. After completion of every 75,000 cycles, 0-h initial signals are the unrelaxed signals from the specimens. In Figs. 3(a1)–3(c1), pristine state signal, 75,000 unrelaxed, 150,000 unrelaxed, and 225,000 unrelaxed normalized time-domain signals are plotted for the 2 Hz, 5 Hz, and 10 Hz fatigue loading rates, respectively. As the fatigue loading cycles and the rates increase, the micro-damage, in terms of micro-cracks, is prone to increase. Understanding the effect of fatigue cycles on the composite damage state requires the transformation of the time-domain signal into a frequency-domain signal. Moreover, as per Eqs. (5)–(8), the acoustic non-linearity is a function of frequency and signal amplitudes. Therefore, fast Fourier transformation (FFT) was performed on time-domain signals and is presented in Figs. 3(a2)–3(c2). The first harmonic frequency (*f _{c}*

_{1}) for all these samples was about 320 kHz, which is inconsistent with the central frequency of the excitation signal. The amplitude of the first harmonic frequency is much higher compared to the higher harmonic frequencies. Therefore, a zoomed-in plot has been shown in Figs. 3(a3), 3(b3), and 3(c3) for 2 Hz, 5 Hz, and 10 Hz FFT signals, respectively, in a range of 2

*f*

_{c}_{1}to 3

*f*

_{c}_{1}. According to the concept of the second- and third-order non-linearity, signal amplitudes in this range have been analyzed. From Fig. 3(b3), it can be clearly seen that the amplitude of the pristine signal is lower than that of all the unrelaxed signals of the second-order and third-order harmonics. It is also evident that the amplitudes of the unrelaxed signals increase as the fatigue cycle increases, which confirms that the material non-linearity increases as the fatigue loading cycle increases. On the other hand, a shift in central frequency is observed in leftward direction as the fatigue cycles increases; i.e., the central frequency of the unrelaxed signal after 225k cycles decreases compared with 150k cycles, which again decreases after 75k cycles. This trend is evident for the case of specimens that went through 5 Hz and 10 Hz of fatigue loading rates. For the case of 2 Hz fatigue loading rate, an increase in amplitude and a decrease in central frequency are observed for third harmonics; however, the samples that went through 150k and 225k cycles followed this trend in second harmonics which is shown in Fig. 3(a3). Therefore, it can be experimentally argued that as the fatigue frequency increases, the micro-damage inside the composite specimen increases, which is obvious, and eventually the amplitudes increase and central frequency of the higher harmonic signals decrease.

Time-dependent stress relaxation can be quantified using the non-linearity concept outlined earlier. Using the Eqs. (7) and (8), the second- and third-order harmonic parameters can be calculated and compared between the 0h and 8h time spans. In this study, the time-domain signals acquired at the pristine state and during the stress relaxation process at the 0h and 8h points are analyzed. In this analysis, for the 2 Hz fatigue loading rate, the amplitudes found from Fig. 3(a3) are utilized as the numerator of Eqs. (7) and (8) and the those from Fig. 3(a2) are denominators. A similar approach is followed for the specimens that went through the 5 Hz and 10 Hz fatigue loading rates. The Non-linear Damage Index (NLDI) parameters are illustrated in Fig. 4. The equations used to determine the NLDI parameters are shown in the first figure of each row of figures. The “Red” color bar or the bottom bar (attached to its pair) represents the NDLI at the beginning of the stress relaxation process as termed as unrelaxed NLDI. To determine the relaxed NLDI, the signals collected at the 8-h point are utilized and performed for the FFT of time-domain signals. After zooming-in upon the second- and third-order harmonics, the amplitude values are determined and thus are eventually used to calculate the relaxed NLDI. The “green” or the top bar (attached to its pair) in Fig. 4 represent the relaxed NLDI. It can be seen from Fig. 4 that the relaxed NLDI is in most cases less than the unrelaxed NLDI irrespective of the fatigue loading rates and cycles. Moreover, as the fatigue cycle increases, the difference between the unrelaxed and relaxed NDLI increases. For example, the length of “Green” bars in the 2 Hz, 5 Hz and 10 Hz specimens are less than that of the “Red” bars observed in second- and third-order non-linearity cases after 225,000 fatigue cycles. Since the NDLI quantifies the damage state at its current stress condition, the smaller value of the relaxed NDLI indicates a lower stress state compared to the unrelaxed stress state. However, the damage state and relaxation are influenced by the fatigue loading rate which is evident by quantifying the lengths of “Green” and “Red” bars among the results of 2 Hz, 5 Hz, and 10 Hz in the second-order harmonic signals. Clearly, the lengths of “Red” and “Green” bars at 75,000 cycles in the 5-Hz specimens are higher than those of the 2 Hz and 10 Hz specimens for the second-order non-linearity. Similarly, the lengths of “Green” and “Red” bars at 150,000 and 225,000 cycles at 5 Hz are higher than those of the bars for the 2 Hz and 10 Hz specimens. It indicates that a higher damage state has prevailed in the 5 Hz specimens compared to the other two fatigue loading rates, which could be an indication of damage-prone loading frequency.

### 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.

Figure 6(a) shows a signal acquired from a point on a specimen that underwent fatigue loading at 10 Hz for 225,000 fatigue cycles. The zoomed-in signal shown in Fig. 6(b) is an illustration of front and back reflection amplitudes. The front reflection of the wave is encountered due to the presence of water–composite interphase, whereas the back reflection is recorded from the composite–water interphase. Therefore, the difference between the front and back reflection is the transmission of the wave through the composite in the thickness direction. It is intuitive that the damage, such as a micro-crack, delamination, or another form of irregularity inside the material, would reduce the wave transmission behavior. Therefore, the ratio of this back and front reflection amplitude provides an indication of the material damage state inside the specimen. As the fatigue cycle increases, the specimen would incur more damage and as such the ratio would decrease. In Fig. 6(c), a distribution of the amplitude ratio for 1500 points at a single instance in time has been presented. It follows a normal distribution with a mean amplitude ratio (MAR) of 11%. Figure 6(f) shows the change of MAR over an 8-h stress relaxation period. The “red” (top), “green" (middle), and “blue” (bottom) lines represent the trend of MAR for a specimen during the stress relaxation period after 75,000, 150,000, and 225,000 fatigue cycles. Clearly, the MAR at 0-h decreases as the number of fatigue cycles increase, which is consistent with our intuitive hypothesis. Again, as the stress relaxation duration increases, the value of MAR increases which is evident from the change of MAR as we go from left to right on the plot. However, the increments on the slope of MAR have steeper values at the beginning that reduce as the relaxation duration increases. The vertical “magenta” line divides the whole relaxation duration into two zones, as termed as Zone A and Zone B, based on the MAR slope values. An arbitrary, at least 0.2% increment of MAR compared with its previous timestamp falls into Zone A, whereas Zone B contains the MAR values which are less than 0.2% increment. Thus, a specimen went through the 10 Hz fatigue loading rate, the zone dividing line was drawn at 2 h of relaxation duration, whereas for 2 Hz and 5 Hz loading rates, these lines are drawn at 1 h and 4 h. The objective of dividing these two zones is to demonstrate the dependence of the fatigue loading rate on the stress relaxation rate. Therefore, it is evident that the specimens that went through 5 Hz of fatigue loading rate have the most sustained MAR slope of at least 0.2% for 4 h compared with other fatigue loading rates. Moreover, for the 5 Hz loading rate, the value of MAR at 0-h is the lowest of the other two loading rates. This indicates the lowest back reflections. This is the evidence of the highest damage state which is at the 5-Hz fatigue rate. As the specimens encounter higher order damage state, at higher order the stress reduction rate and duration are also evident.

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.

## Conclusion

Time-dependent stress reduction in CFRP-woven composites is quantified using non-linear higher harmonic interactions of Lamb waves and traditional bulk waves. To quantify the higher order non-linear parameters, the pitch-catch experiments are performed following 75,000 tensile–tensile fatigue cycles at the loading rates of 2 Hz, 5 Hz, and 10 Hz. These experiments are repeated in three stages where each stage has 75,000 fatigue cycles resulting in 225,000 fatigue cycles, and the experimental data are acquired every 15 min during an 8-h interval. After performing the FFT of the obtained time-domain signals, the calculated second- and third-order harmonics are compared with 75,000, 150,000, and 225,000 fatigue cycles. Keeping the exact configuration of the loading rate and cycles, pulse-echo experiments are performed at 15-min intervals for 8 h. Deploying a parallel experimental process, the mean amplitude ratio of reflected bulk waves traveling through similarly fatigued specimens is determined and compared in the same manner as in the pitch-catch experiment. The results obtained by these two techniques matched closely and quantified the stress state over an 8h period. In both the techniques, the 5 Hz fatigue loaded specimens show a higher damage and strength reduction state compared to the 2 Hz and 10 Hz fatigue loaded specimens. To conclude the findings obtained by the non-linearity in Lamb Waves and traditional bulk waves, the ultimate remaining tensile strengths of the specimens are determined. As predicted by both techniques, specimens with the 5 Hz loading rate had minimum tensile strength compared to the other two fatigue loaded specimens.

## Acknowledgment

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