To model damping effects in structural dynamics, the literature provides a wide range of alternatives. The different models claim different advantages and fields of use where they are useful but, in the end, the compliance of the model with the experimental results, within a given tolerance, is the ultimate criterion for assessing its quality. In the present paper, the behaviour of a simple steel specimen is studied, taking as a focus the time response of the material. Since the harmonic response is well established, the authors propose to submit the specimen to a trapezoidal periodic load and study the behaviour of the response to the load changes. A simple setup was used, using a steel specimen loaded in the linear regime. A carefull test procedure was carried out in order to characterize the dissipation in a quasistatic periodic load. The experimental results show that the response delay to a ramp load depends on the durations of both the previous flat and the ramp itself.
The search for a suitable model for the structural hysteresis has been around for almost a century, at least since Kimball and Lovell work [
The present work is aimed at providing a contribution, based on experimental tests, to better understand the energy dissipation behaviour of metals under variable loads.
To ensure a better control of the load, the tests were performed with a universal testing machine. The load and deformation time history were experimentally measured. As for the periodic load shape, a trapezoidal wave was used with the flat stage increasingly long, so the next ramp could be considered closer and closer to a ramp after rest, that is, a transient load within a periodic signal. This approach provides a loading of successive ramps with variable length flats; the flats allow for the material to settle down and no memory phenomena would appear in the response to the ramp loads. But experience proved other way, so the study was performed in order to present the behaviour of the specimen. Some parameters to quantify the shape of the load wave and the response of the material were defined. The experimental tests were performed maintaining close control over these parameters. The results were plotted and analysed in order to understand how the length of the flats was affecting the response to the ramp loads.
To study the dissipative behaviour of a common material, a reasonable strategy is to apply a load with some characteristics of the harmonic loads (allowing for the frequency domain approach) and the free vibration (for the time response approach). The waveform is, in this case, crucial and its parameterization shall be used as the input for the analysis of the dissipation phenomena.
In the present paper, the load was chosen to be a trapezoidal wave function of time, for three reasons as follows.
It is an intermediate case between the harmonic response, corresponding to an analysis in the frequency domain, and the pure time response, making it possible to draw some comparisons with both cases since.
It provides both ramps and flats, two important cases for studying the response of dynamic systems, before generalising for arbitrary waveforms.
It is a feasible wave for experimental testing, as opposed to ideal rectangular waves or impulses.
The frequency of the periodic wave was chosen so that a quasistatic regimen is achieved during each cycle, allowing for the vanishing of the transient during each ramp and each flat, while the periodic character of the load still permits the use of a manageable number of terms of the Fourier series to compute the forced response. On the other hand, it is a simple enough case to study, as long as the amplitude of the wave is kept well within the linear range, as was the case.
The material damping for metals is usually quite small and since the wave periods were relatively long, the transient response at the beginning of each slice (ramp or flat) is hardly visible when plotting the complete response, so some specially selected parameters shall be used to describe the relations between load and response, both assuming the trapezoidal form.
The trapezoidal wave ratio,
The area,
The time delay,
Trapezoidal wave showing the amplitude,
The area,
The time delay,
For pure trapezoidal curves, the area of the hysteresis unit cycle is easily computed and for a small delay it is directly proportional to the time delay:
To allow for a long flat in the trapezoidal wave, it would not be practical to use a vibrating system; but since the aim is to consider only the elastic and dissipative forces, a universal testing machine was used to apply controlled loads with the required profile.
The material used in this work is the high strength steel 42CrMo4. The chemical composition and monotonic and cyclic mechanical properties are shown in Tables
Chemical compositions of the studied material (in %wt.).
Material  C  Si  Mn  P  S  Cr  Ni  Mo  Cu 

42CrMo4  0.39  0.17  0.77  0.025  0.020  1.10  0.30  0.16  0.21 
Monotonic and cyclic mechanical properties of the studied material.
42CrMo4  

Tensile strength  Ru (MPa)  1100 
Yield strength  Rp0.2%, (MPa)  980 
Elongation 

16 
Young’s modulus 

206 
Cyclic yield strength  Rp0.2%, cyclic (MPa)  640 
Strength coefficient 

1420 
Strain hardening exponent 

0.12 
Fatigue strength coefficient 

1154 
Fatigue strength exponent 

−0.061 
Fatigue ductility coefficient 

0.18 
Fatigue ductility exponent 

−0.53 
Specimen test geometry and dimensions (mm).
In order to study the energy dissipation of this material, cyclic stressstrain tests in tension were carried out. Six different trapezoidal wave forms, as the examples shown in Figure
Experimental tests sequence.
Sequence of tests  Period 
Trapezoidal wave ratio 
Rising time (s)  No. of cycles 

1°  12  6  0.5  20 
2°  10  2  1.25  20 
3°  8  4  0.5  20 
4°  6  3  0.5  25 
5°  4  2  0.5  25 
6°  3  1.5  0.5  30 
7°  6  3  0.5  25 
8°  3  1.5  0.5  30 
9°  12  6  0.5  20 
10°  8  4  0.5  20 
11°  10  2  1.25  20 
12°  4  2  0.5  25 
Examples of applied trapezoidal wave forms. (a)
Tests of uniaxial cyclic tension were performed in a servohydraulic machine Instron 8874; a strain gage was used to measure the strains along the cycles (see Figure
Experimental setup.
The following procedure was applied for the analysis of the experimental data.
For each test, a set of three time histories was recorded, with load, overall displacement as measured by the LVDT sensor and strain as measured by the strain gage. The analysis then went through the following steps.
Plot raw time curves, for test validation or to detect any undesired occurrence.
Select a sequence of 4 to 8 cycles where the signal is stable, namely, avoiding the initial part where the periodic signal may not be fully settled.
Both time curves are normalized to unity, subtracting the offset (average for an integer number of periods) and dividing by the wave amplitude to illustrate this, in the case of the load curve:
Compute a fitting curve of each of the time histories with a normalized trapezoidal curve—here the problem was expressed as a nonlinear least squares problem solved using Matlab “lsqcurvefit” function.
Plot the normalized hysteresis curves,
Plot the rotated hysteresis curve: in order to increase visibility, the former plot is rotated by 45° and the abscissa scale is then changed so that the points of increasing load can be differentiated from the points where it is decreasing.
Compute time delay
The results from the full sequence of sets are finally obtained as the evolution of time delay or hysteretic areas as functions of period, trapezoidal ratio, and rising time.
Since, for pure trapezoidal waves, the area is approximately proportional to the time delay
In order to illustrate and validate the procedure, a sine example is first used and its results are presented in Figure
Sine wave load. (a) Raw data: time histories for load (above) and displacement (below). (b) Selected cycles: raw data and fit. (c) Hysteresis plot: normalized displacement versus normalized force. (d) Rotated hysteresis plot: measured points for increasing (green) and decreasing (blue) force and fitting curve (red).
Here the strain gage output is taken as the displacement measurement for the hysteresis plots.
Despite some scattering mostly due to measurement noise, the rotated hysteretic plot exhibits an elliptic form as predicted by the theory. The values for the area are 0.0645 (computed directly from raw data) and 0.0644 (computed from fitting ellipsis) which are in good agreement.
This case validates the experimental setup for the following cases.
In order to illustrate the procedure for the trapezoidal case, first consider the trapezoidal wave Nr 2 from the above sequence (
Trapezoidal wave (
If compared with the sine case, the increasing and decreasing parts of the hysteretic curve are not so clearly apart and partly overlay due to high noise level.
The next example is the trapezoidal wave Nr 3 (
Trapezoidal wave (
Table
The fitting may first be validated.
Both the trapezoidal ratio
Despite the noise level, the areas from raw data and fitted curves are in very good agreement, with values ranging from 0.023 to 0.034.
Experimental results.
Sequence 







1°  12  5.9  0.51  0.0344  0.0335  5.21 
2°  10  2.0  1.26  0.0291  0.0298  6.50 
3°  8  4.0  0.51  0.0288  0.0298  4.27 
4°  6  3.0  0.51  0.0285  0.0281  3.40 
5°  4  2.0  0.50  0.0261  0.0266  3.85 
6°  3  1.5  0.50  0.0238  0.0247  3.41 
7°  6  3.0  0.51  0.0266  0.0266  4.16 
8°  3  1.5  0.50  0.0231  0.0231  3.32 
9°  12  5.9  0.51  0.0291  0.0281  4.45 
10°  8  3.9  0.51  0.0263  0.0263  3.78 
11°  10  2.0  1.27  0.0264  0.0259  7.89 
12°  4  2.0  0.50  0.0250  0.0244  3.14 
The estimated time delay varies between 3.14 and 7.89 ms.
For those tests where the rising time is
In Figure
Hysteresis cycle area (a) and time delay (b) versus trapezoidal wave ratio.
Note that
despite some scattering due to a reduced signal to noise ratio, both parameters are clearly increasing with the trapezoidal ratio
area and time delay are also sensitive to the rising time
Tests were planned with a simple setup but carefully performed in order to characterize the dissipation in a periodic quasistatic trapezoidal load.
Experimental results show that both the time delay and the dissipation area are depending on the load frequency and rising time. More precisely, for a constant rising time, the response time delay is affected by the flat length between ramps, exhibiting a kind of memory effect.
The material response, as measured by the response time delay, depends on the steepness of the ramp but the material seems to exhibit a memory effect in circumstances that would not usually be taken into account as well, something that invites for further research in this area.
The authors declare that there is no conflict of interests regarding the publication of this paper.