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Laboratory Report on Biomaterial Degradation, Corrosion Potential Measurements and Building a Galvanic Series

Submitted By: Leonardo Mayer

April 15, 2012

Background:

As a result of different chemical compositions, every material known to man has a distinct corrosion potential that is specific to that material and therefore can be used to identify that material. In this experiment, the potential was calculated for seven metal alloys while undergoing Galvanic Corrosion; this electrochemical process is the result of one metal corroding preferentially to another due to electrical contact by means of an electrolytic bridge.

By submerging the metals in the electrolyte, the potential difference causes ions to leave the anodic metal and adhere to the cathode; therefore accelerating the corrosion of the anode, the less noble (Base) metal1. However, it is not easy to measure the potential; in order to do this a Potentiostat must be set up, the electrodes being tested and a reference electrode are connected to an appropriate software like Gamry Framework that will interpret the data.

Because the working electron (the Cathode) must apply an appropriate potential during the corrosion, and because it is difficult for it to maintain a constant potential while passing the current; the inclusion of a reference electrode is crucial (See Annex A.2). The latest has a know potential and thus serves to regulate the working electrode’s potential at all times without passing current or directly affecting the galvanic corrosion.

Differences in potential between materials under the same electrolyte have been measured, and Galvanic Series have been determined, these series list the materials in order of nobility. The purpose of this lab is to create a galvanic series of seven metal/alloys and compare it to the ASTM standards.

Experimental Procedure:

In this experiment, seven different metal/alloys were tested on artificial sea water. Open circuit potentials where measured using a Gamry pentiostat hooked up via USB to the Gamry Framework software, and using saturated calomel electrode as reference electrode3 (See annex A.3).

A beaker was filled with artificial sea water, and the DI water-rinsed reference electrode was introduced. After appropriately connecting the firsts set of electrodes to the pentiostat (refer to reference 3) and inserting them into the beaker with care to avoid physical contact, the Gamry software was set.

From the top menu, Options, Path, Configure Paths and Browse where selected in that order to set the correct folder for saving the files.

Again from the top menu, the Corrosion Potential option within DC Corrosion in the Experiment tab was selected.

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An appropriate name and identifier was selected along with the following parameters: total time of 600sec, sample period of 1sec, stability of 0 mV/s. The area parameter was calculated in cm2 by estimating a diameter and the submerged height of the sample.

The OK button was selected and as the experiment ran, careful monitoring of the Potential vs. Time graph was done. Once the graph stabilized or after reaching the last possible value, the potential value in the upper right corner above the graph was recorded in table 1, then the F2-skip button was once to skip the remaining time and a second time to save the data.

The first test electrode was then removed, and the second sample was inserted into the beaker and hooked to the pentiostat appropriately.

The procedure was repeated for each sample only changing the file names.

Results:

Table 1: Corrosion Potential of metals/alloys

Metal Al Cu Sn C-Steel 304 SS Ti ZnCorrosion Potential

(V)-1.106 -0.142 -0.478 -0.565 -0.153 -0.303 -1.012

Table 2: Ordered list of metal/alloys from least to most noble

Metal Measured Corrosion Potential (V)

Corrosion Potential from ASTM Galvanic Series Table (V)

Al -1.106 -.75 to -1.0Zn -1.012 -.99 to -1.02

C-Steel -0.565 -.6 to -.71Sn -0.478 -.31 to -.33Ti -0.303 10.05 to -.05

304 SS -0.153 -.47 to -.58Cu -0.142 -.3 to -.36

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Analysis of results:

The specimen area was calculated using formula 1 below, where r is the radius and h is the submerged height of the sample specimen.

Specimen area = πr2+ 2πrh (1)

A table depicting the seven metal/alloys and their experimentally measured potentials was constructed using Microsoft Excel software plotting, to obtain the following:

Discussion of results:

A comprehensive analysis of the results in table 2 was done. It was expected that the measured potential values for each material would lie within their corresponding ASME standard interval; however, this was not shown in the results. Only the measured potential for Zn was actually within the interval, nevertheless the values were not significantly far of the range.

It was concluded that several minor factors are responsible for these dissimilarities. Mainly, many experimental errors could have taken place; some of the most prominent sources of error are: dirt residues in the beaker, inappropriate connection of cables, use of artificial sea water instead of flowing sea water and neglecting to clean the set up after each sample was tested.

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Also, it was concluded that the use of non-perfect specimen and minor errors in calculating the area might have a similar effect. In some cases, the test specimen might have had impurities from previous experiments; in other cases, samples did not have perfectly cylindrical shapes, therefore making the specimen area formula inadequate. Also, sometimes the sample would tilt meaning that the submerged height used in the area formula was inaccurate.

Broadly however, the measurements were acceptably approximate to the expected values. So a further analysis was made by comparing the graph obtained in the analysis of results against the ASME standard in annex A.1. With the exception of the places of Al and Zn, and Ti and Cu being switched; It was concluded that the general order agree between the graphs.

Summary and Conclusions:

The prominent

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Annex A1) ASTM standard for Galvanic Series for Metals in Sea Water. By Atlas Steel Technical

Note No. 7 "Galvanic Corrosion."

2) Potentiostat Setup. By Scielo. Geofís Intl. México 2009. ISSN 0016-7169

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3) Layout of a calomel reference electrode. From “Electrochemistry” by R. P. W. Scott, The Calomel Electrode

References:

1) Corrosion Doctors, Galvanic Corrosion website.2) Physics Forum. Galvanic corrosion and electric potential. July 2008.3) Corrosion Potential Measurements and Building a Galvanic Series. Penn State University,

Emch 403, Lab Procedure. 2012.

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Laboratory Report on Material Characterization and Identification between Heat Treated Samples


February 17, 2012

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Background:

Heat treating refers to processes in which a material is exposed to specific temperature changes, with the purpose of altering its chemical properties. Heat treating metallic materials has repercussions on the sample’s physical microstructure; by controlling the rate of diffusion or cooling of the material’s grains, it is possible modify the material’s lattice. There are three stages in heat treating5: Recovery, Recrystallisation and Grain growth.

Recovery is the first stage where atoms start to relocate and existing dislocations in the material get rearranged eliminating residual stresses. Recrystallisation occurs where new grains with virtually no dislocations nucleate and grow giving high ductility but low strength to the specimen.

The crystalline center cubic structure of Steel is well arranged and very specific1. High temperatures for example, cause the atoms within the grains to spread out and move to later regroup at the grain boundaries, allowing for larger grains. It is know that small, well organized grain structure represents better toughness, shear strength and tensile strength. Hence the aforementioned process has significant effects on the physical properties of the sample.

If the steel sample is slowly cooled after being heated, a resulting layered formation of Ferrite and Cementite called soft Pearlite2 results in softer steel than the original form. Slower cooling rates results in finer versions of Pearlite; if allowed to cool fast enough, Bainite forms allowing instead for less ductile but harder form of the steel (See Annex A.1.A, A.1.B) hence the term “cold hardening”.

The purpose of this lab is to comprehensively analyze data and images provided for three steel samples, one with no additional heat treatment, one allowed for air-cooling and another left to cool inside the furnace. By studying tensile stress test results and grain boundary images, a match will be found for each heat treatment processes and its corresponding data and image.


Three specimens of 82 A-2 grade Steel were heat treated, then microscopic pictures of the grain structures were taken and finally a tension test was performed recording load vs. strain data. The images and data were compared and backing with the background information, a match was determined for each heat treatment processes and its corresponding data and image. All the necessary data was collected as follows:

One steel sample was taken as is, two samples were heated on a furnace at 1590F and one of the samples was taken out and left for cooling outside while the other sample was left inside the closed turned off furnace.

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Once the samples were ready and cool, a 10mm piece was sectioned from the ends of each sample.

Each sectioned sample was mounted in the Metallograph and polished. The samples were etched to reveal grain boundaries, the structure was viewed trhough

the Metallograph and a digital image containing 10-15 grains in width was taken. Then a tension test of the three dog bone samples was done. For this, the width and

thickness was measured, the tensile bar was griped and an extensometer was installed on the sample making sure the extensometer was centered and compressed.

Load, extensometer, extension and Crosshead meters were zeroed. Sample measurements and grip separation were inputted into the software. The test was started and ran until the extensometer had to be removed to avoid

damage. Then the test was allowed to keep going until failure of the sample. The sample was removed and the crossheads backed to position.

Having all the data, analysis of the mechanical properties and grain boundary size was done. For analyzing the grain sizes, ASTM standards specifically the comparison method in section 8 of ASTM E1126 was used to visually examine the provided images and order them in increasing order of grain size.

Later the tensile test data was inputted into Microsoft Excel software, firstly the cross-sectional area of each sample was computed, and then the stresses for each corresponding load data point. Then Stress vs. Strain plots were created and used to determine mechanical properties of each sample.

Results:

The grain sizes determined from visual inspection by ASTM standards were in decreasing order:

Austenitic Sample > Normalized Sample > As Is Sample

Table 1: Mechanical properties from Stress vs. Strain curves

Sample Area (in2)

Young’s Modulus (Ksi)

Yield Strength (Klbf)

Tensile Strength (Klbf)

Elongation (%)

Expectation

1 0.025603 191.519 20.802 42.834 42.37 Fast Cool2 0.025049 168.212 16.676 41.767 36.79 As Is3 0.025402 275.996 15.322 41.672 41.34 Slow cool

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From the dimensions of each sample introduced in the tensile test, the cross-sectional area for each specimen was calculated using formula 1. Then the stresses at each load were calculated using formula 2.

Cross-sectional Area = Width * Thickness (1)

Stress = Load/Cross-sectional Area (2)

The last point on the linear portion of the each graph was determined and that slope was defined as the specimens young’s modulus recorded in table 1. Young´s modulus was calculated by:

E = Stress/Strain (3)

From the graphs, the Yield Strength was calculated using a 0.2% offset line (in red), and determining the corresponding stress value for the intercept. The Tensile Strength was determined by the highest stress point reached; and similarly, the Percent Elongation was determined by the rightmost strain value reached.

The graphs obtained using Excel’s scatter plot feature were:

Sample 1 :

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Sample 2 :

Sample 3 :


A comprehensive analysis was done on the results in table 1 and observed grain sizes; it was concluded that the initial assumptions were correct.

The slowly cooled sample presented the largest grain boundary size because the extended period of heat treating allowed for grain growth. Also, this sample presented the least yield and tensile strength values which accord to the initial idea that soft Pearlite was formed, thereof

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reducing the strength of the steel. Moreover, the sample presents the highest Young’s Modulus which makes sense since the soft Pearlite is more ductile.

The Normalized sample was second in grain size; again this makes sense because heat treatment was done allowing some grain growth but fast cooling reduced the expanded grain sizes a little. Furthermore, just as expected, this sample presented the smallest Young’s Modulus and the highest Tensile and Yield Strengths. This is due to the fact that cold hardening took place therefore strengthening the now less ductile steel.

The sample that was left as is presented the smallest grain size, and intermediate values for Young’s Modulus, Tensile and Yield Strengths. This makes perfect sense since the sample did not undergo any kind of grain growth due to increased temperature and because the mechanical properties were not affected neither for increase nor decrease.

The only mechanical property that did not fit the hypothesized result was the percent elongation. It was expected that the slow cool sample would present the highest percent elongation since it was expected to be the most ductile; also the fast cooled sample was expected to have the least percent elongation since it was estimated to be harder and less ductile. Nevertheless, the results showed an increasing percent elongation of as is, slow cooled and normalized respectively; this could be attributed to the fact that chemical changes in the material affected lattice and resulted in a softer structure that allowed for superior elongation of the slow cooled sample.


It was determined that the original expectations were experimentally proven to be correct; sample 1 was the fast cooled, sample 2 was left as is and sample 3 was the slow cooled sample. These correspond to the largest, smallest and medium grain boundary sizes respectively.

The only exception, on which exceptions were not met, was the percent elongation changes due to different heat treatment procedures. Results showed increasing percent elongation in as is, slow cool and fast cool samples respectively instead of the hypothesized increasing order of as is, fast cool and slow cool. This was attributed to the fact that heat treating causes chemical changes in the sample’s lattice which in fact result in a softer structure with superior elongation capability.

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Annex A4) Time/Temperature transformation of steel phases. From Wikimedia.org

A.)

B.)

References:

4) ESABNA handbook. Structure of steel.5) Engineering Network Global. Pearlite, cementite and ferrite.6) Heat treater's guide: practices and procedures for irons and steels. By ASM International 2007.

7) Material Characterization and Identification. Penn State University, Emch 403, Lab Procedure. 2012.

8) Emch 403. The Pennsylvania State University. Lecture on bonds, structure and processing.

9) ASTM E112. Standad Test Methods for Determining Grain Size.

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Laboratory Report on Plastic Deformation of

6061-T6 Aluminum


March 23, 2012

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Background:

Yielding is defined as the point at which a material no longer undergoes elastic deformation and begins to deform plastically. Predicting yielding is crucial most engineering designs because plastic (permanent) deformation very often signifies that the components can no longer function.

It is hard to recreate exact stress conditions and experimentally predict yielding; therefore, many techniques such as the “0.2% offset” have been developed in order to economically estimate the stress a material can handle. However; yield strength is not a property inherent of each material, it depends on the nature of the applied stresses. Thus for complex load situations such as torsion, the Octahedral Shear Stress (DET) and the Maximum Shear Stress (MSST) Theories are used; these are attributed to Von Mises and Tresca respectively.

When normalized with respect to the tensile yield strength, Mises and Tresca criteria can be used to predict an envelope on plane stress axes whose limits represent yielding at different shear stresses, Appendix A.1 depicts Tresca and Mises yield loci on the principal stress axes.

In this lab, an extruded tubing specimen of 6061-T6 Aluminum was subjected to tensile loading and torsion through an MST Axial-Torsional test machine. Throughout the loading, elongation and angle of twist was recorded at several Tensile Force-Torque data points. The objective was to quantify the elastic response, initiation of yielding and plastic deformation for the aluminum tube, and compare experimental results Mises’s and Trescas’s prediction. Furthermore, the plastic portion was related to the Ramberg-Osgood formula for plastic Strain vs. Stress and the formula’s parameters H and n were estimated.


The dimensions of extruded tubing specimen of 6061-T6 Aluminum were taken and tabulated in Table 1 in the Results section.

The specimen was mounted on the MST Axial-Torsional test machine and tension was applied to a specified length portion “L” and Stress-Strain data was collected. Then, Torsion was applied to that same portion L, and this time, torque and angle of twist data was collected.

Having the experimental data; normal and shear Stress vs. Strain plots were created using Microsoft Excel software. For this; values were calculated for normal and shear stresses at each data point in the tensile and torsion tests correspondingly. Consequently, angle of twist was converted from degrees to radians and shear strain values were computed for the torsion test. All this was achieved using Equations 1 through 6 in the Analysis of Results section.

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At this point; also the true Stress and True Strain values were determined for the tensile test using equations 7 and 8, and the data series was incorporated into the previously obtained engineering plot for tension test. Figures 1 and 2 in the analysis of results section depict the resulting plots.

The engineering Stress vs. Strain plots for both graphs were further analyzed by taking fitting a linear regression to the data relating the slope of the linear (elastic) portion the material’s Young’s and Shear Modulus respectively. Also, these values were plugged into equation 9 to calculate the Poisson’s ratio of the specimen. Figures 3 and 4 in the analysis of results section show the elastic portion of both test’s data along with the corresponding regression formula, and the results are tabulated in Table 2 in comparison to Handbook Values.

Moreover, the resulting plots were studied to graphically determine tensile and shear yield strengths using: Proportional limit, 100µ-strain offset, 1000 µ-Strain offset, 0.2 offset and back extrapolation techniques. The first technique consists solely in visually defining were the graph stops being linear; offset techniques consist on tracing a line parallel to the elastic slope at each of those offsets and determining where the traced line hits the graph and finally back extrapolation consists on extrapolating plastic portion and determining where it hits the Y-axis. Appendix 1.2 depicts the three methods and Table 3 shows the calculated values.

The obtained shear yield strength values were then compared to the values predicted by Tresca and Mises yield models as represented by equations 10 and 11 respectively. In addition, percent errors for each experimental yield technique were calculated and the average error between the model’s prediction and experimental results were computed through equation 12.

At this point, the plastic portion of the Stress vs. Strain graphs was considered and a power regression trendline was related to them. The resulting power line function was then associated to the Ramberg-Osgood function in order to determine n and H parameters; the graphs depicting the power trendline are represented in Figures 5 and 6, and correlating line equation through formulas 13b and 14b, the parameters obtained were recorded in Table 5.

Finally; using equations 15 trough 18, effective stress and strain were computed for the Torsion test; Figure 7 shows the resulting plot. In uniaxial stress state, as it was in the Tension test; effective stresses and effective strains are equal to what was previously calculated, so further analysis was unnecessary. Figure 1 was then compared to Figure 7 with the purpose of determining if effective stress and strain definitions are able to reduce the stress-strain response of the material into a single curve, results are depicted in Figure 8.

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Results:

Table 1: Dimensions of the Specimen

Table 2: Engineering Properties of the Material

Property Obtained Value Handbook ValueYoung’s Modulus, E (Ksi) 10000 10000Shear Modulus, G (Ksi) 4000 3770

Poisson’s Ratio 0.25 0.33

Table 3: Yield Strengths per Definition

Technique Tensile Strength (Psi) Shear Strength (Psi)Proportional Limit 40500 13500

100µ-Strain 40554 134501000µ-Strain 42158 14036

0.2% 42845 14524Back Extrapolation 40100 14150Handbook Values 40000 30000

Table 4: Tresca and Mises Error for Shear Yield Strength

Technique Tresca %Error (About 30Ksi) Mises %Error (About 30Ksi)Proportional Limit 10 22.06

100µ-Strain 10.33 22.341000µ-Strain 6.43 18.96

0.2% 3.17 16.14Back Extrapolation 5.67 18.30Avg Percent Error 7.12% 19.56

Table 5: Ramberg-Osgood Parameters

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Dimension Value Outer Diameter (in) 1.004Inner Diameter (in) 0.813

Length Subjected to Stress, L (in) 1.000Original Area, Ao (in2) 0.0287

Line Equation H/Hs Parameter (Ksi) n/ns Parameterε = 47718 σ 0.0208 47.718 0.021γ = 37732 τ 0.0502 37.732 0.050


The specimen’s initial cross-sectional area “Ao” was obtained trough equation 1 below, the normal (σ) and shear (τ) stresses and shear strain (γ ) at each data point were then calculated using equations 2, 3 and 4 respectively:

Ao = π*[(OD-ID)/2]2 (1)

σ = P/Ao (2)

τ = [T*(OD/2)] / J (3)

γ = [Φ*(OD/2)] / L (4)

Radians = Degrees*(π / 180) (5)

Where P is the load, T is the torque, OD is the outer diameter, ID is the inner diameter, Φ is the angle of twist, L is the length subjected to testing and J is defined by equation 5 below:

J = [π*(OD/2)4]/4 (6)

The True Stresses and True Strains for the tensile test were obtained through equations 7 and 8 where σ and ε are the engineering stress and strain respectively:

σt = σ*(1+ ε) (7)

εt = ln(1+ ε) (8)

Poisson’s Ratio was calculated according to equation 9 below:

ν= [E/(2*G)]-1 (9)

Where E is Young’s Modulus and G is Shear Modulus.

Tresca and Mises Yield criteria for pure shear stress respectively are as follows:

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Tresca: σyield = σ0/2 (10)

Mises: σyield = σ0/√3 (11)

Percent Error was calculated as follows:

%error = [(X-Xexpected)/Xexpected ]*100 (12)

Where X is the experimental value and Xexpected is the expected value obtained from the models.

The Ramberg-Osgood parameters were correlated to the line equations by:

ε = σ/E + (σ/Ht)1/nt (13a)

γ = τ /G + (τ /Hs)1/ns (14a)

Considering only the plastic portion, the above equations can be solved for stress:

σ = Ht *ε nt (13b)

τ = Hs *γ ns (14b)

Where Ht, nt, Hs and ns are the Ramberg-Osgood parameters for tensile and shear strain respectively.

Effective Stress and Strain were computed using equations 15 and 16 respectively:

σeff = (1

√2 ) [(σ1- σ2)2+ (σ2- σ3)2+ (σ3- σ1)2] 1/2 (15)

εeff = (1

√2(1+ν )) [(ε1- ε 2)2+ (ε2- ε3)2+ (ε3- ε1)2] 1/2 (16)

Where, for the Torsion test:

σ1 = τ , σ2= σ3= 0 (17)

ε1 = γ/2, ε2 = 0, ε3 = - γ/2 (18)

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By plotting the obtained results, the graphs depicted on Figures 1 through 8 were obtained:

Figure 1: Engineering Stress vs. Strain and True Stress vs. Strain for the Tension Test

Figure 2: Engineering Stress vs. Strain for the Torsion Test

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Figure 3: Elastic Region for Tension Test

Figure 4: Elastic Region for Torsion Test

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Figure 5: Plastic Region for Tension Test

Figure 6: Plastic Region for Torsion Test

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Figure 7: Effective Stress vs. Effective Strain Plot for the Torsion Test

Figure 8: Superimposed Effective Stress vs. Effective Strain Plots for Both Tests

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After careful analysis of the obtained results; it was determined that the material is in fact isotropic since the data shows that it´s mechanical properties depend in part to the direction across the material in which they are measured.

At large, the results did resemble the handbook values. The value for Young’s modulus was an exact match, and even though this was not the case for Shear Modulus; the value was still fairly similar. Do to the aforementioned dissimilarity; the Poisson’s Ratio value was thrown off by .08; however, this can be attributed to calculation and round-of errors or more even to the fact that Microsoft Excel’s Trendline function does not yield a precise regression formula.

Nonetheless, R2 values for all trendlines and seem to be fairly close to one. It was therefore concluded that the models are quite good and that the major source of error comes from having non-ideal experimental conditions and round-of errors.

The five techniques for estimating shear yield strength gave results fairly close to those predicted via Tresca and Mises models. A percent error analysis allowed to determine, that on average, the models were off by 7.12% and 19.56% respectively; which is more than acceptable considering the numerous sources of errors that may affect the experimental calculations. Because the techniques used to predict yield, are prone to human and approximations mistakes; these two were regarded as the main cause of errors.

Moreover, it was concluded that Tresca’s model does a better job at predicting actual yield strength for every yield definition; this makes sense since Mises model is more conservative and tends to deviate from actual yield values with the purpose of having a safer design.

Finally; an analysis of Figure 8, allowed determining that Effective Stress and Effective Strain concepts serve to relate the trends of both experiments. The elastic portions of both trends are accurately similar to each other; however, yield occurs at a lower stress level in the Tensile plot. Also, there appears to be significantly less plastic deformation in the Torsion test. It was concluded that effective stress and strain concepts are only able to reduce stress-strain response to a single plot for elastic deformation; however, yielding will begin at different point for each stress state and thereon the graphs are not reduced to a single plot.


The specimen appears to be isotropic, and results demonstrate that Tresca’s model is a better predictor of yielding for every yield definition. Effective Stress and Strain definitions allowed reducing the elastic portion of the graphs into a single plot but once yielding occurs, the plots deviate from each other.

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Appendix A5) Tresca and Mises Yield Loci - from the New World Encyclopedia -

6) Yield Definitions

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Offset

Extrapolation

Proportional Limit

References:

10) Plastic Deformation of 6061-T6 Aluminum. Penn State University, Emch 403, Lab Procedure. 2012.

11) Mechanical Behavior of Materials, 3rd Edition, N.E. Dowling, 200712) The New World Encyclopedia, Structural Engineering 2008.

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Laboratory Report on Ultrasonic Measurement of

Engineering Properties


February 24, 2012

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Background:

By definition, a Wave is an oscillation that travels trough time and space; it is known that waves are affected by the medium on which they travel. Parting from this idea, the study of wave´s properties as they travel allows us to infer physical properties of that medium.

Ultrasonic Non-Destructive Testing (NDT) makes use of this principle; by using acoustically coupled piezoelectric transducers to generate high frequency sound pulses, pulses are sent at one end of the material being studied and received on the other1. It is good practice to use a couplant between the transducers and the material, this will allow for easy wave travel into the material, thus avoiding unnecessary signal distortion. The sound wave travels through the material and the received signal is amplified and analyzed.

In this experiment, longitudinal and shear ultrasonic waves were used to carry a non-destructive analysis of stainless steel. Relations were drawn between the inspected wave travel time through the sample and the sample’s physical properties such as elastic modulus (E), Poisson’s ratio (ν), shear modulus (G), bulk modulus (K) and Lame’s constant (λ).


Using electrical calipers and balances, three measurements were made of the sample’s diameter, height, weight and calculated density and recorded in Table 1; the height was then determined to be the distance between the transmitter and receiver.

The transducers were coupled at the end of the sample using a specialized couplant for the longitudinal and honey for the shear wave transducers. Finally, computer software was used to measure the travel time for the 1st peak (wave reaching receiver) and 2nd peak (echo of wave back into transmitter) and results were recorder in Table 2.

Having collected all the necessary data, longitudinal and shear wave velocities denoted CL and CT respectively were calculated using equation 3 in the analysis of results section and average values of the results were recorded in Table 3. It was taken into consideration that the travel distance for the second peak is three time the height.

Stiffness coefficients were calculated using equations 4, 5 and 6, and a matrix was formed to then be inverted using Microsoft Excel’s “=minvers” function (see Annex A) with the purpose of obtaining a resulting compliance matrix. The engineering properties were determined using equations 7 trough 11 respectively. The obtained results in Table 5 were compared to estimated values using formulas 12 through 14 in Table 6, they were then compared to handbook values in Annex B, and finally a measurement error analysis was performed through Taylor series expansion to determine uncertainties in the calculated values.

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Results:

Table 1: Physical Properties Measurements and Calculations

Duplicate Weight (Kg) Height(m)

Diameter (m) Volume (m3) Density (g/cm3)

1 0.101 0.0254 0.0254 0.00001286384524 7.8512 0.1 0.0254 0.0254 0.00001286384524 7.7733 0.101 0.0254 0.0254 0.00001286384524 7.851

AVERAGE 0.1007 0.0254 0.0254 0.00001286384524 7.825ERROR (%) 0.99 0.0 0.0 0.0 0.99

Table 2: Measured Longitudinal and Traverse travel times

DuplicateLongitudinal Traverse

1st peak (µsec) 2nd peak (µsec) 1st peak (µsec) 2nd peak (µsec)1 4.57 13.8 8.61 25.012 4.57 13.8 8.68 24.973 4.57 13.85 8.65 25.01

Table 3: Calculated Wave Velocities

DuplicateLongitudinal Traverse

CL 1st peak (m/s) CL 2nd peak (m/s) CT 1st peak (m/s) CT 2nd peak (m/s)1 5557.99 5521.74 2950.06 3046.782 5557.99 5521.74 2926.27 3051.663 5557.99 5501.81 2936.42 3046.78

AVERAGE 5536.54 2992.99ERROR 1.01 4.19

Table 4: Stiffness Coefficients and Compliance Matrix Coefficients

C11 C12 C66 S11 S12 S66

239.86e6 70.10e6 99.67e6 4.17e-9 -1.22e-9 1e-8

Table 5: Resulting Mechanical Properties using Stiffness Matrix calculations

E (Gpa) ν G (Gpa) K (Gpa) λ (Gpa)239.81 .292 100 192.16 130.28

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Table 6: Resulting Mechanical Properties using Wave Velocity Relation Formulas

E (Gpa) ν G (Gpa)181.29 .294 70.1

Analysis of Results:

The material physical properties were determined as follows:

Volume = π*radius2*height (1)

ρ = mass/volume (2)

Wave velocities were calculated using formula 1 below:

[Δ2 U/ ΔX2] = [1/C2]*[ Δ2 U/ Δt2] (3)

Stiffness coefficients were calculated from the data as follows:

C11 = ρ*CL2 (4)

C66 = ρ*Ct2 (5)

C12 = ρ*[CL2 – (2* Ct

2)] (6)

The engineering properties were obtained from the compliance matrix by:

E = 1/ S11 (7)

ν = - [E *S12] (8)

G = 1/S66 (9)

The Bulk modulus and Lame’s constant were calculated using formulas 10 and 11 respectively:

K = E/[3*(1-2ν)] (10)

λ = [νE]/[(1+ν)*(1-2ν)] (11)

Relations between engineering properties and wave velocities:

E = ρ*{[Ct2*(3CL

2-4CT2)] / (CL

2-CT2) (12)

ν = {1-[2*(CL2-CT

2)]} / (CL2-CT

2) (13)

G = ρ*CT2 (14)

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For the error analysis, the difference between the largest and smallest measured values for a certain quantity was taken; this difference denoted as the Uncertainty for that measurand was then divided by the average of that measurand’s replications to get the error, and recorded in corresponding table.


A comprehensive analysis was done on the results in Table 5 and 6 was done in comparison to the handbook values stipulated in Annex B. It was concluded that at large, both the stiffness matrix and the velocity relation methods yielded extremely accurate approximates for Poisson’s ratio. Bulk Modulus, Shear Modulus and Lame’s constant results were considered to be within a reasonable error discrepancy; however, further examination of uncertainties had to be done. Moreover, the estimates for Young’s Modulus were off by a large unreasonable difference.

The stiffness matrix yielded a Young’s Modulus value of twice as expected the wave velocity formula a value sixty units below. These approximates were regarded as illegitimate and a measurement error Taylor series analysis was performed. It was determined that the calculations of all physical properties and travels time presented negligible to non errors. The assumption that discrepancies resulted from these errors was disregarded.

Further uncertainty study was done on the wave velocity measurements; the Longitudinal still presented an error just above 1% hence was disregarded a source of error. However, the Traverse Wave velocity measurement presented an error of about 4.2%; this could have had a significant impact in the final determination of mechanical processes.


After thorough consideration, it was concluded that measurement uncertainty, precision and repeatability errors may have had a minimal causation effect on the mechanical properties calculations. Nevertheless these were not the major source of blunder; miscoupling the transducers, matrix inversion mistakes, imperfect samples and rounding/approximating errors all played a bigger role.

Be that as it may; with the exception of large Young’s Modulus discrepancies, most of the results obtained through both Stiffness Matrix and Velocity Relation formulas yielded acceptable results as compared to the handbook. The experiment, and thus the Ultrasonic NDT technique were regarded as a successful, affordable and precise (if enough time and dedication are put into it) way for studying material mechanical properties.

32

Annex AA.)

- Compliance Matrix

- Stiffness Matrix (Inverse of Compliance Matrix)

4.17e-9 -1.22e-9 -1.22e-9 0 0 00 4.17e-9 -1.22e-9 0 0 00 0 4.17e-9 0 0 00 0 0 1e-8 0 00 0 0 0 1e-8 00 0 0 0 0 1e-8

B.) Handbook Values for Stainless Steel Mechanical Properties2

E (Gpa) ν G (Gpa) K (Gpa) λ (Gpa)193 .305 77.2 163 115.66

References:

13) American Society for Nondestructive Testing, Nondestructive Testing Handbook, Volume 7, Ultrasonic Testing (ASNT, 1991).

14) The Engineering Toolbox. Metal and Alloys Properties Handbook.

33

239.86e6 70.10e6 70.10e6 0 0 00 239.86e6 70.10e6 0 0 00 0 239.86e6 0 0 00 0 0 99.67e6 0 00 0 0 0 99.67e6 00 0 0 0 0 99.67e6

Laboratory Report on Anisotropic Response of an Orthotropic Sheet Material


February 17, 2012

34

Background:

The mechanical properties of any material are strongly related to that material’s composition and structure; and in general these properties do not depend on the direction in which they are measured across the material. However, there are cases as fibrous composites for example, which present greater strength and stiffness along the direction of the fibers than perpendicular to them.

This concept of properties varying when measured in different directions across the material is defined as Anisotropy, and materials that present this quality are called Anisotropic. Furthermore, special cases that present symmetry about three orthogonal planes are called orthotropic materials.

In this lab, a thin orthotropic sheet of glass fiber reinforced epoxy was tested; the sheet was subjected to uniaxial tensile stress in both along the fibers direction and at 45-degrees with respect to the fibers. Using the collected data, Hooke’s law for a thin orthotropic sheet was used to determine four independent elastic compliance coefficients and from there a stiffness matrix was constructed for the sheet.

The purpose of the lab was to use the obtained stiffness coefficients in order to determine the material’s Poisson’s Ratio, Shear Modulus, and Young’s Modulus. The latest was calculated for both directions in which the sheet was tested; and results were compared to, as previously discussed, an expected rise in strength and stiffness as measured parallel to the fibers.


Tensile specimens from a glass fiber reinforced epoxy were fabricated; one of which had fibers oriented parallel to testing direction and another which had fibers oriented at 45-degrees with respect to testing direction.

Three measurements for width and thickness were taken, and the averages were used together with equation 1 in analysis of results section, to calculate the cross-sectional area for both specimens. Results were recorded in Table 1 on the results section.

Afterwards, the tensile test experiment was set up; longitudinal and traverse gages were glued to each specimen and connected to junction box 9 -output 6 and junction box 10-output 7 respectively. Then, gages were balanced and calibrated by turning the balance knobs until the red light on output extinguished. Finally, the shunt cal switched was turned to A-position and the gain on the Labview program was adjusted to 2.0V.

The first sample was installed in the MST grips; the strain gages were zeroed at load zero, and the DAQ program was started. Then the MST program (micro profiler #11) was started in load

35

control mode, using a 20%FS level and a 1%/sec rate. Throughout testing, load was incrementally applied allowing for at 98 strain reading, but with care not to exceed the proportional limit of the sheet. When testing was over; the DAQ test was stopped, the data was copied into a text file, the sample was unloaded and the procedure was repeated for the 45-degree sample.

The data collected was analyzed and for this, the first step was to convert load and strain (measured in volts) into the more familiar units. For this, Microsoft Excel software was employed, and equations 2 and 3 on the analysis of results section were used to convert the load and strain units into pounds and micro-strain respectively. Subsequently, stress values for each data point were calculated using equation 4 and results were plotted. Finally a linear least-squares fit line was added using Excel’s Trendline function and added to each plot. For each sample respectively, Figures 1 and 2 in the Analysis of Results section depict a single graph containing plots of both Longitudinal and Traverse Strain vs. Stress.

The obtained trendline functions were then used to determine the four independent compliance coefficients S11, S12, S’11 and S’12. This was achieved by relating the slopes of each trendline to one of the coefficients as explained in relations 5 through 8 in the analysis of results section. Furthermore; the remaining compliance coefficients S22 and S66, were calculated by solving equations 9 and 10 accordingly. Results were recorded in Table 2.

At this point, the stiffness coefficients Q11, Q12, Q22 and Q66, were calculated using the previous results and plugging them into equations 11 through 14 respectively. Results were recorded in Table 3.

Further analysis was done in order to calculate the elastic engineering properties of Poisson’s Ratio, Shear Modulus, and Young’s Modulus in both directions. For this, the matrix relations described by relation set 15 in the analysis of results section were used (see Appendix A.1 for data).

Finally, the experiment was concluded by performing an analysis of the results in comparison with the aforementioned expectations. It was hypothesized that an increase in stiffness due to anisotropy would be represented in a slightly higher Young’s Modulus for the 0-degree test.

Results:

36

Table 1: Dimensions of Specimens

Specimen 0-degrees 45-degrees

Replicate 1 2 3 Avg 1 2 3 AvgWidth (in) 1.004 1.001 1.005 1.003 0.940 0.952 0.952 0.948

Thickness (in) 0.216 0.219 0.217 0.217 0.215 0.217 0.214 0.215Cross-sectional Area (in2) 0.218 0.204

Table 2: Compliance Coefficients

Table 3: Stiffness Coefficients

Table 4: Engineering properties


37

Compliance Coefficient ValueS11 1.1554S12 -0.1775S’11 1.8731S’12 -1.0413S22 2.2920S66 4.3999

Stiffness Coefficient ValueQ11 0.8759Q12 0.0678Q22 0.4415Q66 0.2273

Property Value (Msi)E1 0.8655E2 0.4363ν12 0.2273ν21 0.4068G12 0.2051

The specimen’s Cross-sectional area was calculated using equation 1 below, where T is the thickness and W is the width of the sample.

Specimen area = T*W (1)

Load and Strain units were converted from data in volts to Pounds and Micro-strain by:

1V = 400lb (2)

2V = [(2.085/2)*1000] µε (3)

Stress was computed using equation 4 below, where P is load and A is cross-sectional area:

σ = P/A (4)

By plotting the previous results, the graphs depicted on Figures 1 and 2 were obtained:

Figure 1: Longitudinal and Traverse Micro-Strain vs. Stress for the aligned (0-degree) sample

Figure 2: Longitudinal and Traverse Micro-Strain vs. Stress for the unaligned (45-degree) sample

38

The logic relating the Stain vs. Stress graphs to the compliance coefficients was as follows:

S11 = Slope of Longitudinal Strain plot of aligned sample (5)

S12 = Slope of Traverse Strain plot of aligned sample (6)

S’11 = Slope of Longitudinal Strain plot of unaligned sample (7)

S’12 = Slope of Traverse Strain plot of aligned sample (8)

Further arithmetic analysis was used to solve equations 9 and 10 below for S22 and S66, Where θ is 45-degrees. (The reasoning for behind these relations can be found on Appendix A.2)

S’11 = [S11*cos4(θ)] + [S22*sin4(θ)] + [(2S12 + S66)*cos2(θ)*sin2(θ)] (9)

S’12 = {S12*[cos4(θ) +sin4(θ)]} + [(S11 + S22 - S66)*cos2(θ)*sin2(θ)] (10)

The stiffness coefficients were then calculated using the following equations

Q11 = S22/ (S11*S22 – S212) (11)

Q12 = -S12/ (S11*S22 – S212) (12)

Q22 = S11/ (S11*S22 – S212) (13)

Q66 = 1/S66 (14)

Relation set for matrix comparison between elastic compliance and engineering properties

39

S11 S12 S13 S14 S15 S16

S12 S22 S23 S24 S25 S26

S13 S23 S33 S34 S35 S36

S14 S25 S34 S44 S45 S46

S15 S26 S35 S45 S55 S56

S16 S26 S36 S46 S56 S66

Where corresponding boxes between the matrices represent equalities, and all the terms that appear in red are equal to zero.


After careful analysis of the obtained results; it was determined that at large, the results did show the expected trend. E1 was found to be about 0.4Msi greater than E2 to, which resembles an expected 0.2Msi rise; nonetheless, the magnitude of these values did not emulate what was originally expected.

Initially; E1 was expected to be around 3.3Msi and E2 was expected to linger somewhere around 3.1Msi. Experiment results yielded values of 0.8655 and 0.4363 respectively. Because the values for all mechanical properties were within reasonable range, and because the general trends of these results were as expected; the discrepancy between calculated and expected magnitudes was attributed to calculations and fit errors.

All throughout the experiment calculations had to be done, and most of the time results were rounded down; to some extent this may have caused results to stray away from expected and into those lower values.

Moreover; early in the experiment, linear fits were performed using Microsoft Excel’s trendline function and the slope of the resulting lines were used to calculate stiffness coefficients. However, it was later found that some R2 values for the fits where as low as 0.73; meaning that the models were not accurate in predicting the trends and the slope values were not good predictors of the actual stiffness coefficients. Again this may have thrown result magnitudes astray from what was expected without affecting the relations among them.


It was concluded that at large, the magnitude of all results were low but somewhat within a reasonable range. Discrepancies were attributed to calculation and computation errors. Nonetheless, it was concluded that expected relations between results as E1>E2 were represented by the results in both magnitude and direction.

Appendix A40

=

1/E1 -ν21/E2 -ν31/E3 0 0 0

-ν12/E1 1/E2 -ν32/E3 0 0 0

-ν13/E1 -ν23/E2 1/E3 0 0 0

0 0 0 1/G13 0 0

0 0 0 0 1/G31 0

0 0 0 0 0 1/G12

7) Elastic compliance coefficients matrix:

1.1554 -0.1775 0 0 0 0-0.1775 2.2920 0 0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 4.3999

Relations to Hooke’s Law:

1/E1 = 1.1554 (a)1/E2 = 2.2920 (b)1/G12 = 4.3999 (c)

-ν21/E2 = -0.1775 (d)-ν12/E1 = -0.1775 (e)

8) Relations between S and S’ matrices:

ε= S*σ (a)

Because S’ matrix belongs to a sample tested with fibers oriented 45-degrees from the basic coordinate system, Stress and Strain formulas must be adapted by:

ε’= S’*σ = T2*σ (b)σ’= T1*σ (c)

T1 and T2 are determined as follows, where θ is 45 degrees, m is COS(θ) and n is SIN(θ):

Similarly, the stiffness matrix S’, can be obtained through:

S’= T2*S*T1-1 (d)

From simple linear algebra, the multiplication of said matrices yields equations 9 and 10 in the analysis of results section.

References:

1) Anisotropic Response of Orthotropic materials. Penn State University, Emch 403, Lab Procedure. 2012.

2) Mechanical Behavior of Materials, 3rd Edition, N.E. Dowling, 2007

41

T1= T2=m2 n2 2mn

n2 m2 -2mn

-mn mn m2-n2

m2 n2 mn

n2 m2 -mn

-2mn 2mn m2-n2

Laboratory Report on Fatigue Crack Propagation Measurement


April 15, 2012

Background:

42

In material science terminology, fatigue is defined as the progressive structural damage that takes place on a localized portion of a material when subjected to cyclic loading. Cyclic loading refers to the process on which a material is loaded and unloaded copious times; and if the applied loads are above a certain limit, the procedure will eventually induce microscopic cracks in the material which slowly but surely grow with respect to further loading.

P.C. Paris and F. Erdogan were the first to characterize the structural parameters that fatigue crack propagation in metallic alloys. The “Paris Law” successfully relates the rate at which a crack length (a) changes in response to accumulated load cycles (N).

Both the material and the geometry of the specimen subjected to cyclic loading will greatly influence the specimen’s fatigue life. Stress raisers like holes or sharp corners are prone to allow for crack formation and potentially failure. Therefore it is critical to take fatigue into account in any engineering design whatever its purpose may be.

In this laboratory; a notched 7075-T6 aluminum specimen was subjected to cyclic loading, and data of crack propagation vs. number of cycles was collected in order to estimate the appropriate Paris Law corresponding to this material.


The dimensions of a notched 7075-T6 aluminum alloy specimen were taken and the area was calculated using equation 1 in the Analysis of Results section; results were then recorded in Table 1 in the Results Section. Moreover, a razor blade was run from the tip of the notch and across the width. The specimen was then mounted into a servo-hydraulic testing machine in order to induce fatigue crack growth of the now sharpened single edge-notched plate (See Annex A.1).

The specimen was cycled in load control under approximately 800lbs to allow for a reasonable crack growth rate; the minimum load was set at 10% of this maximum load (80lbs) for a fatigue ratio (R) of 0.1. Using a telescope; progressive measurements of crack length were recorded as cyclic loading took place.

The collected data was inputted into Microsoft Excel software in order to create a scatter plot of crack length as a function of number of cycles, as shown by Figure 1 in the Analysis of Results. Furthermore; the slope with respect to the origin was calculated at five of the original data points using equation 2, consecutive points that resembled the most linear relation were selected for this. Table 2 presents the resulting slopes for each of the points.

43

The next step was to calculate the range in stress intensity factor (ΔK) at each of the aforementioned points by means of equations 3 through 7 in the Analysis of Results. Results were recorded in Table 3.

Furthermore; resulting values for the slope at the selected points were plotted against corresponding results for ΔK on a Log-Log scale as depicted by Figure 2; from there, the trend equation was found and hence, according to equation 8, the Paris Law parameters. The results were tabulated in Table 4 and finally an assessment with the intent of comparing them with the handbook values.

The last part of this laboratory consisted on taking SEM images (See Annex A.2) of one of the fractured halves of the specimen, and studying the appearance of fractured cross-section at about 10000 magnifications. Both the average spacing between striations and the distance from the striation to the tip of notch were measured for two points. These results were converted to US units using equation 9 and then tabulated in Table 5. An analysis allowed determining that each striation represents a mark left by one cycle; thus, the striation spacing was related to the rate of change in crack length in terms of one cycle. From here; equation 8 was solved for the respective ΔK values, and results were added to Figure 2 (in purple) were studied for agreement to previous results.

Results:

Table 1: Specimen Dimensions

Table 2: Slope Measurements at Five Points

44

Dimension Measured ValueThickness, t (in.) 0.0625Width, W (in.) 1.00

Initial Crack Length, a (in.) 0.1875Cross-sectional Area (in.2) 0.0625

Point (# of Cycles, a) Slope (in/cycles e-5)(8458 , 0.1074) 1.27

(10458 , 0.1645) 1.57(11458 , 0.2010) 1.75(12458 , 0.2508) 2.01(12958 , 0.2898) 2.24

Table 3: Specimen Dimensions

Point (# of Cycles, a) Calculated F Calculated ΔK (Psi√¿ .)(8458 , 0.1074) 1.1940 7989.885

(10458 , 0.1645) 1.2931 10708.47(11458 , 0.2010) 1.3730 12569.08(12458 , 0.2508) 1.5033 15371.83(12958 , 0.2898) 1.6248 17859.84

Table 4: Paris Law Parameters

Table 5: Striation Measurements

Striation Spacing (in.) Distance From Notch Tip (in.)

ΔK (Psi√¿ .)

1 1.33e-5 3e-4 160002 2.47e-5 0.109 25700


The specimen’s cross-sectional area (A) was calculated using formula 1 below, where t is the thickness and W is the width of the sample specimen.

A = t * W (1)

Plotting the experimentally obtained data yielded the graph depicted by Figure 1 below.

45

Parameter Experimental ValueC 2e-8

m 0.701

Figure 1: Crack Length vs. Number of Cycles

Having selected five linear points within the plot in Figure 1, their respective slopes were calculated about the origin according to equation 2 below.

Slope = (a2 – a1) / (N2 – N1) (2)

Where the subscript 2 represents the value for crack length and number of cycles respectively at the point being evaluated, and the subscript 1 similarly represents the quantities for the origin which in this case were zero. Parting from this, equation 2 was simplified in the form represented by equation 2.b below.

Slope = a2 / N2 (2.b)

The stress intensity factor for these five points was calculated according to equation 2 below

ΔK = Δσ * F * √π∗a (3)

Where Δσ is the range of applied stress calculated trough equations 3 and 4, a is the instantaneous crack length and F is determined as a polynomial function of a according to equations 5 and 6.

Δσ = σM – σm (4)

σ = P / A (5)

F = 1.12 – 0.231 X + 10.55 X2 - 21.72 X3 + 30.39 X4 (6)

X = a / W (7)

46

Where A is the cross-sectional area, P is the applied load; σM is the stress at maximum applied load and σm the stress at minimum applied load.

Plotting results for slope against results for ΔK, Figure 2 below was obtained.

Figure 2: Change in Crack Length over Number of Cycles vs. Stress Intensity Factor Range

Paris’ Law equation can be represented as equation 8 below.

dadN

= C*ΔKm (8)

Where dadN

represents the rate of change in crack growth with respect to number of cycles, and

C and m are constant parameters specific to each material.

Unit conversion of Meters to Inches is governed by equation 9 below.

1m = 39.37in. (9)

Where m represents units of Meters and in. represents units of Inches.

47


Even though experimental results for Paris’ Law constants C and m for the 7075-T6 aluminum appear to be underestimating the actual trend, they seem to efficiently the overall tendency up to a reasonable degree. It was determined that there may be several sources of error that lead to this discrepancy.

The most prominent source of error was found to be the fact that, when measuring the crack length as a function of cycle number; the crack did not grow in an infinitely horizontal manner. In fact; as soon as cyclic loading was applied to the specimen, it was experienced that the crack grew diagonally upward and across the width of the specimen. Furthermore; because the microscope could only take horizontal readings of crack length, simple arithmetic explains that the (horizontal) measurement assumed to the crack length at that moment, is in fact smaller in magnitude that the actual (diagonal) crack length.

This phenomenon would certainly lead to underestimation of the crack growth rate due to cyclic fatigue; hence driving our Paris’ law constants estimates away from their actual values.

Another important source of error was found to be the lack of accuracy of equation 6; this relation has been found to be within 0.5% accuracy for a crack length to width ratio of less than 0.6. Nonetheless; in our experiment, a specimen having slightly larger ratio of 0.0625 was used. This fact would also contribute to driving the results away from what was originally expected.

The aforementioned effects are visually represented in Figure 2. It is noticeable that ΔK values from the SEM results do not fall along the experimental data trend, but rather appear to be smaller (underestimated). This “agreement exercise” helps us relate the mathematics world to the physical world; it is very useful for linking the data to its physical (visual) meaning.


It was concluded that at large, results did yield a good representation of the expected dadN

vs.

ΔK trend; however, Paris’ Law constants were slightly underestimated. This was due to two main causes; the crack length measurement error resulting from the inclination of propagation, and the inaccuracy of equation 6 for the geometry of the specimen used.

48

Annex A9) Single edge-notched plate under cyclic load P.

10) SEM Top View Pictures of the Fractured Surface.

Striation 1:

Striation 2:

References:

15) Fatigue Crack Propagation. Penn State University, Emch 403, Lab Procedure. 2012.16) Mechanical Behavior of Materials, 3rd Edition, N.E. Dowling, 2007

49

Laboratory Report on Fracture Toughness of an

Aluminum Alloy


April 7, 2012

50

Background:

Among the many mechanical properties that must be considered for a safe, the linear-elastic Fracture Toughness may be the most important. This quality represents the ability of the cracked material to resist fracture, and it can be determined by means of stress intensity factor (KIc) at which a crack in the material begins to grow.

In this laboratory, a three point bend test was done on a cracked Aluminum Alloy specimen with the purpose of determining the fracture toughness. Following ASTM E 399 standards, an experiment was design on which cyclic loading was applied to the specimen in order to generate a pre-crack and further loading was later applied to fracture the specimen. Data was collected for Load and Crack Opening Displacement (COD) until fracture occurred; and from the data, the stress intensity factor was computed by isolating the elastic portion of the Load-COD relation and using the established elastic stress analysis equations.


According to ASTM E 399 standards, a notched aluminum alloy specimen was pre-cracked by cyclic loading. First; the sides of the notched were polished to enhance observation of the fatigue pre-crack, then the average of three specimen thickness and width measurements was calculated, results were recorded in Table 1 in the results section. The load point and the pre-crack limits (.45 to .55 times the width) were marked on the specimen, a COD gage was installed and cyclic loading was applied.

After roughly 9000 cycles the crack initiated and 500lbs compressive force was used to open the crack. The pre-crack was observed with a telescope and its tips were checked to be within the marked limits. At this point the specimen had been successfully pre-cracked and an average of three measurements for crack length was taken.

Later, a three point bend fracture toughness test (See appendix A1) was done under E 399 limits; stress intensity factor of 30-150 Ksi√¿ ./min, 0.3displacement, 4000lbs load and 0.1in. COD were established as ranges. Similarly, the experiment was set to run at 5000 cycles fatigue rate, 4Hz frequency, 100-1000lbs amplitude, 4in. span and 400lbs/min load rate. Finally load was applied to the specimen until it fractured, and data for load and COD was collected.

Having the collected data, a plot of Load versus COD was created using Microsoft Excel’s trendline function; results are depicted in Figure 1 on the analysis of results section. At this point, a visual inspection allowed defining the linear portion of the experimental trend.

51

Furthermore, the initial slope (mo) of the resulting trend was calculated using equation 1 in the analysis of results section, and a new set of Load data was calculated. This new data set represented a linear trend between Load and COD having 95% of the previously calculated mo.

Values for the new Load data set were hence obtained through equation 2, and a plot was created depicting a range equivalent to that of the linear portion on the experimental trend. Finally; as shown on Figure 2, the experimental elastic portion and the equivalent 95% slope range were plotted on the same graph in order to analyze their discrepancies.

The next step was to calculate the provisional stress intensity factor corresponding to the load at which the slope deviates from the linear portion. For this, equations 3 through 5 were used and results were compared to expected KIc values in order to assess if these were equal or why they might have not been equal.

Finally the last step was to conduct a validation test in order to determine is the determination of the KIc was valid. Because complete data was not available and the maximal load (Pmax) was not known, only one validation test was done as explained by equation 6 and considering a load of 3200lbf. Validation test results are tabulated in Table 3.

Results:

Table 1: Dimensions of Specimens

Table 2: Critical values

52

Dimension Measurement

Replicate 1 2 3 AvgWidth (in.) 0.995 - - 0.995

Thickness (in.) 0.506 0.500 0.500 0.502Crack length, a (in.) 0.1630 0.1685 0.1615 0.1643

Property Value

α (in./in.) 0.165f(α) 1.064

Load, PQ (lbf) 3200Stress Intensity Factor, KIc (Ksi√¿ .) 27.365

Table 3: Validation Test Results


The initial slope of the Load vs. COD curve was obtained by studying two points within the linear portion of the trendline and plugging into equation 1 below.

m0 = (Pfinal – Pinitial) / (CODfinal – CODinitial) (1)

Where mo is the initial slope, P is the Load at each respective point and COD is the crack opening displacement.

Furthermore, the linear load data set was obtained through equation 2 below, where P.95 is the corresponding new Load value at each original COD data point.

P.95 = (0.95* mo) * COD (2)

The Stress Intensity Factor KQ was calculated using equation 3

KQ = f(α) * [(PQ*S) / (B*W1.5)] (3)

Where PQ is the load in the force-COD curve, S is the span length, B is the specimen’s thickness, W is the specimen’s width and f(α) is a function of α defined as:

f(α) = {(3*√α)*[1.99 – α*(1 - α)*(2.15 - 3.93α + 2.7α2)]} / [2*(1 + 2α)*(1 – α)1.5] (4)

Where α is the ratio of crack length (a) over specimen width as shown in equation 5 below.

α = a / W (5)

The test was deemed valid as long as equation 6 below proved to be true

2.5* (KQ / σys) 2 ≤ W - a (6)

Where σys represents the yield strength of the specimen studied.

53

Property Value

σYS (Ksi) 68W – a (in.) 0.8307

2.5* (KQ / σys) 2 0.4049

As explained in the experimental procedure section; plotting respective results yielded Figures 1 and 2 accordingly.

Figure 1: Experimental Load vs. COD plot

Figure 2: Elastic portion of Experimental and 95% slope plots

54


Careful analysis of the results was done in order to determine whether the provisional KQ was in fact the specimen’s Fracture Toughness KIc or not. Handbook values estimate KIc for aluminum alloys to be around 24 Ksi√¿ .; however, test results yielded a value of 27.365 Ksi√¿ . which can be regarded as significantly larger.

However; the validation test led us to believe that the test was in fact valid. It was therefore determined that the difference between the experimentally obtained value and the handbook value was not too large, and that the results were approximately representative of the originally hypothesized values. Nonetheless, further analysis had to be performed in order to determine the source of the discrepancy.

After careful consideration of the testing conditions on which the experiment took place, several facts were categorized as possible sources of error. Firstly, the validity of this method depends strongly on having a sharp-crack condition at the tip of the fatigue crack; therefore the pre-cracking of the specimen must be conducted at low stress intensity, and the size of the specimen must be adequate with respect to the specimen’s toughness to yield strength ratio. Minimal errors in the pre-cracking procedure may have lead to detrimental crack geometries which might have thrown off the results. Furthermore; from calculation mistakes to accumulated approximation inappropriateness, it is clear that computation and human errors also led to inaccuracy.

Another critical possible source of error which could invalidate the test would be the use of an inappropriate PQ value; that is, with respect to the maximal load the specimen was able to withstand. Maximal load data was not available and therefore a validation test could not be conducted, but if it were to be that maximal load to PQ ratio was greater than 1.10, then the test would be invalid and results should be disregarded.


It was concluded that results were valid, but may have been thrown off by any or all of the aforementioned causes. Specially; if the maximal load to PQ ratio relation was violated, the test should be deemed invalid.

More accurate results could be obtained if the pre-cracking procedure was done at lower stress intensity or perhaps the test was done on a more adequate (bigger) specimen size.

55

Appendix A

11) Three Point Bend Test Setup:

References:

17) Fatigue Toughness of Aluminum Alloy. Penn State University, Emch 403, Lab Procedure. 2012.

56

18) Mechanical Behavior of Materials, 3rd Edition, N.E. Dowling, 200719) ASM Metals Reference Book, Third edition, 1993.

Analysis of the Yield Surface

For Aluminum

57


March 23, 2012

Having the collected data for the 16 yield points; an algebraic analysis was done in order to relate the data points to an equation of elliptical form that could later be used to calculate the overall yield point “σ0”. The first step was to define the elliptical equation for the yield surface in a Normal Stress vs. Shear Stress plane; the resulting relation was determined to be as shown in equation 1 below, where σ is normal stress and τ is shear stress:

(σ/a)2 + (τ/b)2 = 1 (1)

At this point, variable transformations of X = σ2 and Y = τ2 in combination with algebraic manipulation, yielded equation 2 below where a and b are model parameters:

Y = - (b2/a2) * X + b2 (2)

The data for σ and τ was transformed into X and Y respectively and graphed to obtain the graph depicted in Figure 1.

Figure 1. Normal vs. Shear Stress in X-Y form

The regression trendline function was related to equation 2, and it was determined that “a” equals σ0.The resulting value of a was calculated to be 220.8481Mpa and this value was then plugged into Mises and Tresca’s yield models as depicted in equations 3 and 4 respectively.

σ02 = σ2 + 3*( τ2) (3)

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0 10000 20000 30000 40000 50000 600000

2000400060008000

100001200014000160001800020000

f(x) = − 0.335236108688439 x + 16349.2337609429

Normal vs. Shear Strain in X-Y form

Normal Stress in X-form (Mpa^2)

Shea

r Str

ain

in Y

-form

(Mpa

^2)

σ02 = σ2 + 4*( τ2) (4)

Plugging experimental data of σ and the calculated value for σ0 into equations 3 and 4 allowed solving for Mises and Tresca’s prediction of τ at each corresponding data point. With care to apply the correct sign to τ depending on which quadrant it was; the results for experimental, Mises and Tresca’s prediction of yield surface where plotted as depicted in Figure 2.

Figure 2. Yield Surfaces

Finally, an analysis was done to determine whether the 50µm strain consideration was in fact representative of yielding and to further define which criteria better fits the experimental data.

Using the 0.2% offset yield technique, it was determined that the non-linear strain was approximate to the plastic strain, but not a very good estimate. This is because, with the purpose of saving costs and avoiding variation; all the loading phases of the experiment were done on a single sample. Meaning that, in order to avoid changes in the material’s mechanical state from one test to the other, very small plastic deformation was allowed on each yield probe. Such small offset yield strength definition (in the magnitude of 50 µm) may have led to underestimation of the yield point.

Finally, it was concluded that Mises criteria yields a better fit to the experimental data, this could be attributed to the fact that Tresca’s criteria involves an aspect ratio of 4 instead of 3 for

59

-300 -200 -100 0 100 200 300

-150

-100

-50

0

50

100

150Yield Surfaces on Stress vs. Shear Stress

Experimental Yield SurfaceMises Yield Sur-faceTresca Yield Sur-face

Normal Stress (σ in Mpa)

Shea

r Str

ess (

τ in

Mpa

)

Mises, which implies that Tresca gives a more conservative approach for calculating yield. Therefore, Tresca’s model will usually underestimate the experimental data as in this case.

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