Ultrasonic Lamb Waves Research Articles

The Methodology for the Reliability Evaluation of the Signal Processing Methods Used for the Dispersion Estimation of Lamb Waves

The applications of ultrasonic Lamb waves are becoming more and more frequent and diverse in industrial fields. These waves are used in non-destructive testing (NDT), structural health monitoring (SHM), for the assessment of the quality of materials, etc. However, due to the unusual properties of these waves, new or adapted signal processing methods are necessary to use for the analysis of such wave signals. Based on the requirements of ISO 17025, the work presents the methodology for verification of the signal processing methods used to reconstruct the dispersion curves of the phase and group velocities of Lamb waves. The key sequence of the mathematical and experimental stages, which must be passed to evaluate the reliability of the methods are developed. The present methodology includes the algorithm to optimally quantify the uncertainty, as to avoid additional testing, saving cost and time. The main characteristics and sources of the uncertainties, that mainly affect the accuracy of the obtained results are identified, analyzed, and presented. New standards for certification testing, inspection, and monitoring can be developed more easily and quickly based on the provided methodology in the future.

An Emat Design with Track Coil and Periodic Permanent Magnets for Enhanced Ultrasonic Lamb Wave Generation in Non-Ferromagnetic Plates

An Emat Design with Track Coil and Periodic Permanent Magnets for Enhanced Ultrasonic Lamb Wave Generation in Non-Ferromagnetic Plates

Recurrence analysis of friction based dry-couplant ultrasonic Lamb waves in plate-like structures

In this study, the effect of friction on the generation of dry-coupled Lamb waves is experimentally investigated. Recurrence analysis is performed to analyze the complex behavior of friction based dry-coupled Lamb waves. In particular, the effect of the normal force, which is necessary for a stronger dry-coupled Lamb wave generation and the friction, on the transmission of mechanical energy and determinism characteristics of Lamb waves are investigated. The results verify that larger friction coefficient and friction force are crucial for generation and propagation of strong Lamb waves supporting the fact that the main mechanism to transfer mechanical energy using dry-couplant is friction. The sensitivity of Lamb waves to the friction coefficient, highlights the importance of designing specific pads with respect to condition of the surface. Besides, the results show that the normal force and friction coefficient can change the determinism characteristics behavior of multimode Lamb waves. Furthermore, it is shown that the determinism value is sensitive to the friction coefficient and normal force. A similar trend is observed in the determinism values and friction coefficient. In general, a smaller friction coefficient indicates smaller determinism value. Additionally, it is shown that a normal load can change the behavior of a system, as observed from recurrence plots, owing to changes in the Lamb waves trajectories in the phase-space domain. In addition, it is shown that recurrence plots enable the detection of mode transitions in multimode Lamb waves. Recurrence analysis is a complementary tool to frequency domain methods for accurate analysis of multimode Lamb waves behavior.

Bayesian compressive sensing for recovering the time-frequency representation of undersampled Lamb wave signals

The characteristic extraction of ultrasonic Lamb wave is the prerequisite for its efficient utilization in the structural health monitoring. In the situation of intentional signal compression or unexpected data missing, the accurate recovery of contained information is challenging. To address this problem, this work proposes the time-frequency representation (TFR) reconstruction scheme for undersampled Lamb wave signal. Unlike the conventional method, both the sparse prior and structural sparse prior in the two-dimensional plane are considered in the design of Bayesian compressive sensing. The simulated signal is adopted to validate the effectiveness of the proposed method. Furthermore, different ratios of available samples are investigated to analyze the recovery ratio of TFR. Even if the available samples are smaller than those from the Nyquist rate, the TFR recovery ratio can reach 70%. The experiments using array transducers in noisy environments are also conducted. The time-of-flight information extracted from the recovered TFR is accurate and the relative error is smaller than 3%. Besides, the comparisons with conventional schemes for compressive sensing are carried out to demonstrate its superiority.

Multi-defect detection based on ultrasonic Lamb wave sign phase coherence factor imaging method

The ultrasonic Lamb wave total focusing method (TFM) only uses the amplitude of the defective scattered signal for virtual focused imaging, while ignoring the phase information of the scattered signal and the dispersion characteristics of the Lamb wave, resulting in low imaging resolution and easily produced artefacts in imaging. To solve this problem, an ultrasonic Lamb wave imaging method based on phase coherence is proposed in this paper and the sign coherence factor (SCF) in the full matrix scattering signal is extracted. Moreover, the method uses the SCF to weight the amplitude of the full matrix scattering signal, suppresses the side lobes of the defect echo signal and the Lamb wave dispersion effect, improves the ultrasonic Lamb wave imaging resolution and weakens the artefacts. Finally, single- and multiplehole defects in aluminium plates are detected for experimental validation using an ultrasonic phased array. The array performance indicator and signal-to-noise ratio are used as indicators for quantitative assessment of imaging performance. The results show that compared with the TFM imaging, the SCF imaging can effectively suppress the noise and scattered signal side lobes, improve the array performance indicator (API) by 69.1% and improve the signal-to-noise ratio (SNR) by 73.9%. In addition, the SCF imaging can effectively weaken the interference of scattered signals between multiple through-hole defects, resulting in fewer artefacts in imaging.

Advances in theoretical, experimental and applied studies on second-harmonic generation of ultrasonic Lamb waves

<p indent=0mm>Nondestructive evaluation and testing of material degradation or micro-damages in an early stage are essential for the maintenance of the safety-critical structures. Therefore, it is important to detect defects in a very early stage or even better to detect changes in the microstructure before the defect initiation even occurs. Linear ultrasonic characteristics are not sufficiently sensitive to the kind of microscopic degradation of the material. However, on the other hand, even very small imperfections can produce very significant excess nonlinearity, which can be orders of magnitude higher than the intrinsic nonlinearity of the intact material. The excess nonlinearity is produced mainly by the strong local nonlinearity of microcracks. In order to assess the damage in components at an early stage, the use of nonlinear acoustics has been shown to be the leading and one of the most promising techniques. Second harmonic generation is a typical acoustic nonlinear effect of ultrasonic waves, which can be used to assess the material degradation and detect the micro-defects in their early stages. Furthermore, ultrasonic Lamb wave has been widely used in large-scale industrial structures for health monitoring and material nondestructive testing. Possibility to inspect inaccessible or hidden areas and great cost-effectiveness are the key advantages of ultrasonic Lamb wave-based damage detection approaches. Second harmonic Lamb wave, which combines the advantages of Lamb wave and the high sensitivity of second harmonic waves to micro damage, is drawing increasing attention in recent years. The measurement of second harmonic Lamb waves is investigated as a promising feature extraction technique to characterize material nonlinearity in plate-like structures. However, the second harmonic fields of Lamb waves are much more complex due to the dispersion and multi-mode nature of propagating Lamb waves. In general, the effect of second harmonic generation is often very small and can easily be overlooked experimentally. Consequently, proper mode tuning with physically based feature is essential to enhance the efficiency of second harmonic Lamb wave generation and reception. In experimental work, it is also difficult to observe and measure the second harmonic Lamb waves. Although significant development has been made in theoretical analysis and experimental investigation of second Lamb waves, the use of second harmonic Lamb wave for structural damage assessments is still limited by a series of scientific and technical difficulties. In this paper, the theoretical and experimental advances concerning second harmonic Lamb wave are systematically reviewed. First, theoretical analysis of second harmonic Lamb waves based on nonlinear reflection of acoustic waves is reviewed. Then, a general approach of perturbation method and normal modal expansion for theoretical study of the second harmonic Lamb waves propagation in a plate is introduced. The physical insight of nonlinear feature among different Lamb modes is clarified. As revealed in these pieces of theoretical work, the second harmonic modes of Lamb waves display an accumulative effect when conditions of phase matching and non-zero power transfer from the fundamental mode to the second harmonic mode are satisfied. The conditions for strong nonlinear effect of Lamb wave are discussed. Experimental observation and measurement of second harmonic Lamb waves are reviewed. Cumulative effect and efficiency of second harmonic generation of Lamb waves are discussed. The guideline for choosing the Lamb modes with higher sensitivity to material nonlinearity is provided. Based on the theoretical and experimental studies, the applications of second harmonic Lamb waves for material damage evaluation are summarized. In addition, some obscure understanding and disputed issues during the study in this research area are clarified. Finally, prospective priorities and research trends of second harmonic Lamb waves are discussed.

Non-contact phase coded excitation of ultrasonic Lamb wave for blind hole inspection

The combination of air-coupled ultrasonic testing (ACUT) and ultrasonic Lamb wave is featured with long-distance propagation and high sensitivity to discontinuities, which is a promising method for rapid and accurate inspection of plate-like materials and lightweighted structures. However, dispersive nature of Lamb wave, signal attenuation plus inevitable noises would lead to low signal-to-noise ratio (SNR). To address this problem, phase coded excitation and pulse compression technique are proposed in this paper to achieve higher SNR by over 10 dB in received signals. 13-bit and 1-carrier-period Barker code is employed as both main lobe peak and Peak Side-lobe Level (PSL) are relatively high. It is demonstrated that A0 mode Lamb wave has good localization ability for defects based on these SNR-enhanced signals. Furthermore, Damage Index (DI) and modified Reconstruction Algorithm for the Probabilistic Inspection of Damage (RAPID) are applied to realize ultrasonic imaging based defect evaluation. Results show that the imaging results agree well with the actual artificial defects in terms of size and shape. Lamb-wave-based air-coupled ultrasonic testing, combined with DI and ultrasonic imaging algorithm, could be a potential way in the NDT of lightweighted structures.

In situ monitoring of polymer 3D printing using ultrasonic Lamb waves generated by a 3D printed wedge

The additive manufacturing (AM) of polymers has become increasingly popular in industry to create complex geometries for parts while maintaining low manufacturing costs. Rapid nonuniform cooling introduced during the printing process can lead to part warping and delamination which are precursors to part failure and/or poor part quality. Previous researchers have integrated online monitoring techniques such as optical or thermal sensors. These techniques are limited to monitoring exposed areas of the part leaving unexposed areas unmonitored. In this work, ultrasound has been implemented to monitor the inaccessible areas where defects can originate. An in situ guided wave-based ultrasonic inspection system was implemented into the test setup to detect delamination during printing. A 3D printed ultrasonic wedge was designed to generate S0 guided waves that were received by a broadband pinducer. The wedge was printed onto a plate of 6061 aluminum acted as the print bed for the experiments. A 3D printed part was printed between the transducers, which was continuously monitored for signs of delamination. This presentation will report on precursors to part delamination that are difficult or impossible to sense using previous methods. This work seeks to reduce 3D part failures and, thus, polymer waste associated with 3D printing.

Finite Element Simulation of Axisymmetric Elastic and Electroelastic Wave Propagation Using Local-Domain Wave Packet Enrichment

Abstract A local-domain wave packet enriched multiphysics finite element (FE) formulation is employed for accurately solving axisymmetric wave propagation problems in elastic and piezoelastic media, involving complex wave modes and sharp jumps at the wavefronts, which pose challenges to the conventional FE solutions. The conventional Lagrangian interpolations for the displacement and electric potential fields are enriched with the element-domain sinusoidal functions that satisfy the partition of unity condition. The extended Hamilton’s principle is employed to derive the coupled system of equations of motion which is solved using the simple Newmark-β direct time integration scheme without resorting to any remeshing near the wavefronts or post-processing. The performance of the enrichment is assessed for the axisymmetric problems of impact waves in elastic and piezoelectric cylinders and elastic half-space, bulk and Rayleigh waves in the semi-infinite elastic domain and ultrasonic Lamb wave actuation and propagation in plate-piezoelectric transducer system. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for all problems, including those involving multiple complex wave modes and sharp discontinuities in the fields at the wavefronts.

Sparse Array Ultrasonic Lamb Wave TDTE Imaging Method

The current Lamb wave topology imaging methods are limited to the ideal model of through-hole defects, ignoring the influence of mode coupling. However, blind hole defects, such as structural plate corrosion and cracks, are the most likely defects in practical applications, and Lamb waves are bound to appear mode-coupled at these defects. The use of the full matrix data calculations of the direct and concomitant sound fields is time-consuming and limits the industrial application of the ultrasonic Lamb wave time-domain topological energy (TDTE) imaging method. A TDTE imaging method of a sparse ultrasonic Lamb wave is proposed in this paper to solve the aforementioned two problems. The sparse array is designed based on a genetic algorithm, then the direct sound field and the accompanying sound field are calculated using the sparse array data. The mode conversion of the Lamb wave is considered when calculating the accompanying sound field, and then the accompanying sound field is reversed in time so that the direct sound field and the accompanying sound field are adaptively focused at the defect. The influence of Lamb wave mode conversion and dispersion on the imaging results is overcome. Finally, the TDTE values of the direct sound field and the accompanying sound field are used as the imaging pixel values to improve the imaging accuracy. The experimental results show that for asymmetric blind hole defects, the TDTE imaging method is more accurate than the total focus (TFM) imaging method; the sparse array designed by the genetic algorithm has good sidelobe characteristics and can effectively reduce sidelobe noise assuming the array aperture is unchanged. The TDTE imaging results based on the sparse array data show that the imaging efficiency is greatly improved compared with the full array when the receiving array is four elements under the condition of ensuring the imaging quality.

Using Lamb wave TDTE method to realize ultrasonic array super-resolution imaging of multiple asymmetric defects

Lamb waves are characterized by frequency dispersion and mode conversion. The imaging results obtained with traditional ultrasonic array imaging methods suffer from artifacts, reducing the imaging accuracy. In addition, traditional methods are subjected to the Rayleigh criterion and cannot image multiple defects with spacing smaller than the resolution threshold of ultrasound array imaging. In this paper, an ultrasonic Lamb wave inverse scattering topological imaging method based on time-domain topological energy (TDTE) was proposed. In this method, the topological progressive process of the inverse scattering topological imaging method was transformed into solving the direct and the adjoint sound fields. Then, by inverting the time of the adjoint sound field, the direct and the adjoint sound field can focus adaptively at the defect, which overcomes the influence of Lamb wave mode conversion and dispersion on the imaging results, and improves the imaging accuracy and resolution. Finally, the direct and the adjoint sound field are fused, and the TDTE value was taken as the pixel value of imaging. Accordingly, the limitation of the Rayleigh criterion was broken; the resolution of imaging was improved; the multi asymmetric defect imaging with the interval was realized less than the resolution threshold of the ultrasonic array imaging. The experimental results show that the TDTE method for multiple asymmetric defects can achieve super-resolution than the total focus method (TFM) and time-reversal (TR) method and can accurately distinguish the asymmetric defects when the defect spacing is smaller than the resolution threshold of the ultrasonic array imaging.

Influence of cyclic-loading induced fatigue micro-crack growth on generation of nonlinear ultrasonic Lamb waves

Influence of cyclic-loading induced fatigue micro-crack growth on generation of nonlinear ultrasonic Lamb waves

Unidirectional emission and detection of Lamb waves based on a powerful and compact coils-only EMAT

This study investigated a compact noncontact electromagnetic acoustic transducer (EMAT) without bulky magnets or separate electromagnets for unidirectional transmission and reception of ultrasonic Lamb waves in 1 mm thick steel plates. The transducer head consists of only two, closely adjacent and elongated flat coils. The powerful sensor technology bridged air gaps up to 1.5 mm. Two phase-adjusted 1 MHz pulses at 1 kV amplitude enhanced the sideways-projected ultrasound field in one direction; others were attenuated by destructive interference. A 5 mm ultrasound wavelength with 1 mm thick steel plate was used for testing; wave field calculations were in good agreement with results. As a possible application, an inaccessible edge of a steel plate was scanned, where the difficulty was increased by an additional edge at a similar distance representing competing interfering reflections. More general progress beyond this specific non-contact application is the demonstration of phase-accurate ultrasound transmission and sensitive echo reception with a compact system of mutually coupling inductors at high power and continuous high-bias currents (1 kA), allowing compact and powerful EMAT phased arrays based on single-layer densely arranged coils.

Automated fatigue damage detection and classification technique for composite structures using Lamb waves and deep autoencoder

This paper presents the development of a robust automatic damage diagnosis technique that uses ultrasonic Lamb waves and a deep autoencoder (DAE) to detect and classify fatigue damage in composite structures. Piezoelectric (PZT) transducers are installed on carbon fiber reinforced polymer (CFRP) composite plate specimens to interrogate structural integrity under uniaxial fatigue loading. Fatigue damage evolution from matrix cracking to delamination is monitored by periodically acquiring the ultrasonic wave response. A deep autoencoder (DAE) model is adopted for effective tracking of ultrasonic response variations and for diagnosing fatigue damage in the composite specimens. The ultrasonic signals collected from pristine specimens are processed and used for training the DAE model. To improve the accuracy and sensitivity of the damage diagnosis, the architecture and hyperparameters of the DAE model are optimized, and a statistical detection baseline is defined to capture damage indicators. The ultrasonic signals obtained after applying additional fatigue cycles are introduced into the trained DAE model to validate the damage detection and classification capabilities. The damage sensitive features automatically extracted from the bottleneck layer of the DAE model are used to classify the fatigue damage mode. Singular value decomposition (SVD) is used to further reduce feature dimensionality. The patterns in the reduced features are then analyzed using a density-based spatial clustering of applications with noise (DBSCAN) algorithm. The results show that the proposed technique can accurately detect and classify the fatigue damage in composite structures, while removing the need for manual or signal processing-based damage sensitive feature extraction from ultrasonic signals for damage diagnosis.

An advanced Wigner–Ville time–frequency analysis of Lamb wave signals based upon an autoregressive model for efficient damage inspection

It is quite straightforward to generate and acquire ultrasonic guided waves in metallic or composite structures to investigate structural defects; however, the interpretation and evaluation of the reflected/transmitted signal to extract useful information is a challenging task. This is primarily due to the dispersion and multimodal behavior of Lamb waves, which are dependent on the frequency of the excitation wave and the thickness of the material under investigation. These multimodal and dispersive behaviours lead to a complex waveform structure, and therefore require an advanced signal processing technique to decipher useful information in the time and/or frequency domains. For this purpose, due to its desirable mathematical properties, the Wigner–Ville distribution (WVD) is considered to be a powerful tool for generating time–frequency (TF) spectra and estimating the temporal and spectral features of this type of complex signal. However, due to its quadratic nature, undesirable cross-terms and spurious energies are also generated, which limit the readability and the interpretation of the spectrum. To suppress this effect, this paper describes the use of an autoregressive model based upon Burg’s maximum entropy method to modify the kernel of the discrete WVD. This technique was applied to ultrasonic Lamb wave signals obtained using various numerical and experimental configurations to extract the useful discriminatory spectral and temporal information required for the mode identification, localization, and quantification of structural damage. When used for damage localization, the proposed method precisely estimated the distance between two closely spaced notches in a metallic plate from different simulated noisy signals with a maximum uncertainty of 5%, based upon an excellent TF energy distribution. Moreover, the energy concentration (EC) of the TF energy distribution, in combination with the variation of its instantaneous frequency (IF) curve, was also effective in identifying the overlapping modes of the Lamb wave signal. Finally, for damage quantification, three TF-based damage indices, namely EC, TF flux, and IF, were extracted from five sets of specimens using the proposed TF scheme and used to train the regression model. Model testing proved that the damage indices have the potential to predict crack sizes precisely and reliably.

High-resolution Lamb waves dispersion curves estimation and elastic property inversion

Ultrasonic Lamb waves have been widely used for non-destructive evaluation and testing. However, the inversion from the measured guided signals to the material properties is still a challenging task in terms of multimodal dispersive signal processing and parameter estimation. This paper presents a robust strategy including the high-resolution extraction of the multimodal dispersion curves and model-based elastic property estimation. First, the estimation of signal parameters via rotation invariant technique (ESPRIT) is employed to extract the dispersion curves of the Lamb waves in the plates. Then, the particle swarm optimization (PSO) algorithm is used to retrieve the optimal model parameters by maximizing the objective function built from the dispersion equations. The elastic properties (i.e., two independent constants for the isotropic plate and four constants for the transversely isotropic plate) can thus be determined. Results of the steel, aluminum and composite plates demonstrate that the estimates are in agreement with the references. The root mean squared errors (RMSEs) between the estimated and theoretical dispersion curves calculated by the inversed model parameters for simulation, steel, aluminum and composite experiments are 0.027 rad/mm, 0.032 rad/mm, 0.033 rad/mm and 0.102 rad/mm respectively. The estimated error of thickness is less than 1%. The proposed model-based inversion strategy offers several advantages: (1) the high-resolution estimation of dispersion curves allows the objective function built for the parameter inversion without a peak-finding process. (2) The ESPRIT based dispersion curves extraction strategy offers a sharp objective function in the parameter space. (3) The inverse problem for ultrasonic waveguide characterization is solved using the PSO optimizer which can be implemented with ease and few parameters need to be tuned.

Reusable piezo-sandwiched PTFE film for in-process monitoring of advanced composites

Environmental and cost-saving advantages derived from the use of composites attract aerospace and automotive industries as these materials offer significant structural and aerodynamic advantages over traditional metal structures. The composites industry, however, is concerned with the manufacturing processes as they cannot provide fast enough cycle time to match metal alloy processes. Our research aims to develop a sensing technology in the form of a reusable in situ cure monitoring and assessment system that can predict the formation of manufacturing defects and monitor the degree of cure. Thin-film material is chosen from various PTFE-based material by prioritizing the debonding effect and signal transmission through the composite part. Then, the film is used to sandwich piezoelectric actuators and sensors to monitor out-of-autoclave carbon fiber composite plates using ultrasonic Lamb waves by temporarily adhering to the manufactured part creating an effective electromechanical coupling between the sensing film and the laminate. Initial results, through the analysis of the fundamental antisymmetric A0 mode at low frequencies, indicate that analyzing the velocity and amplitude of these waves over cure time determines gelation and vitrification points. Experimental results have also proved the feasibility of using such a reusable film for different curing cycles, always determining certain cure parameters.

Investigation of Surface Damages in Composite Materials Using Ultrasonic Lamb Waves

In this research, it was aimed to figure out the damages in composite materials using air-coupled ultrasonic method. The ultrasonic lamb waves were used to obtain C-scan images of damaged materials. A couple of planar air coupled transducers (500 kHz-Ultran Group-USA) and an air-coupled scanning system (MISTRAS-EPA, France) were used for obtaining the data from the surfaces of materials used in this research. These transducers were located in different incidence angles for obtaining the lamb waves. The data obtained as A-scan signals was saved in txt data. This txt data was processed with the MATLAB codes for drawing C-scan images of the surfaces of materials. The results obtained from this research showed that the surface damages in composite materials can be characterized with using ultrasonic lamb waves.

Total Focusing Method Damage Imaging in Frequency Domain Using Laser-Ultrasonic Lamb Wave Based on Time-domain Filtering in Multi-band

A fully non-contact experimental platform for ultrasonic Lamb wave damage detection was constructed, where laser exciting and the scanning laser Doppler vibrometer were used to realize the high-resolution pickup of the Lamb wave field in the structure, which has overcome the disadvantages of low spatial resolution caused by the conventional contact Lamb wave transducer. In order to suppress the dispersion effect of broadband laser-ultrasonic signal, we proposed time-domain filtering in multi-band method based on wavelet analysis to decompose the broadband signal into multiple narrowband ones and separate the scattering signals effectively without reference signal. On this basis, the total focusing method (TFM) was used for damage imaging. However, when the traditional TFM was applied to image based on ultrasonic Lamb wave, the inherent dispersion characteristic of ultrasonic Lamb wave could lead to the miscalculation of time delay, thus reducing the imaging precision. Therefore, the frequency-domain TFM was developed by applying phase delay in the frequency domain. The logical AND was introduced to synthesize the damage imaging results of multiple narrowband signals to obtain high-precision damage imaging. Our study has shown that the method of time-domain filtering in multi-band combining with frequency-domain TFM can realize non-contact and accurate damage detection in isotropic plate structures, and it is a potential effective method for application in engineering practice.

Adaptability of Ultrasonic Lamb Wave Touchscreen to the Variations in Touch Force and Touch Area.

Previous studies on Lamb wave touchscreen (LWT) were carried out based on the assumption that the unknown touch had the consistent parameters with acoustic fingerprints in the reference database. The adaptability of LWT to the variations in touch force and touch area was investigated in this study for the first time. The automatic collection of the databases of acoustic fingerprints was realized with an experimental prototype of LWT employing three pairs of transmitter–receivers. The self-adaptive updated weight coefficient of the used transmitter–receiver pairs was employed to successfully improve the accuracy of the localization model established based on a learning method. The performance of the improved method in locating single- and two-touch actions with the reference database of different parameters was carefully evaluated. The robustness of the LWT to the variation of the touch force varied with the touch area. Moreover, it was feasible to locate touch actions of large area with reference databases of small touch areas as long as the unknown touch and the reference databases met the condition of equivalent averaged stress.