In ultrasonic fatigue testing of steels one can observe rapid local heating due to crack initiation and propagation caused by non-metallic inclusions and in addition also an overall heating of the gauge length portion of samples which is attributed to dissipational effects. The computations performed in this study are based on a three-dimensional, fully-coupled linear thermoelastic continuum model, where dissipation is included by employing a volumetric heat source. In the numerical computation the temperature distribution evolution in the geometry of interest is the result of a combination of initial conditions, boundary conditions and the heat source contribution. The heat source function's geometry and intensity are deduced by comparing computed temperature profiles to data obtained experimentally using a thermo-camera. It turns out that the modeling approach, making extensive use of thermography data, yields computational results that are in agreement with the experimental heat evolution, and additionally the amount of heat generated is in agreement with results found in literature.
26.1 Introduction
In ultrasonic fatigue testing non-metallic inclusions in steels can cause fracture under cyclic loading, where crack initiation and propagation in the bulk of samples induce a local temperature increase, see e.g. [1‐3]. Further, heating can be observed along the gauge length of samples without failure. There exist studies to characterize dissipation from a quantitative and qualitative point of view, based on thermomechanics, using thermography data from ultrasonic fatigue testing, see e.g. [4‐7]. In Boulanger et al. [7] and the references therein, the heat equation is used as a basis, which includes terms specifying different sources of heating of the material, capturing intrinsic dissipation, thermoelastic coupling, microstructural coupling and an external heat source term accounts for heat exchange with the environment. By introducing certain assumptions these contributions can be condensed to just one collective heat source, see [7].
In all of these analyses the evaluation of the heat equation is based on a reduction of the spatial dimension, treating boundary and initial conditions via introduction of constants, to be determined using experimental data, and assessing dissipation in a simplified manner. As a diffusion-type equation may be subject to boundary conditions of Dirichlet, Neumann and Robin types, see e.g. [8, 9], the intricacies of these formulations are avoided in the investigations introduced before. In contrast, in Yang et al. [6] natural convection and radiation are introduced in the 1D heat equation, based on [10]. In this procedure boundary conditions are applied in terms of parameters that are fitted to experimental temperature data. Furthermore, a finite element computation of pure heat transfer is performed, accounting for heating induced by dissipation, applying a constant inward heat flux at the gauge length portion of the model geometry.
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In the present study a fully-coupled linear thermoelastic model is used to capture a complete pulse phase of an ultrasonic fatigue experiment, excluding fracture and major plastic effects. Numerical simulations have been performed on the JURECA general-purpose supercomputer [11]. The experiment comprises 15 pulse–pause phases in total, each of which exhibits approximately 1800 load cycles. Each load cycle is modeled applying sinusoidal excitation. Thermal boundary conditions are derived from experimental data. Further, from the difference of a computational reference data set and experimental temperature data the intensity and geometry of a volumetric heat source are deduced, applying a fitting procedure. This allows for a detailed assessment of the evolution of heat generation, interpreted as dissipation in the bulk of the sample.
26.2 Experimental Setup
The heating in steel as a consequence of dissipative effects due to high frequency and fully reversed loading (R = -1) was studied, using an ultrasonic fatigue testing equipment (UFTE, University of Natural Resources and Life Sciences, Vienna, Austria) for very high cycle fatigue (VHCF) experiments. The UFTE, as shown in Fig. 26.1, is operating by resonant vibration of the sample at a frequency of about 20 kHz. An ultrasonic transducer is generating an ultrasonic wave which is transferred via an amplification horn (titanium) to increase the amplitude of the ultrasonic wave in the fatigue sample. The length of the sample was designed via a modal analysis of the resonant conditions of the investigated material. Young’s modulus E, Poisson’s ratio υ and the velocity of sound for longitudinal waves cL in the material under investigation are needed. The ultrasonic wave propagating through the sample is reflected on the free end of the sample generating a standing wave due to the resonant conditions resulting in longitudinal fatigue of the sample. Thus, the resulting stress amplitudes are zero at both end faces of the sample and have a maximum value at the centre of the gauge length, illustrated in Fig. 26.1. This technique allows for the application of up to 109 cycles within a reasonably short time. The temperature of the sample increases during resonant vibration due to material damping (dissipation), the magnitude of which depends on the applied stress amplitude and material behaviour. Therefore, a favourable pulse/pause ratio has to be chosen to limit the temperature increase to 10 K over the entire experiment, i.e. vibration of the sample and rest alternate during the pulse and the pause stage, respectively. In this study, the influence of compressed air cooling does not have to be considered and, therefore, a pulse/pause ratio was chosen, which led to an effective test frequency of about feff = 1 kHz.
3 parts. A close-up of a mechanical testing apparatus with its schematic representation, and detailed technical drawing. The apparatus consists of a metallic structure, an amplification horn, a specimen, and various sensors and components.
Fig. 26.1
Ultrasonic fatigue testing setup with a overview, b corresponding stress and displacement amplitude distribution along the entire loading path and c sample geometry
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The test material was 42CrMo4 steel in quenched and tempered condition, see [12]. The microstructure after quenching and tempering treatment consists of tempered martensite. The sample geometry designed according to the resonant condition at 19.3 kHz using material parameters E = 210 GPa, υ = 0.285 and cL = 5179 m/s is shown in Fig. 26.1 with a constant gauge length. A parallel gauge length of 9 mm and a diameter of 4 mm were used.
Ultrasonic fatigue testing is a vibration-controlled method. Thus, the stress amplitudes applied for fatigue tests have to be calibrated based on the chosen vibration amplitudes. For this purpose, two strain gages were glued on opposite sides at the centre of the gauge length. In the next step, the vibration amplitude, which correlates with the amplitude of the generated ultrasonic wave of the ultrasonic fatigue testing equipment, was set at three distinct supporting points and the resulting strains ε at the strain gages were recorded. In addition, displacement measurements Δx were performed at the free end of the samples using a fibre optic sensor (MTI 2100 Fotonic-Sensor, MTI Instruments Inc., New York, USA). Consequently, a linear relation of strain ε and displacement ∆x, respectively, vs. vibration amplitude VA was obtained. Then, the applied stress σ can be calculated from strain ε by Hooke's law using the value of Young’s modulus of E = 210 GPa. Finally, both linear relations, i.e. ε vs. VA and ∆x vs. VA, were merged into the following linear equation with [σ] = [E ε] = [B] = MPa, [∆x] = μm and [A] = MPa/μm:
Here, A = 9.42 MPa/μm and B = 1.39 MPa were used for cylindrical samples.
Infrared thermographic measurements were conducted in situ during ultrasonic fatigue loading. To this end, a long wave range (7 to 14 µm) thermal camera Vario hr head (Infratec Dresden) with a 640 × 240 pixel focal plane array detector was used, enabling a lateral resolution of 25 µm at a thermal resolution of 0.03 K. Thermography measurements at a sampling rate of 50 Hz were performed at higher stress levels in order to observe measurable heat output during ultrasonic load. To ensure a defined value of the coefficient of emission of 0.96, black lacquer (Dupli-Color® SUPERTHERM) was applied to the samples. Since the thermal camera just captured the front view of the sample, a special mirror thermography was adopted to capture the entire circumference of the sample, see [1]. For this purpose, two mirror-polished aluminium plates were placed at an angle of 45° behind the sample, shown on the left in Fig. 26.2. Thus, three thermograms were recorded at the same time, i.e. the real thermogram of the front view of the camera and two reflectograms of the infrared radiation of the two mirrors behind the sample. The temperature deviation between the front thermogram and reflectrograms was less than 2 K. Finally, a sample was subject to the desired stress amplitude for a relatively short
period, i.e. 100 ms (see Sect. 26.3.2) and the thermogram. The considerations, therefore, refer to heat dissipation during the period of the ultrasonic pulse rather than from the increase of temperature due to the crack growth and final fracture, as shown in the reflectograms in Fig. 26.2.
A schematic of the mechanical testing setup, a photograph of the actual testing equipment, and a thermal image indicating stress points. The photograph highlights the clamps, rods, and cables of the equipment.
Fig. 26.2
In situ infrared mirror thermography setup and associated infrared thermograms. The mirrors as well as their reflections are marked as (1). The sample is marked as (2) and the infrared camera is marked as (3). As an example, the reflectogram of a 42CrMo4 sample showing a hot spot due to an internal defect was used
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26.3 Modeling Approach
The goal of this study is to use experimental information combined with a well-established constitutive theory to mimic temperature evolution observed in an ultrasonic fatigue testing setup of \(42{\text{CrMo}}4\) steel samples, discussed in Sect. 26.2. Section 26.3.1 provides the model’s equations as well as the corresponding weak form and time discretization. Section 26.3.2 deals with the kind of data obtained in experiments and with a way to explore this data in order to gain some insight into the nature of the temperature evolution. In Sect. 26.3.3 a closer look at the data and an in-depth explanation of the methods applied are presented. To model material behavior, different quantities are derived from thermography data and employed via initial and boundary conditions. The results of finite element computations of the temperature evolution during pulse and pulse-decay phases are compared with the experimental outcome.
26.3.1 Mechanical Model
The mechanical framework under consideration is linear thermoelasticity. Here, just a very brief outline of the underlying relations is given. For a comprehensive derivation see e.g. [13]. The free energy density is of the form
where \(\rho\) is the mass density, \(\mu\) denotes the shear modulus and \(K\) is the bulk modulus of the material. The parameter \(\alpha\) is referred to as the coefficient of thermal expansion, \(c\) is called the specific heat capacity. The symbol \({\varvec{\upvarepsilon}}\) denotes the small strain tensor and \(\widehat{{\varvec{\upvarepsilon}}}\) represents its deviatoric part. The difference between the local temperature \(\theta\) and the constant reference temperature \({\theta }_{0}\) is introduced via the variable \(\vartheta =\theta -{\theta }_{0}\). The elastic energy contribution is represented by the first two terms on the right-hand side of (26.1). The third term is a coupling term relating elasticity and temperature and the last term in the free energy is exclusively associated with temperature. Based on the free energy the corresponding constitutive relations for entropy and stress can be derived. Using the Gibbs relation and introducing the Fourier model of isotropic heat conduction
where \({\varvec{\upsigma}}\) denotes the stress tensor and \(\mathbf{b}\) is the body force vector per unit mass. From the right-hand side of (26.3) it can be seen that inertia effects will not be considered in what follows, see e.g. [4, 7]. To implement this model within the framework of the finite element library FEniCS, see [14, 15], the weak form of Eqs. (26.2) and (26.3) is needed. Therefore, the equations are multiplied by test functions and integration is performed over the problem domain. For Eq. (26.2) the associated test functions are denoted \(\delta \vartheta\) and the \(\vartheta\) are referred to as the trial functions. Then, performing integration by parts on terms containing second-order spatial derivatives and applying the divergence theorem yields the variational statement with boundary terms, associated with Neumann and Robin boundary conditions. As (26.2) contains time derivatives of the solution variables, a time discretization procedure has to be applied; here the backward Euler scheme is used, see e.g. [16]. The fully coupled bilinear form is given by
where superscript \(n+1\) denotes the current time step, \(n\) refers to the previous time step and \(\Delta t\) is the step size. Superscripts have been omitted for the sake of readability on trial functions. Subscripts \(N\) and \(R\) are used in the boundary terms to indicate the \(i\) terms accounting for Neumann and Robin boundary conditions, respectively, where \(\left(\cdot \right)=\left\{h,\varepsilon \hspace{0.17em}\sigma \right\}\) and \(\left(\star \right)=\left\{\mathrm{1,4}\right\}\) have been defined to introduce one general Robin boundary condition term, accounting for convection and radiation. The problem to be solved using the finite element method can then be expressed as
The experimental data consist of a time series of arrays of pixel values, captured by a thermal camera, see Fig. 26.3.
3 parts. A. A photograph of a laboratory setup and 2 contour plots. The contour plots has a color gradient scale ranging from 21 to 28 and label theta. B. A scatter-line combo graph of theta versus t plot fluctuating trend plots. C. A graph of u 3 versus phi plots 5 data plots and sine wave.
Fig. 26.3
Experimental setup with thermography plots of pulse and pause phases (a), time evolution of the mean temperature of the sample’s gauge length (b) and the displacement evolution throughout a single pulse (c)
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It is assumed that in the experiment under investigation the temperature at the sample’s surface at fixed positions along the gauge length’s axis is approximately equal to the mean temperature on the associated cross sections, see e.g. [4]. Hence, temperature distribution in the sample must be the same in the front and rear views. Therefore, just a small rectangular part of each of the thermal maps, comprising the gauge length of the sample’s front view, is considered. The mean temperature on cross sections is computed as the mean value of pixel values taken along a horizontal row in the sub-arrays defined for each frame.
The mean temperature values of the gauge length can be plotted versus time for each instant, illustrated in Fig. 26.3b. It can be seen that there are \(15\) pulse–pause phases, each having the same qualitative behavior, where a steep temperature increase during pulse operation is followed by a temperature decrease during pause mode. Each pulse phase is comprised of a pulse and pulse-decay period with a decay of oscillations. In pulse there is an excitation period of \({t}_{\text{pulse}}=100\hspace{0.17em}{\text{ms}}\), after that there is a pulse-decay of \({t}_{\text{pulse-decay}}=200\hspace{0.17em}{\text{ms}}\). Each pause phase has a duration of \({t}_{\text{pause}}=2000\hspace{0.17em}{\text{ms}}\). In addition, it can be observed that the maximum temperature increases with every excitation phase, similar to what can be found in loading of Stage I type, see e.g. [16].
After inspection of the overall temperature evolution of the sample, the third pulse–pause phase is selected as the basis for the following investigation. Therefore, the experimental data of interest will be taken from this phase. In what follows, a \(1{\text{D}}\) temperature difference representation along the sample’s gauge length will be used, obtained by defining equidistant cut planes orthogonal to the sample’s axis. The mean temperature difference values on each of these slices is plotted against coordinate \({x}_{3}\). As it is presumed that temperature on the sample’s cross sections is near-constant, this is a means of reduction of the \(3{\text{D}}\) surface temperature at defined positions along the gauge length.
The material parameters for \(42{\text{CrMo}}4\) steel required in the continuum model outlined in Sect. 26.3.1 are given in Table 26.1.
As the temperature differences encountered in the experiment are small, a linear model formulation with constant parameters is used. In addition, experimental data can be employed in the numerical computation in terms of initial and boundary conditions.
26.3.3 Methods and Results
For the simulation only the gauge length part of the sample is considered, where a cylinder geometry with radius \(r=2\hspace{0.17em}{\text{mm}}\) and height \(h=9\hspace{0.17em}{\text{mm}}\) is assumed. The cylinder axis is parallel to the \({x}_{3}\)-direction and the bottom base is at the origin of coordinates. Hence, boundary conditions may be applied at the top or the bottom base or on the lateral surface.
The mechanical boundary conditions are prescribed via displacements, based on experimental meta data. In the experiment the top surface is fixed and the bottom surface experiences a sinusoidal excitation in pulse operation. Therefore, Dirichlet boundary conditions for the bottom base of the cylinder geometry for a single pulse–pause phase can be formulated as
The starting point is set to a reference time of \(t=0\hspace{0.17em}{\text{s}}\). In (26.6) and in what follows subscript \(b\) and superscript \(D\) are used to denote the cylinder’s bottom surface at \({x}_{3}=0\) and Dirichlet boundary conditions, respectively. The end of excitation is not immediately obvious as in pulse-decay there still is excitation of the sample, but a value of \(\tilde{t }\approx 0.18\hspace{0.17em}{\text{s}}\) can be identified as the time, where heating of the sample stops. An illustration of the deformation, according to Eq. (26.6), for a single loading cycle is given in Fig. 26.3, where the color plots of the sample indicate the magnitude of displacement in the \({x}_{3}\)-direction. The amplitude \(a\) can be found from the experimental setup, given in Sect. 26.2. In the experiment there are approximately 1800 full load cycles per pulse phase followed by some pulse-decay. Assuming four time steps in each load cycle, this gives a total number of \(n=7200\) time steps, giving a step size of \(\Delta t=\tilde{t }/n=2.5\cdot {10}^{-5}\hspace{0.17em}{\text{s}}\) for an even distribution of cycles on \(\left[0,\tilde{t }\right]\) with a constant time step value. Here, decay is not considered explicitly since there are no experimental details available. It is observed that in tension a decrease in temperature in the bulk of the sample occurred, whereas during compression the temperature increased and at the end of each loading cycle. No influence of displacements on temperature was seen, cf. [7]. At the top surface of the geometry, displacements are set to
$${{\text{u}}}_{t}^{D}=0,$$
where here and in what follows subscript \(t\) refers to the top surface of the cylinder geometry at \({x}_{3}=9\, {\text{mm}}\). For the mechanical contribution to the coupled problem no additional types of boundary conditions will be explicitly prescribed and no body forces will be considered. Thus, the associated terms in the linear form (26.4) vanish.
It is not immediately obvious which kind of thermal boundary conditions can mimic material behavior as found in the experiment and how the respective quantities in the boundary expressions can be derived from thermography data. As indicated above, the sample’s geometry has been modeled as a cylinder, characterizing the gauge length of the sample. Hence, thermal boundary conditions at the bottom and top of the real sample cannot be immediately employed in the computation. What is more, the top of the sample is connected to the amplification horn and the bottom of the sample is free, as pointed out in Sect. 26.2. This means that heat transfer across the top and bottom bases of the cylinder geometry is based on different mechanisms than in the experiment. In what follows, the third combined pulse and pulse–decay portion of the experiment will be analyzed to inspect the underlying heat generation and transfer mechanisms. The interval of interest starts where temperature difference increases, associated with pulse operation. Next, the different aspects of modeling will be examined in detail. For the simulation the point of departure is when the lowest temperature difference value is attained in the second cooling phase, being defined as reference time \({t}_{0}=0\).
Next, temperature boundary conditions will be defined, based on experimental data. Heat conduction in the bulk can be modeled using Dirichlet or Neumann boundary conditions at the bottom and the top of the cylinder geometry. For interaction with the environment, Robin conditions may be introduced at the lateral surface. It can be observed that the \(1{\text{D}}\) experimental temperature profile along the gauge length is parabolic, where the maximum value can be found at the center of the sample, increasing continuously in the course of the experiment. Heating of the sample’s gauge length must be due to dissipation during excitation, captured in a phenomenological fashion in the computation. From the temperature distribution along the gauge length it can be deduced that there is no inward heat flux from the sample’s bottom and top boundaries. In addition, it is difficult to quantify the heat loss via outward heat flow, convection or radiation. Since temperature differences are small, it will be assumed negligible. Hence, neither Neumann nor Robin boundary conditions will be considered during pulse operation. The definition of Dirichlet boundary conditions, based on thermography data, is the more accessible alternative. To use experimental information the mean values of horizontally aligned pixel values at the desired boundaries are computed from gauge length temperature arrays for all points in time captured during pulse and part of pulse–decay operation. To use experimental boundary data in the computation, a function fit must be performed for the bottom and top boundary, respectively. The discrete mean values over time follow a sigmoid type distribution. Hence, a logistic function fit seems to be a natural choice, see e.g. [18]. The corresponding functions, used for the bottom and top boundaries, are given as
where the \({L}_{i}\) are associated with the maximum values, the \({k}_{i}\) represent the logistic growth rates and the \({t}_{i}^{0}\) are the midpoints of the curves. The plots of functions (26.7), shown in Fig. 26.4, are shifted upwards by values \({\vartheta }_{b}^{\text{min}}\) and \({\vartheta }_{t}^{\text{min}}\), respectively.
2 scatter-line combo graphs of v versus t plots a sigmoid curve for v D t along with a data points for experimental data.
Fig. 26.4
Dirichlet temperature difference logistic fit for the a bottom and b top of the sample
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In addition, an initial state of the system at the reference time must be defined, where for displacements this corresponds to the undeformed configuration of the sample, i.e. \({{\text{u}}}^{0}={\text{u}}\left({\text{x}},t=0\right)=0\) holds, where here and what follows the supersript \(0\) indicates an initial condition. The initial temperature state is prescribed as the \(1{\text{D}}\) distribution along the gauge length at \(t=0\hspace{0.17em}{\text{s}}\). The polynomial
is employed to fit the initial temperature difference distribution, used in the simulation, see Fig. 26.5.
A scatter-line combo graph of v versus x 3 plots an open downward parabola for V 0 pulse along with the data points for experimental data.
Fig. 26.5
Initial temperature difference polynomial fit for the pulse phase
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Based on the reasoning in Blanche et al. [4], it is assumed that the temperature evolution observed in the experiment can be reproduced as the result of the superposition of proper boundary conditions and a volumetric heat source in the bulk of the cylinder geometry. The idea is as follows: In ultrasonic fatigue testing sample heating may have diverse characteristics. In the three-stage model in [19] the first stage is characterized by a near-linear temperature increase. After that, a plateau-like temperature evolution can be found, followed by a steep increase in temperature, observed shortly before failure. If there is failure due to non-metallic inclusions in the bulk of the sample this is accompanied by a large local temperature rise, appearing as a hot spot on the sample’s surface, see e.g. [1, 2]. In the following, just heating attributed to dissipation without damage, related to Stage I, described before, will be inspected, where the experiments discussed in Sect. 26.2 have been designed to not exhibit major plastic effects, fracture or damage.
Mimicking heating in the bulk of the sample is be done using a phenomenological approach, defining a heat source function. In the simulation the evolution of the temperature distribution in the sample is the outcome of the setup with initial and boundary conditions and the heat source in Eq. (26.4). The difference of \(1{\text{D}}\) temperature profiles based on a reference computation and the parabolic \(1{\text{D}}\) experimental distributions is used to deduce the associated geometry along the \({x}_{3}\) direction and intensity of the volumetric heat source. The experimental data series is shown in Fig. 26.6.
Eight scatterplots of v versus x 3 for t equals 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, and 0.16 seconds. All graphs plot wide open downward parabola-shaped plots.
Fig. 26.6
Experimental \(1{\text{D}}\) temperature difference profile evolution in the pulse phase
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The reference computation is performed without prescribing any additional heating in Eq. (26.4). Thermal initial and Dirichlet boundary conditions are used as specified in Eqs. (26.8) and (26.7). Computation is done in a \(3{\text{D}}\) finite element setting, where in a postprocessing step results are transformed to a \(1{\text{D}}\) representation, computing mean temperature values on cross sections of the sample geometry at fixed positions, coinciding with positions of cut planes used for the experimental data. The resulting series of data points is then plotted along the \({x}_{3}\)-axis of the cylinder. The computational reference data series is given in Fig. 26.7. It can be seen immediately that some kind of heat supply in the bulk of the sample is needed to turn these profiles into parabolic shape, as found in the experimental data.
Eight scatterplots of v versus x 3 for t equals 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, and 0.16 seconds. Graphs a to d plot bell-shaped plots. Graphs e to h plot W-shaped plots.
Fig. 26.7
Computational \(1{\text{D}}\) reference temperature difference profile evolution
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Hence, the difference of discrete experimental and computational reference data, provided in Fig. 26.8, is used as a basis to deduce the geometry and intensity of a volumetric heat source. At the beginning of the time series, the distribution of points is noisy, but turns into a geometry with a symmetric plateau about the center of the gauge length and a sharp fall-off towards the sides. Presuming the geometry of the scatter plots shown below approximately correspond to the geometry of a heat source in the bulk of the sample in the \({x}_{3}\)-direction, the super Gaussian formulation
is used to model the flat-top temperature profiles, see e.g. [20]. The associated function fit is illustrated in Fig. 26.8. Function (26.9) is defined in terms of parameter \(a\), denoting the maximum value of the associated curve, \({x}_{3}^{0}\) is the center point, \(b\) represents the value of half the width of the flat-top and the order of the super Gaussian is prescribed via \(2{n}_{1}\), where parameters \(a\) and \(b\) will be determined using curve fitting. It turns out that a value of \({n}_{1}=4\) is a proper choice to fit the data, using (26.9), shown in Fig. 26.8.
Eight scatter-line combo graphs of v versus x 3 for t equals 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, and 0.16 seconds. All graphs plot invertered U-shaped curves. The data points are plotted along the curves.
Fig. 26.8
\(1{\text{D}}\) experimental-computational temperature difference data and associated super Gaussian fit
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To get some insight into the evolution of parameters \(a\) and \(b\) in the course of the pulse phase the discrete parameter values are plotted versus time in Fig. 26.9.
2 line graphs. Graph a of a versus t plots the increasing trend curve. Graph b of b versus t plots a curve in decreasing trends initially then increases and decreases.
Fig. 26.9
Evolution of parameters \(a\) and \(b\)
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To identify a formulation for the intensity of the heat source, the evolution of parameter \(a\) is inspected. At the beginning of the pulse phase the heat generated in the bulk of the sample is close to zero, followed by a near-linear increase of \(a\) for \(0.04\hspace{0.17em}{\text{s}}<t<0.14\hspace{0.17em}{\text{s}}\). Eventually, there is still some increase in intensity, but at a reduced rate. From the experimental setup these characteristics seem reasonable, since at the beginning of the experiment, a standing wave is induced by the testing device, where it takes some time to produce oscillations at the desired frequency. This complies with the very low amount of heating observed at the beginning of the pulse. Then, there is a steady excitation of the sample, continuously heating up the material. The delayed decrease in the rate of evolution of \(a\) after \(t=0.14\hspace{0.17em}{\text{s}}\) can be attributed to pulse-decay. Assuming a piecewise-linear time dependence of \(a\) on intervals \(\left[{t}_{i},{t}_{i+1}\right)\), there is a correlation with the respective heat source strength, corresponding to a piecewise-constant intensity of the heat source function introduced in (26.9). To employ this relation, a normalized representation of the slopes \({\gamma }_{i}\), defined on intervals \(\left[{t}_{i},{t}_{i+1}\right)\) for \(i=0,\dots ,8\), is chosen, given as \(\tilde{\gamma}_{i}={\gamma }_{i}/{\gamma }_{\text{max}}\), using the maximum slope value \({\gamma }_{\text{max}}\). As data fitting has been performed in terms of temperature difference values, a constant factor must be introduced to relate the internal rate of heating per unit mass to the resulting temperature difference in the sample, given by
Then, \(\rho \hspace{0.17em}r\approx 5.8\cdot {10}^{4}\hspace{0.17em}{\text{W}}/{{\text{m}}}^{3}\) for the maximum \({\check{a}_{i}}\), where \(\left[{\check{a}_{i}}\right]=\left({\text{kg}}\hspace{0.17em}{{\text{mm}}}^{3}\right)/\left({\text{N}}\hspace{0.17em}{{\text{s}}}^{5}\right)\). Using [7] as a reference, this appears reasonable, compared with the value of \(2.0\times {10}^{5}\hspace{0.17em}{\text{W}}/{{\text{m}}}^{3}\) stated, where it must be borne in mind that the temperature increase encountered in this study has a value of approximately \(11\hspace{0.17em}{\text{K}}\). Therefore, the piecewise constant representation
Since parameter \(b\) is immediately related to the geometry of the volumetric heat source along the cylinder axis, the interpretation of its evolution is different. The right part of Fig. 26.9 reveals the variation of \(b\) over time, where for simplicity a linear time dependence is assumed, giving
In function (26.10) the \({\alpha }_{i}\) denote the discrete \({b}_{i}\) and the \({\beta }_{i}\) represent the values of slope on intervals \(\left[{t}_{i},{t}_{i+1}\right]\). The derived relations constitute the part of the volumetric heat source definition that can be obtained from experimental thermography data. To get a spatial formulation, the \(1{\text{D}}\) super Gaussian function of coordinate \({x}_{3}\) must be combined with the geometry of the heat source on cross sections at fixed values of \({x}_{3}\). Assuming that heat generation in the bulk of the sample on cut planes is constant in the interior and has a sharp fall-off towards the lateral surface a super Gaussian formulation may be employed, again, where this kind of circular distribution can be defined as
representing the spatial and temporal evolution of heat in the bulk of the sample, induced by dissipation, see e.g. [8].
Then, using (26.11) in (26.4) with initial and boundary conditions as in the reference computation, it turns out that the resulting computational \(1{\text{D}}\) temperature difference distribution along the gauge length resembles a parabolic profile and is in good agreement with the experimental result, shown in Fig. 26.10.
Eight graphs of v versus x 3 for t equals 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, and 0.16 seconds plots 2 data points for numerical results, and experimental data. Graphs a to d plot bell-shaped curves. Graphs e to h plot W-shaped curves.
Fig. 26.10
Comparison of experimental and computational \(1{\text{D}}\) temperature difference profiles for the pulse phase
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26.4 Summary and Outlook
In this study experimental thermography data are used in the initial and boundary conditions of the fully-coupled linear thermoelasticity model. Based on experimental data and metadata the construction of a volumetric heat source, associated with dissipation, is provided in detail. It is shown that the outcome is in good agreement with experimental findings for the temperature evolution in pulse operation of an ultrasonic fatigue experiment. The rate of heat and geometry associated with the heat source definition are based on fitting. It was found that the resulting intensity can be interpreted in an intuitive way. For the evolution of the width of the distribution in the \({x}_{3}\)-direction the interpretation turned out to be more intricate. The distribution of experimental-computational temperature difference values is pretty noisy up to a time of \(t=0.06\hspace{0.17em}{\text{s}}\), explaining a quite strong variation in parameter \(b\) at the beginning of the pulse. What is more, heat generation might be more susceptible to a decay in amplitude of excitation, illuminating the decrease of \(b\) after \(t=0.1\hspace{0.17em}{\text{s}}\).
The investigations presented before are meant as a preliminary study, discussing dissipation from a phenomenological point of view. To allow for a physical interpretation of dissipation, the equations derived and implemented in the FEniCS finite element library can be enhanced to capture thermoplasticity and damage. Information on the size of non-metallic inclusions, see [21], will be considered to define a proper resolution of the numerical mesh. It is envisaged that more complex material behavior can be inspected in the following analyses based on experimental thermography data exhibiting characteristics of damage.
Acknowledgements
This work was supported financially by the German Research Foundation (DFG) within the framework of the Collaborative Research Center “Multi-Functional Filters for Metal Melt Filtration” (CRC 920, project number 169148856, subproject C04). The authors gratefully acknowledge the computing time granted by the JARA Vergabegremium and provided on the JARA Partition part of the supercomputer JURECA at Forschungszentrum Jülich. Special thanks go to Dr. Rhena Wulf (Institute of Thermal Engineering, TU Bergakademie Freiberg).
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Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
