High‐speed fatigue testing of high‐performance concretes and parallel frequency sweep characterization

Cycling loading of brittle materials like ultra‐high‐performance concrete (UHPC), which is often used in marine and civil structures, results in unexpected failures. When a material is subjected to cyclic loading, its mechanical properties change due to the evolution of (micro‐)fractures often denoted as damage. To better understand the effective material's properties under such kind of fatigue load and to relate the material's properties to the specific time‐dependent loading characteristics, the mechanical response of the material shall be characterized at characteristic harmonic excitations.

standard DMA, effective complex mechanical properties of the material in tangential space are obtained in frequencies between 0.01 and 1000 Hz; while the observed mechanical properties of these materials change with increasing frequency.In the case of materials' behavior, by increasing the frequency, Young's modulus increases and Poisson's ratio decreases.Experimental fatigue results will be presented for UHPC samples.Harmonic experimental data include (direct) strain measurements in axial and circumferential directions as well as forces in axial directions.In addition, the resulting complex Young's modulus and evolving damage-like "history" of UHPC will be shown.

INTRODUCTION
In recent years, there has been a notable rise in the quantity of concrete structures exposed to fatigue loading.This can be attributed to advancements in cement and concrete technology, which have resulted in the development of high-and ultra-high-performance concretes (HPC and UHPC) that possess enhanced strength and excellent manufacturing (e.g., flow) characteristics.As a result, structural engineers are now able to design reinforced and prestressed concrete structures with more slender profiles.Due to the reduced cross section of slender concrete structures, the ratio between permanent and variable loads decreases, making them more vulnerable to oscillations.Therefore, it is necessary to conduct fatigue verification during the design process [1].
The loading scenarios that building structures endure during their lifetime vary in frequency and intensity [2], and there is a lack of research on how the rate of loading affects concrete's ability to withstand fatigue under compression.
Despite the varying frequency and intensity of loads that buildings experience throughout their lifetime, the effect of loading rate on concrete's fatigue resistance under compression has not been extensively studied.Research suggests that loading frequency plays a significant role in determining the number of cycles before failure, particularly for high-stress and high-amplitude fatigue loading [3].While buildings are subjected to varying levels of stress and intensity over time, still there has been limited research on how loading rate affects concrete's ability to withstand fatigue when compressed.Previous studies suggest that the frequency of loading is a significant factor in determining the number of cycles before failure, particularly in cases of high-stress and high-amplitude fatigue loading.As a result, the design principles outlined in the CEB-FIP Model Code 90 [4] and fib-Model Code 2010 [5] do not adequately address the impact of frequency on concrete fatigue.Research on the material behavior of high-strength concrete was conducted through small-scale test specimens, examining the static, cyclic, uniaxial, and multiaxial material properties [6].In this work, the impact of loading frequency on fatigue behavior was particularly investigated.Recently, researchers mostly investigate how loading frequency impacts the number of cycles before failure and the strain development of concrete.In development of concrete.This is the first step in a more detailed analysis of fatigue behavior [7].
There is a lack of comprehensive research on how the frequency of loading affects the fatigue behavior of high-strength concrete.Moreover, the majority of studies have only investigated the number of cycles needed to cause failure, without examining the strain development in depth [8].Furthermore, there are limited findings on how maximum stress level, loading frequency, and waveform impact strain and stiffness development in high-strength concrete [9].This shortage of data is particularly apparent when compared to the abundance of research conducted on normal-strength concrete.As a result, our understanding of the degradation process that occurs in concrete due to fatigue loading remains still limited [10].
In this study, as a part of the DFG Priority Program SPP-2020, the behavior of an HPC and UHPC under transient compressive loading and its fatigue behavior in relation to loading frequency are investigated.The main focus is on the number of cycles to failure and also the evolution of strain.It is aimed to analyze the test results to understand the damage development in HPC and UHPC.Thus, the results of the high-speed fatigue test are examined, and, in parallel, the results of the strain changes, the changes in the topology of the samples using micro x-ray computed tomography (µXRCT), and the mechanical properties of the material are examined and sequentially characterized with the help of the dynamic mechanical analysis (DMA) method.

EXPERIMENTAL METHODS
A novel experimental fatigue testing method is developed, allowing for faster cyclic fatigue experiments compared to classic fatigue tests at lower frequencies (which are typically at around  = 10 Hz).In addition, the complex mechanical properties of the samples are extracted.In each step, the microstructure of the samples can be simultaneously characterized using µXRCT to finally obtain a fracture network from the initiation phase of cracks in the sample till the final failure of the sample.

Description of the high-speed fatigue testing
In classic fatigue tests, the excitation frequency is between 0.1 <  < 10 Hz.The maximum frequency is given by technical limitations of the often applied servo-hydraulic testing devices.Here, significantly smaller samples than usual are investigated, which allows for the application of smaller maximum forces as well.Thus, a high-voltage piezoelectric actuator (Piezosystem Jena PSt 1000-150) has been used for fatigue cyclic loading.In the setup developed for this purpose, excitation frequencies are generated in a frequency generator (Physik Instrumente P-517).The continuous sinusoidal signal is amplified (100) with a high-voltage linear amplifier (Physik Instrumente E-482) and sent to the piezoelectric actuator.The piezoelectric actuator can convert the voltage to a displacement up to 150 µm and can apply a maximum excitation force of  = 30 kN.According to Figure 1, the setup is mounted on a universal testing machine (UTM) and, additionally, on a rotary table.The UTM device is responsible for applying preload; while the actuator is responsible for applying cyclic loads.A highly sensitive piezoelectric force sensor (HBM CFW/50KN) is connected below the actuator, which will be responsible for accurate force measurement.The sample is mounted in the middle of an x-ray transparent triaxial cell made out of polyether ether ketone and located on the rotation stage of the µXRCT device.It should be noted that the scan part must be placed directly in front of the CT scan detector so that the sample can be scanned simultaneously.Two strain gauges (HBM K-CXY1 T-Rosette) are placed on the sample to measure the axial and circumferential strains.Also, the internal temperature of the cell and the outside environment are measured during the experiment (by PT-100 sensors).All instruments and sensors are connected to a high-speed data acquisition system (HBM-HBK Transient recorder GEN2tB and GN1640B).The cell is designed for samples with a diameter of  = 6.35 mm = 1/4'', to obtain an optimal resolution.The excitation frequency selected for the fatigue test is 20 <  < 200 Hz.

Description of the DMA
This DMA setup has the capability to measure the evolving effective mechanical properties in real-time and, assuming isotropy, directly.Regarding this, the longitudinal and transverse strains, as well as the applied forces, are directly measured, which permits the direct calculation of Young's modulus and Poisson's ratio.The proposed DMA setup is shown schematically in Figure 1.Further information about this setup and also other DMA configurations utilized in geophysics is available in [7].A high-voltage piezoelectric actuator is used for harmonic excitations.It should be noted that the use of piezoelectric actuators is advantageous due to their sensitivity to slight voltage changes and ability to produce precise oscillations.Furthermore, piezoelectric actuators are fast in responding and therefore allow for high-frequency investigations, and, consequently, the use of threshold voltage is not required.Typically, the displacement amplitude especially at high frequencies ( > 100 Hz) cannot be controlled (open loop) but is influenced by factors such as excitation frequency, static preload, and the cumulative stiffness of the system.However, it is worth mention that actuator displacements can result in highly nonlinear changes in the input voltage received.A piezoelectric force sensor has been incorporated into the system, allowing for direct measurement of the axial force especially at higher excitation frequencies.The setup for each HPC and UHPC sample includes two strain gauges-one for lateral strain and one for longitudinal strain-connected to a diagonal Wheatstone bridge.A transient recorder was used to record the motion of the piezoelectric actuator, forces, strains, and temperature.The experiment (applied frequency sweeps and data acquisition) was controlled via MATLAB scripts.

RESULTS AND DISCUSSION
In the following, the characterization of UHPC properties will be examined using various frequency-dependent, preloads.The interpretation of the experimental results obtained in the high-speed fatigue tests (HSFT) should be discussed.

Effect of preload on fatigue tests
This research aims to explore the mechanical properties of UHPC being developed within the priority program SPP 2020 [1].HPC and UHPC have superior properties while being lighter in weight compared to regular concrete.Nonetheless, regarding the intricate composition and more challenging manufacturing requirements of this concrete variant, it is crucial to evaluate its effectiveness and investigate its mechanical properties.As depicted in Figure 2, one could observe a dispersion effect (frequency dependence) of Young's modulus.Further Young's modulus of UHPC is decreasing when the applied axial stresses (preload) are increasing from 1 to 10 MPa.This is attributable to the "nonlinearity decreasing" pattern observed in stress-strain curves of concrete [11].As a matter of fact, when the level of strain increases, the rate of change in the stress-strain curve decreases, resulting in a lower slope of the curve and consequently, a lower Young's modulus value.Furthermore, one could observe in Figure 2 only the slightly dispersive behavior of Poisson's ratio.In addition to the absolute values of the complex Young's modulus and Poisson's ratio, the intrinsic attenuation of the sample was analyzed by the loss factor 1∕ = tan .Note that   is the phase shift between axial stress and axial strain in the case of Young's modulus .In the case of Poisson's ratio, the phase angle   is calculated from the phase shift between the lateral and longitudinal strains (cf. Figure 2).It could be observed that for UHPC loss factors (phase angles) are small and only slightly dispersive.At around  = 300 Hz, a resonance effect of the setup could be observed.Such resonance effects are inherent in those experimental investigations and could not be avoided.Modifying the setup's stiffness and mass, it has been possible to shift it above 100 Hz.
Consequently, the higher the preload, the stiffness of the material will decrease and will further more reduce the number of cycles until the material fails.Therefore, the amount of preload has a direct relation to the fatigue test results.For this research, a preload equal to 45% of the final compressive strength failure limit has been selected for the tests.

Results of fatigue failure with HSFT
Next, the results of the (high-amplitude) fatigue test using the HSFT method are presented.The fatigue test results of UHPC samples under different test frequencies (f = 50 Hz and f = 100 Hz) are shown in Figure 3.As pointed out in the Introduction already, the applied excitation frequencies are up to one order of magnitude higher than in "classical" high cycle fatigue experiments.As shown, at an excitation frequency of  = 100 Hz, the number of cycles leading to failure is around 140,000 cycles while it increases to 350,000 cycles at a decreased excitation frequency of  = 50 Hz.So far, a physical explanation of the frequency-dependent number of cycles to failure has not been given.From the DMA results shown in Figure 2, a small frequency dependency of the small-strain properties of the material could be observed in the range from 50 to 100 Hz.It could also be observed in Figure 3 (right) that especially the lateral strains significantly change which could be an indication of failure.
In addition, for the same load level, the longitudinal and lateral strain amplitudes a higher excitation frequencies approaching a higher value (cf. Figure 3).Within the experiment, the strain level in the longitudinal and transverse modes (for 100 Hz excitation frequency) becomes about 3 times the level of the lower excitation frequency ( = 50 Hz).By increasing the excitation frequency, the duration of each test is significantly reduced.Applying  = 50 Hz excitation frequency, the experiment has taken 115 min while at  = 100 Hz the test has taken only about 23 min.

F I G U R E 4
The results of the changes in complex mechanical properties of UHPC based on frequency dependence in the first and third phases of a high-frequency fatigue test with µXRCT scanning in both phases.
One of the main goals of this investigation has been to simultaneously characterize changes in mechanical properties and morphological changes in the microstructure of the sample within the fatigue test.Since after a number of loading cycles in the fatigue test, (micro-)fractures evolve, the effective behavior of the material also evolves.
Also, in this setup, for each step when the fatigue test is stopped, a DMA test is taken and at the same time, the sample is scanned using µXRCT to record three-dimensional (3D) data image stacks from the initiation and growth of fractures inside the sample.Finally, the results of the HSFT, DMA, and µXRCT scans can be compared with each other to fully understand the cause of any strain change in the fatigue test.
For example, in Figure 4, the results of phases 1 and 3 are extracted from a fatigue test ( = 50 Hz).It is shown that in the first phase of the fatigue test the stiffness value of the sample is about 47 GPa and in the third phase it is decreased to 43 GPa.The amount of intrinsic attenuation related to these two phases in a complete sample is different.Based on the evolving 3D morphological information of the sample obtained from µXRCT in these two phases, microcracks in phase 3 are observable.These microcracks (material's damage) cause a decrease in material stiffness and a change in the shape of the effective material's properties.

CONCLUSIONS
In this investigation, a new method for high-speed fatigue testing has been developed.The classical fatigue tests have their own limitations when it comes to frequency excitations, as they often can only exert lower frequencies of around  = 10 Hz.However, with the present setup, the frequency range of the test can be extended to a larger frequency domain from 0.1 <  < 1000 Hz.The major advantage of the presented fatigue testing approach is its ability to characterize the material with small amplitudes (DMA) within the fatigue test (high amplitudes).This allows for a sensitive, that is, highresolution characterization of the material without mounting/demounting the sample from the UTM.Another benefit of this method is its capability to measure the impact of temperature on the fatigue resistance of the material.Higher excitation frequencies lead to higher thermal strains of the material.Further, the sample heats up during the fatigue test, influencing the total number of cycles to failure.In order to characterize the evolution of the morphology of the sample during the fatigue test and to characterize microfracture evolution, the fatigue test is stopped at various times, without removing the (static) preload.The sample was characterized by µXRCT, and mechanical properties were examined.Finally, the test is resumed again.
The fatigue crack growth network was extracted from the beginning to the end on the basis of µXRCT data.The setup developed in this research aimed to perform multiaxial tests and the sample sizes selected to have the best voxel size to conveniently track the microcracks from their initiation.These investigations, especially the correlation of fracture evolution on the microscale and the evolution of effective mechanical properties on the macroscale at uniaxial and triaxial preloads (stress states) are ongoing.

A C K N O W L E D G M E N T S
We acknowledge the funding by the German Research Foundation (DFG) through the project STE-969/12-2 (project No. 353921616) within the SPP 2020 "Temperature and humidity induced damage processes in concrete due to pure compressive fatigue loading".Also, we acknowledge the funding by the German Research Foundation (DFG) through the project STE-969/16-1 (project No. 424876160) within the SPP 1897 "Calm, Smooth and Smart-Novel Approaches for Influencing Vibrations by Means of Deliberately Introduced Dissipation." Open access funding enabled and organized by Projekt DEAL.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The datasets generated during and/or analyzed during the current study are available from the corresponding author on request.

F I G U R E 1
High-speed fatigue test setup added to the µXRCT setup.

F I G U R E 2
Dispersive, that is frequency-dependent Young's modulus and Poisson's ratio of UHPC with different preload.

F I G U R E 3
Evolution of longitudinal and lateral strains at excitation frequencies of  = 50 and  = 100 Hz.