Fracture Spectroscopy.pdf

Fracture Spectroscopy as Done by D.Passoja in1983 “A Symbiosis of Fractography and Mechanical Testing”

Some General Comments on Spectroscopy Spectroscopy, in general, is concerned with the analysis of electromagnetic radiation or sound waves. A listing of common wavelength ranges that are used are shown below. The sources displayed are principally used for the analysis of molecular structures.

The table below lists various spectroscopic techniques that are in use for the analysis of atomic structure.

Modern Approaches to Spectroscopy 1. The first is done by passing a selected source of light through an unknown and scanning a filter (monochrometer) through a selected wavelength range. This is known as single wavelength spectroscopy 2. The second is done by passing white light (sometimes called “white noise”) through an unknown which resides in one leg of an interferometer. The interferogram is analyzed by means of the Fast Fourier Transform in order to obtain a spectrum. This is known as FT-(IR)-(UV)-(NMR) depending on its use. 3. The third is a variation of the second but is mathematically equivalent to it- the input is an impulse and the output is analyzed by means of the FT.

Fracture Spectroscopy

Fracture Spectroscopy Sequence of Relevant Papers over the years by Dann Passoja

Distribution of Energy in the Ductile Fracture of high Strength Steels

Comparison of the Inclusion Size and Spacing in 2.XX Space

Fracture Profile Analysis by Fourier Transform Techniques

Fourier Transforms, Fracture and Fatigue

Fractal Character of Fracture Surfaces of Metals

Quantitative Analysis of Brittle Fracture Surfaces using Fractal Geometry

Fundamental Self-similar crack Relationships between propagation in brittle energy and geometry in materials fracture

Fractal Dimension as a characterization of the free volume during the fracture of brittle materials

From fracture of ductile metal alloys (Fe Al) to fracture of brittle ceramics, all observations indicate their having fractal characteristics

There were two test methods that I used in performing fracture spectroscopy. •

Fatigue Tests done under servo-hydraulic load control. In comparison to standard spectroscopy this is essentially a monochromatic input-the test frequency, a wave, the crack, that encounters barriers having unknown characteristics (amplitudes and frequencies). It’s desirable to know how this process progresses and what its spectral description is. Equipment should include: a dual channel FT analyzer 100 KHz, two phase lock amplifiers, servo-hydraulic testing apparatus, optical and scanning electron microscopes an optical digitizer and FT software.

•

Bend Tests in a bend test a crack generates a pulse. In fracture spectroscopy speak this is the crack’s excitations and interactions with its own acceleration field. This was proven to be a very difficult test to perform. Vibrational resonances of any kind destroyed any coherency that was present. Approximately 150

experiments were performed before success was achieved. The specimen geometry was always the same. The two variables were the crack starting notch and the test fixture. Success was achieved when a crack was started from a razor blade cut and the sample was gently held in my hands.

Fracture Accelerometry Fracture accelerometry is a bend test done on a pre-cracked specimen under especial vibration isolation conditions that utilizes two small rigidly mounted accelerometers symmetrically mounted on ends of a specimen that are drilled and tapped. The accelerometers measure the longitudinal accelerations emitted from a fracture occurring midway between them in a bar of PMMA having dimensions of 1 cm X 1 cm x 10 cm. The accelerometers were capable of measuring longitudinal vibrations having frequencies of up to approximately 30 kHz and could withstand shocks of 50 (g=9.8 m/sec^2). The test sample, a bar of PMMA 1cm X1cm X10 cm, was drilled and tapped at both ends in order to mount small accelerometers, is precracked. In the test the bar is loaded, crack movement is initiated and the specimen is broken into two pieces. The initial test mass consists of the fracture specimen plus the accelerometers totaling approximately 12.5 grams. The test is concluded when the assembly fractures with each part of the assembly separating apart. The test appeared as is shown in the following figure:

The Accelerometers: Accelerometers were chosen for the fracture measurements because of their high frequency response and of their ability of providing low noise velocity and displacement signals. Force measurements were also considered as an alternative but it would not have been possible to determine velocity and /or displacement measurements because of the complications associated with the inertial mass of the system. Mass: In this study mass was considered to be constant i.e. it is defined: as that amount of “matter” that can be treated as a proportionality constant existing between force and acceleration -the inertial mass. The mass is considered to be adjustable in these experiments. It is possible to use either force or acceleration languages by using the equation: F(t)=ma(t). In the analytical section of this work I use both the measured acceleration(s) and the computed momenta where the velocity is acceleration x time.

Experimental Method: Experimental Information Accelerometers: PCB, longitudinal detection, shock-up to 50 Kg, measurement to 30 kHz (resonant frequency at 40 K Hz) Test Material: Plexiglass- v- sound=1858 m/sec, elastic modulus=4.18 x 109 Pa Sample size: 1 cm x 1 cm x 10 cm Assembly mass: 12.5 grams-when fractured~6.25 grams each section Signals Observed: All acceleration signals observed taken from a fracturing specimen exhibited an initial large pulse ( 1p) followed by a damped sinusoid. This was observed from either accelerometer and all types of materials including PMMA glasses and ceramics. The “signature or characteristics” -1p- pulse-its height and width: for PMMA the time of the pulse half width was 2 µ seconds and the pulse height was 20 KGs. The computed velocity is .392 M/second having a displacement of 7.84 x 10^-7 M. During the fracture event, before separation the momentum was 4.92 x 10^-3 kg-M/sec. the kinetic energy was ~1.92 x 10-3 Joules. Taking the crack velocity to be .8 csound it was estimated that the sample would separate into two pieces in 6.7 x 10^-6 seconds. The acceleration signal was associated with crack formation, the emission of longitudinal pulses, movement of the masses of the fractured specimen after separation and interaction of the crack with its own stress waves. The longitudinal waves generated came from the displacements of the cracks’ faces during its formation. The pulses, formed as the crack moved occurred within ~ 6x10-6 seconds then propagated outward from the crack faces and remained as stable standing waves before being attenuated. It was determined that these waves contained ~ 200 nodes in the case of the large unbroken sample and 2 x ~100 nodes in the case of the broken samples. The cross spectrum exhibited several characteristic spectral features: there was a peak at ~ 11 k Hz another at ~17k Hz ( both of these peaks could be excited by an impulse response (by tapping with a BB) and were due to longitudinal standing waves. There was a broad rounded background region from ~ 0 to 17 k Hz that had a maximum that was

the same as the 11k Hz peak- in other words, the peak “split” the broad band in half. There were no phase shifts detected between the detectors. In other words this didn’t occur because of phase shifts between detectors 1 and 2. The impulse response of the assemblies both before and after fracture identified the sharp peaks to be sample resonances: the peak at 11 k Hz came from the unfractured sample and the peak(s) at 17 kHz came from the fractured halves of the sample. The 17 kHz peak was calculated to be 14.14 kHz but it could not be determined where the discrepancy came from. It was evident that the broad peak between 0 and 17 kHz was related to the crack’s history. From the width of the peak it is possible to estimate the lifetime of the fracture “event” this is 1/ω= 1/17 kHz or ~5.88x 10^-5 seconds so this represents a crack history that occurred after the fracture event since separation occurred in 6.7x10^-6 seconds. This would include any remaining actions that the crack would have had after separation. What the crack did: The history of crack’s lifetime can be deduced, in part, from the spectrum and the CCF. Fracturing imparted an impulse load to the system and the crack traveled across the sample, exciting longitudinal standing waves in both the partially fractured and completely fractured, separated pieces. The broad spectral peak was created from the 11 kHz peak in the following manner ( this information was obtained by obtaining the impulse response of partially fractured samples). The 11 kHz peak split into two peaks, of which, one moved downward and disappeared (>> 0 hZ) with the other one moved upward, sharpened, and finally remained at ~17 kHz. (See figures for more information). The mechanics behind this are as follows: the downward movement of the peak ( reduced ω from 11kHz >> 0 kHz) occurs due to a compliance change caused by the crack. The figures included outline this point further and include some differential equations that you can study that elucidates the behavior of the partially cracked mechanical “system”. The movement from he center position upward from the 11 kHz position to>> 17 kHz also comes about from the separation of the sample into two parts but, in this case the shift comes about due to mass separation.

Cross Correlation Function The figure shown below is typical of the cross correlation function output obtained from the fracturing of PMMA. Mathematically the cros correlation function is defined as: t

( aL ∗ aR ) = ∫ aL∗ (t )aR (t + τ ) dt to

is a product AL* (ω ) A (ω ) in Fourier space

The CCF appeared to be an interferrogram similar to that obtained in optical interferometry. At first, this result was completely unexpected since the coherency requirement for these experiments was not conformed to. Contradicting: any such results were the facts that: 1) the loading conditions, used for some of these tests were noisy and sometimes

severe, destroying any hopes for coherent signal detection 2) the detectors were not matched so their outputs could be slightly different under the same conditions 3) the fracture process itself was thought to be far from coherent and was thought to be a “white noise source” 3) “path lengths” were not varied in order to get interference. Confirming: the experiments showed that the detectors had detected coherency in the fracture acceleration signals taken from the opposite directions of the fracture surfaces. Apparently, there were longitudinal standing waves excited in both parts of the broken samples that were initiated by the initial 1p crack pulse, that were excited coherently by the fracture source. The signals were correlated by the cross correlation function and were shown to be interdependent. These results were obtained under the strictest control of outside noise sources-in other words they didn’t influence the test-in fact the test couldn’t be done without applying these controls. Discussion: Shown below is a summary of some of the relevant data in this work. I’m including it here as a whole because I’ve found myself referring to it quite often. Some of the numbers do have meaning. Item

Value

1p pulse width

2.x10^-6 seconds

1p pulse height

20x10^6 m/sec^2

Velocity Displacement Mass of assembly Momentum (µ) Momentum (α)

~4000 cm/sec 8x10-3cm (80µ) 0.0125 kg 3.308 x 10^-3 kg-m/sec 5.486x10^-27 kg-m/sec

Item Value Kinetic energy (µ) Kinetic energy (KE ) Atomic mass of the assembly Density PMMA Wavelength(λ=h/p)

0.192x10^-3 joules 3.00x10-24 joules 1.87x10-5 eV 2.075x10-26 kg/atom 1180 kg/m^3 1.207x10^-7 m

On Acceleration Measurements It’s got some relevance since it applies to the numbers of resonant frequencies that are available to a fracturing body. Among other things, fracture is a partitioning process. The measurement of the accelerations were done with parity symmetry with a probability amplitude of ψ(x)=ψ(-x). This was done in order to evaluate the symmetry of the forces surrounding crack while the crack initiated, propagated and disappeared. The placement of the detectors was based on the fact that there were forces on a crack, perpendicular to its faces, causing it to open and to propagate. Crack Symmetry and History The symmetry associated with the fracture event was associated with the excitation and alteration of the sample geometry by the crack. The crack initiates the process with a force pulse and moves it as it travels along its path. The force pulse excites resonant frequencies of objects in the crack’s surroundings. The crack’s structure is altered as a result of changes that occur in these resonances. The crack’s “signature” is that of a moving force pulse (probably having higher frequencies than those of the resonances of the test sample), coupled with an alteration of the sample geometry that entails partitioning the sample and changing its compliance. The frequency content of the broad 0-17 kHz band is composed of many different frequencies not unlike that of “white noise”. This would indicate that as the 11kHz resonance changed, it did so in a continuous manner both positively and negatively. The system did not lose energy because there was both positive and negative change in the frequencies. Obviously, these changes were associated with the establishment of the

sample’s higher resonant frequency. The crack mediated the change in the frequencies by changing the mass (ω ↑ ) and by changing the compliance (ω ↓). A record of the crack’s history was established as it was eliminated from the sample. Nevertheless, the record was still in existence encoded on the longitudinal vibrations that it had established in the sample during its short lifetime. Our detection methods were optimized to detect coherent signals, so that it might be possible to determine the record of the crack’s history. If such a history were present it would be encoded on the resonant frequency of the unbroken sample (the “carrier” frequency). It would most likely be an amplitude modulated signal having a distinctive spectrum like the one that was observed. Our ability to observe it was greatly improved due to the analytical methods that we used. It is entirely possible that part of the crack’s history was heterodyned down from higher frequencies via the 11 kHz carrier frequency. This would provide a poor fidelity record of the cracking events because of the low carrier frequency. Nevertheless moving the history down to lower frequencies makes amenable to analysis even though it might be of poor quality. Measurements and Quantum Interference The two detector-parity method that was used for these measurements was difficult to develop because of outside noise sources. Having solved these problems, then what else needs to be considered? It was never clear that we’d be able to measure anything but noise-and that might have been an acceptable answer-a crack is a “white noise acceleration source” having a band width of 10 kHz, for example. Our detection arrangement would’ve identified that. Furthermore, we were hoping to measure some coherence knowing that under ideal conditions a crack’s stress wave probably is, in some sense, symmetrical enough to have spatial symmetry. We also knew that spatial symmetry is wonderful but we needed both spatial and temporal symmetry to achieve coherent detection with our accelerometers. As it came to be things worked quite well. The coherency issue was helped by the generation of longitudinal standing waves in our test samples. We got interference mathematically (instead of physically) by using the cross correlation function and two detectors. A good, quite relevant, question to be asked about the detection method –is will it be possible to measure interference, in general, by this

method? Furthermore is anything that was measured due to quantum interference? Can you tell if fracture is a quantum phenomena? Fracture is a very complex phenomena that takes place in D dimensional space it is irreversible, it occurs in solids (ceramics, glasses, metals, amorphous solids, plastics, polymers, rocks and many more things, some are relevant some are not. When things are placed on an atom basis this work shows that it is possible to obtain some agreement between classical mechanics and quantum mechanics. Momentum, p , scales nicely with Nav. Without the change in scale factor the various factors, such as wave length, become unreasonably small ~ 10^-48 meters or so. The measurement of acceleration interferences that occurs during fracture clearly demonstrates that a coherence exists between the (+/-) acceleration detectors but there is, in all probability, an atomic interference phenomena that underlies the observed behavior. The acceleration signals are presented as momenta p, subsequently, for convenience. The momentum measurements are p1(t) and p2(t) taken during fracture.