Torsion Lab

By

Charisse Lewis

0729543

ME 354 AC

Date of this Lab Exercise: February 11, 2010

Date of Lab Report Submission: February 25, 2010

1

Executive Summary:

Torsion testing is one of the basic mechanical tests that are conducted to obtain material

properties, such as the elastic modulus, strength, shear modulus, and shear strength. During this

torsion lab the Technovate model 9041 Torsion Tester was used to conduct the testing. The

materials used in this lab were 6061-T6 Aluminum, and A36 Steel. The setup for the lab took a

lot of attention to detail, making sure that the specimen was tightened into the grip just right, and

time, if the specimen was not clamped properly or failed inside the grip the test would have to be

redone.

The main purposes of the lab were to observe how the specimens of two materials that were used

reacted under torsion, and to verify the accuracy of the Power-Hardening and Ramberg-Osgood

applied torque predicting models. While observing the specimens of Aluminum and Steel data

was collected as well, so as to compare the two samples numerically to each other and to the

torque predictor models.

When the data for aluminum and steel are compared, the aluminum sample is more ductile than

steel is, which is shown in Figure C.1 in the results. This is evident because the aluminum

sample underwent more angle of twist per unit length than the steel specimen did.

The results show that the Power-Hardening and Ramber-Osgood torque predicting models work

well for Aluminum, which was the only material that had data applied to the models. The

maximum percent error, shown in Table C.3 in the results section, % Error, for the Power-

Hardening model was 0.706 % and for the Ramberg-Osgood it was 1.40 %. These two models

had the % Error maximums at opposite ends of the data set. The Power-Hardening model had

the greatest % Error when θ/L was a maximum, and the Ramberg-Osgood had the greatest %

Error when θ/L was a minimum.

2

A. Introduction

Torsion testing is widely used for evaluating the elastic modulus, strength, shear modulus, shear

strength and other properties of materials. The main difference between torsion testing and

tensile testing, as discussed in previous labs, is that the stress is not uniform over the cross

section of the test specimen. The only useful exception to that is a round thin-walled tube, when

the wall is sufficiently thin, but this lab did not use thin walled specimens, so that is that will be

said about that.

In this lab the apparatus that was used was the Technovate model 9041 Torsion Tester

(Appendix A) with the specimen being held in by grips. The force that was applied was

measured at the force gage via a lever arm. A threaded horizontal drive rod was used to adjust

the angle of twist in small increments, and when it was necessary to adjust the angle in large

increments there was a lever arm, that could adjust the angle up to 90 degrees. The specimens

that were used for this lab were constant-diameter gage section torsion specimens of 6061-T6

Aluminum, and A36 Steel. The diameters of the specimens were approximately 4.77 mm (about

3/16 inch).

There were many purposes for doing this lab, the most important ones being to examine the

responses of metals, like the steel and aluminum that were used, to applied torsion, and to verify

the accuracy Power-Hardening and Ramberg-Osgood models in predicting the applied torque.

Also, this lab was conducted so that students learned about torsion testing, the apparatus that is

used, and the information that can be obtained from the tests.

At the beginning of the experiment a straight line had to be drawn on the specimen. This was

done so that as the lab was progressing the students were able to tell if the set of data that was

being collected was a good set or not. If it was a good set, then the line that was drawn on the

specimen at the beginning of the lab would wrap around the sample in a uniformly spaced

progression. The less uniform the spacing the worse the data set was. This is known to hold true

because a radial line which is straight before loading remains a straight radial line after loading.

3

B. Procedure

The first thing that was done in the lab was the measuring of the diameter of the specimen gage

section using calipers, and to record that value for later calculations. After that, it was necessary

to draw a straight longitudinal line on the specimen so that the angle of twist of the specimen can

be observed during the test.

When installing the specimen in the machine it must first be installed in the bottom grip, and the

specimen has to be very tightly camped in the grip to ensure that it will not slip. Next, the pin

behind the lower grip that engages the ratchet mechanism must be located and make sure the pin

is pulled forward (this ensures that the ratchet is engaged). Then, the horizontal drive rod must

be unthreaded until the end of the threaded region has reached the threaded nut, just before the

threaded rod is disconnected from the base. Then the lever arm must be rotated counter-

clockwise, moved to the right, as far as possible.

At the top of the apparatus identify the wire ropes that transfer the torque to the top grip and

specimen, and loosen the two nuts as far as possible. Rotate the top grip clockwise, so as to

remove the ‘slack’ in the wire ropes, and while holding the top grip in this position as firmly as

possible, tighten the grip. Make sure that the specimen is very tightly fastened in the top grip.

Remove any remaining slack in the wire ropes by retightening the two nuts that were loosened

before, but monitoring the force sensor while that is being done. If the nuts are completely

tightened and slack still remains, tighten the threaded horizontal drive rod until an increase in

load is sensed.

After the specimen is tightly clamped into the machine, measure and record the distance between

the grips, but make sure to account for how far the bottom face of the upper jaw is recessed into

the upper plate in the measurement. Adjust the pointer so that it indicates “zero” degrees, and

zero the output of the force sensor.

All of the prior steps were the setup for the lab, and with those complete the data collection for

the lab may begin. Start applying known angles of twist to the lower grip, beginning with 2o

4

using the threaded drive rod, and record the force displayed on the force gage. After that

increase the angle of twist by 2o until the total angle of twist is 30o, recording the force shown on

the force gage at the end of each increase. Next, increase the angle of twist by 5o. Continue to

take measurements at this increment until the ratchet in the lower grip “clicks”. This usually

occurs around 45o, but it can occur at a total angle of twist larger than that, up to about 60o, just

keep on increasing the angle of twist until the “click” occurs. If the “click” doesn’t occur until

well after 60o, it may be that the specimen was not correctly installed, and therefore may not give

a good data set, so it might be advisable to start over.

After hearing the “click”, record four measurements at increased angles of twist, in increments of

45o. Record four more measurements, this time at increments of 90o, and four more

measurements at increments of 180o. Finally, record measurements for increased angles of twist

in increments of 360o, until specimen failure occurs.

After failure remove the broken halves of the specimen. Carefully examine the specimen and the

fracture surface, record what is necessary to discuss later if it is required.

C. Results

The data that was collected during the torsion testing lab was force (kg), and angle of twist.

Comparison of the measured values for torque (Appendix B) versus the angle of twist divided by

the length, θ/L, is shown in Figure C.1 below:

5

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7Measured Torque vs. θ/L

A36 Steel

6061-T6 Aluminum

θ/L (rad/m)

Mea

sure

d To

rque

(N-M

)

Figure C.1: Measured Torque vs. θ/L for the A36 Steel and 6061-T6 Aluminum specimens that were tested.

On the next page, in Figure C.2, is an example of what one of the A36 Steel specimens looked

like after fracture. This example was representative of most of the steel specimens, so it is used

for determining how brittle or ductile the material was just based on the appearance of the

fracture.

6

Figure C.2: Steel specimen at the point of failure, after fracture.

The values for force converted to Newtons, and the θ/L for aluminum is shown in Tables C.1

and C.2. The other data that is shown in Table C.1 is for predicting torque according to the

Power-Hardening model. The data that is shown in Table C.2 other than the θ/L is the calculated

values necessary for the Ramberg-Osgood model for predicting applied torque.

Table C.1: The Power-Hardening model (Appendix D) for predicting torque with values that are necessary for calculation

Angle (deg) Force (N) Angle (rad)θL

(rad/m)

Measure Torque (N-m)

Predicted Torque(P-H) (N-m)

0 0 0 0 0 02 10.3 0.0349 0.137 1.35 0.1794 10.5 0.0698 0.275 1.38 0.3576 11.3 0.105 0.412 1.48 0.5368 12.0 0.140 0.550 1.57 0.714

10 13.0 0.175 0.687 1.70 0.89312 14.0 0.209 0.825 1.83 1.0714 14.5 0.244 0.962 1.90 1.25

7

16 16.8 0.279 1.10 2.20 1.4318 16.9 0.314 1.24 2.22 1.6120 17.4 0.349 1.37 2.28 1.7922 18.1 0.384 1.51 2.38 1.9624 19.6 0.419 1.65 2.57 2.1426 20.4 0.454 1.79 2.67 2.3228 21.6 0.489 1.92 2.83 2.5030 22.4 0.524 2.06 2.93 2.6835 24.5 0.611 2.40 3.21 3.1240 26.5 0.698 2.75 3.47 3.5745 28.4 0.785 3.09 3.73 3.9390 35.6 1.57 6.18 4.66 4.81

135 37.5 2.36 9.28 4.92 5.01180 40.2 3.14 12.4 5.27 5.12225 40.7 3.93 15.5 5.34 5.19315 41.7 5.50 21.6 5.46 5.31405 41.7 7.07 27.8 5.46 5.40495 41.7 8.64 34.0 5.46 5.47585 41.7 10.2 40.2 5.46 5.52765 42.7 13.4 52.6 5.59 5.62945 42.9 16.5 64.9 5.62 5.69

1125 44.4 19.6 77.3 5.82 5.761305 44.4 22.8 89.7 5.82 5.811665 45.2 29.1 114 5.93 5.902025 46.2 35.3 139 6.06 5.982385 46.6 41.6 164 6.11 6.042745 47.1 47.9 189 6.17 6.093105 47.1 54.2 213 6.17 6.143465 47.2 60.5 238 6.18 6.183825 47.6 66.8 263 6.24 6.224185 47.6 73.0 288 6.24 6.264545 47.9 79.3 312 6.27 6.294905 48.2 85.6 337 6.31 6.325265 48.3 91.9 362 6.33 6.355625 48.8 98.2 387 6.40 6.385985 49.1 104 411 6.43 6.406345 49.1 111 436 6.44 6.426705 49.1 117 461 6.44 6.457065 49.1 123 485 6.44 6.477425 49.1 130 510 6.44 6.497785 49.1 136 535 6.44 6.518145 49.5 142 560 6.49 6.538505 49.3 148 584 6.47 6.548865 50.0 155 609 6.56 6.569225 50.0 161 634 6.56 6.58

Table C. 2: The Ramberg-Osgood model (Appendix E) for predicting torque with values that are necessary for calculation

8

~τ c (MPa) ~γ pc ~γec ~γc θ /L (rad/m)Predicted Torque (R-O) (N-m)

0 0 0 0 0 050 3.33E-14 0.00196 0.00196 0.820 1.07

100 4.64E-08 0.00391 0.00391 1.64 2.13150 0.000182 0.00587 0.00605 2.54 3.27200 0.0646 0.00782 0.0724 30.3 5.57250 6.13 0.00978 6.14 2580 6.99270 29.5 0.0106 29.5 12400 7.55280 62.0 0.0110 62.0 26000 7.83290 127 0.0113 127 53200 8.11300 253 0.0117 253 106000 8.39305 355 0.0119 355 149000 8.53310 495 0.0121 495 207000 8.67315 686 0.0123 686 287000 8.81320 945 0.0125 945 396000 8.95325 1280 0.0127 1300 544000 9.09330 1770 0.0129 1770 743000 9.23335 2410 0.0131 2410 1010000 9.37340 3260 0.0133 3260 1370000 9.51345 4390 0.0135 4390 1840000 9.65350 5890 0.0137 5890 2470000 9.78355 7860 0.0139 7860 3300000 9.92

The comparison of the two predictive models, Power-Hardening and Ramberg-Osgood, to the

measured values of applied torque is in Figure C.3. This was done so that the students could

visually see how the predicted models compared to the measured values, instead of just seeing

data values.

9

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

Comparison of Measured Torque to Predictive Models for 6061-T6 Aluminum

Ramberg OsgoodPower Hardening Measured Torque

Torque (N-m)

(θ/L

) ( ra

d/m

)

Figure C.3: The comparison of measured torque to the predictive models for the 6061-T6 Aluminum specimen.

Data values are still an excellent way of representing the differences between the measured

values and the predicted values. This is done in Table C.3 below, which shows the values for the

measured torque at four different values of θ/L, the corresponding predicted torque for the

Power-Hardening and Ramberg-Osgood, and the % Error corresponding to differences from the

measured values.

Table C.3: The percent error of the predictive models, Power-Hardening (P-H) and Ramberg-Osgood (R-O), to the measured values for torque

θ/L (rad/m)Measured

Torque (N-m)

Predicted Torque (P-H)

(N-m) % Error

Predicted Torque (R-O)

(N-m) % Error100 5.86 5.85 0.222 % 5.95 1.40 %250 6.21 6.20 0.111 % 6.23 0.307 %400 6.41 6.39 0.391 % 6.38 0.596 %550 6.47 6.52 0.706 % 6.45 0.403 %

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D. Discussion

In this experiment the main goals were to observe the metals specimens to see how they act

when under torsion, and to verify the accuracy of the Power-Hardening and Ramberg-Osgood

predictive models for determining torque.

Observing the data collected for each of the metals in a graph of measured torque vs. θ/L, Figure

C.1, it can be seen that the Aluminum specimen is much more ductile than the steel specimen.

This is shown by the fact that the aluminum withstood a greater amount of angle of twist

compared to its length, and the steel endured a greater amount of torque prior to fracture.

Although, the steel was more brittle than the aluminum, the steel did behave as mostly ductile.

This is shown in figure C.2 in the Results section, where the steel specimen exhibited mostly

ductile behavior at fracture, which is shown by the flat fracture surface, and the small amount of

tearing at the edge. The aluminum sample looked very similar to that of the steel, with the

biggest difference being around the area of fracture where on the steel specimen it was course

like sand paper along the sides that were originally smooth (Figure C.2 in the Results). Both

samples initially had the straight line that was drawn on them before starting the testing, and by

the end of the lab it formed a spiral around the outside of the sample. The more uniform the

spacing that the spiral had, the better the data for the specimen would be.

As stated above the other main goal was to verify the accuracy of the Power-Hardening and

Ramberg-Osgood models for predicting torque. This can be shown by Figure C.3 in the results

section, which is a graph with the measured torque and the predictive models all on the same

graph. The predictive models for torque appear to be very close to the measured values, and they

are as shown in Table C.3, the percent error, % Error, of the predictive models as compared to

the measured values for torque. The values for the Power-Hardening had an increasing error

percentage as the values for θ/L increased, which means that this model is better for predicting

the torque being applied for smaller values of θ/L. With the Ramberg-Osgood, the opposite

thing happened, where the % Error was greatest in the beginning and then it became smaller as

the value for θ/L increased. Even though the % Errors varied from beginning to end, the values

for the % Error is still very small, with a max error for the Power-Hardening model being .706 %

and for the Ramberg-Osgood it was 1.40 %. Since the max percent errors are so small, it is a

11

good assumption that both models are a good at predicting the applied torque, at least for a

ductile material.

The best way to test how applicable the Power-Hardening and Ramberg-Osgood models are for

predicting torque would be to test them with a lager variety of materials, and run more than one

test for each material. The materials used in this lab were both mostly ductile, and this showed

in the lab results, but this only gives results for ductile materials for the behavior of the applied

torque predictive models. If there was a more brittle material used in the lab alongside the

ductile materials it could be seen whether or not the predictive models apply. The minimum that

could have been done was try to apply the predictive models to steel, which as discussed is more

brittle than the aluminum.

Another problem that many groups experienced was tightening the specimens in the grips. If the

grips were too tight, the material would get scratched, and would tend to fail in that region. On

the other hand if the specimen wasn’t tightened into the grips tightly enough, then the specimen

would slip, and the data set would be poor. The best way to resolve this would be to find a way

to gage how tight the specimen is in the grip, and have a specified grip strength (how tight the

grip is on the specimen) so that students know how much tighter it has to be, or if they have gone

too far.

Also, the machines that were used were very old, and the force gage was in kilograms (kg). The

force gage being in kg, which is a unit of mass and not force, was not terrible because it is not

difficult to convert kg to N, but it was also very touchy, probably due to how old it was. If

anyone hit the table at all the gage usually changed values, whether it increased or decreased the

value was not consistent either. This could also be a source of error between the measured and

predicted values.

Other possible sources of error include; not tightening the specimen into the grip enough,

tightening the specimen into the grip too much (as discussed above), the specimen being bent at

all when installed, and adding angle of twist at too rapid a pace. The first two sources of error

were already discussed above, but if the specimen was bent at all, this would create other forces

12

other than the shear stress and shear strain. The predictive models would not be as accurate

because the initial assumption that was made about there only being shear stresses and forces

would be incorrect; all of the equations would be off and would not account for the additional

forces that are present. Also, if the angle of twist was added to the specimen too rapidly, then the

planes in the crystalline structure of the material would not have as much time to slide past each

other. Therefore the material would fracture sooner than a specimen that had the angle of twist

applied at a slower rate.

13

Appendix A: The Technovate Torsion Test Machine

During this torsion lab the Technovate model 9041 Torsion Tester was used to test specimens

made of A36 Steel, and 6061-T6 Aluminum. In order to know how the lab works, it is necessary

to know the setup that is being used. Therefore, below in Figure A.1 is a picture of the top view

of the torsion tester with the key components labeled.

Figure A.1: To view of the Technovate model 9041 Torsion Tester, with key components labeled.

There is also a schematic of the above picture in Appendix B, which is used for discussing how

to calculate torque from the force reading collected. Below are two photos that display the

bottom half of the testing apparatus with the key components labeled. These pictures are for

reference throughout the report so that it is clear what is being discussed.

14

Top Lever

Wire Rope

Wire Rope

Force Gage

Top Grip

Figure A.2: A view of the bottom half of the testing apparatus, and the key components labeled.

Figure A.3: An alternate view of the bottom half of the testing apparatus, with key components labeled.

15

Driving Rod

Ratchet Pin

Bottom Grip

Bottom Lever Arm

Driving Rod

Bottom Grip

Bottom Lever Arm

Angle Indicator

Appendix B: Converting the Force Reading to Torque

The setup of the torsion test needs to be known in order to be able to calculate the torque being applied to the specimen, and this is shown in Figure B.1 below:

Figure B.1: Schematic top view of the Technovate Torsion Test Machine.

As shown above the grip and specimen are held in place by the wire ropes, one that is attached to

one side of the machine and one that is attached to the lever arm. The lever arm is held in place

at the pivot point and the force gage, both of which cause a force at either end of the beam. The

wire rope is the only other force acting on the lever arm, which is acting on the lever arm slightly

off center. This is shown in the form of a Free Body Diagram (FBD) in Figure B.2, on the next

page.

16

Figure B.2: FBD for the lever arm with the three forces acting on it.

During the torsion experiment the data that was collected was the force at the force gage, and the

force was in kilograms (kg). First the force in kg had to be converted to Newtons (N), which is

done by multiplying the force value, in kg, by 9.81m

s2 and then to calculate the torque on the

specimen it was necessary to calculate the force pulling on the wire rope. This was done using

static equilibrium moment equations centered at the pivot point, pp, shown here:

∑M PP=0=¿ Fwire rope∗.085m−F force gage∗.213m¿ (Equation B.2)

With Equation B.2, Fwire rope is able to be obtained since F force gage is the measured data from the

lab that was converted to N. With the force in the wire rope the torque applied to the specimen

can be calculated by multiplying the force in the wire rope times the radius of the top grip, (the

distance to the application of the torque). This is shown in Equation B.3 below.

T wire rope=Fwire rope∗0.02615m (Equation B.3)

The FBD for the forces acting on and the torque being applied to the top grip are shown in

Figure B.3 below.

17

Applied Force (F force gage)

Fwire ropeReaction Force

Figure B.3: FBD of the grip with the wire ropes as the forces causing torque.

This is just the torque caused by one of the wire ropes, therefore the T specimen has to be doubled in

order to obtain the actual value for the torque applied to the specimen. In other words;

T specimen=2∗T wire rope (Equation B.4)

This is the measured torque in the specimen that is compared to the predicted torque from the Power-Hardening model (Appendix D), and the Ramberg-Osgood model (Appendix E).

18

Torque

Appendix C: True and Effective Stress and Strain

The equation for effective stress is listed below as Equation C.1

σ= 1

√2√(σ x−σ y)

2+(σ y−σ z )+(σ z−σ x )+6 (τ xy2+τ yz

2+τ zx2) (Equation C.1)

Since this lab was interested in finding predicted values for torque beyond yielding, the use of

true stress was used in calculations. Thus the equation for effective stress becomes:

σ= 1

√2√(~σ x−~σ y )2+(~σ y−~σ z )+ (~σ z−~σx )+6 (~τ xy2+~τ yz

2+~τ zx2 ) (Equation C.2)

The effective stress equation can be rewritten to get the effective strain equation, as shown in

Equation C.3.

ε= 1

√2√(~ε x−~ε y )2+(~ε y−~εz )+ (~ε z−~εx )+6 (~γ xy2+~γ yz

2+~γ zx2 ) (Equation C.3)

During the torsion test there was only one stress, ~τ xy, and one strain, ~γ xy, applied. Knowing that

there is only uniaxial stress, and uniaxial shear strain, Equation C.2 and Equation C.3 become:

σ=√3~τ xy (Equation C.4)

ε= 1

√3~γ xy (Equation C.5)

Equation C.4 an Equation C.5 are important for the application of the power-hardening and

Ramberg-Osgood torque predicting models as discussed in Appendix D and Appendix E,

respectively.

19

Appendix D: Power-Hardening Model

Using the data collected from the torsion test that was conducted, a power-hardening model,

(Equation D.1), was fit to the measured force, (assuming true axial stress and true axial strain),

response for 6061-T6 Aluminum.

σ x=H ε xn (Equation D.1)

The coefficients in Equation D.1 are H, the strength coefficient, which is 413 MPa, and n, the

hardening exponent, which is .0633. Also, assuming the use of true stress and strain, as

discussed in Appendix A, and using Equations C.4 and C.5, a new equation for ~τ xy can be

developed that relates it to ~γ xy using the coefficients H and n, as shown in Equation D.2.

~τ xy=H√3 ( ~γ xy

√3 )n

(Equation D.2)

With this model the goal is to predict torque versus (θ/L) response, and for this it is necessary to

find equations relating the two. The total applied torque is equal to the sum of the torque acting

on the elastic inner-region and the torque acting on the plastically-deformed outer-region, shown

below:

T total=T elastic+T plastic (Equation D.3)

The elastic inner-region extends over0<r<r y, and in this region~τ xy=rr y

~τo, where~τ o=σo√3

.

Therefore:

T elastic=∫0

r y

(¿~τ xy)(r ) (2π )dr=π ~τo r y

3

2¿ (Equation D.4)

The plastic outer-region extends overr y<r<c, and in this region Equation D.2 becomes

~τ xy=H√3 ( rθ

√3 L )n

(Equation D.5)

T plastic=∫r y

c

(¿~τ xy) (r ) (2 π )dr= 2 πH√3(n+3) ( θ

√3 L )n

[cn+3−r yn+3 ] ¿ (Equation D.6)

20

Using Equation D.4 and D.6 the torque vs. (θ/L) response can now be predicted based on the

Power-Hardening model. During this lab the Power-Hardening model was only applied to the

Aluminum 061–T6 sample data.

Appendix E: Ramberg Osgood Model

An alternate approach to the Power-Hardening model for fitting data is the Ramberg-Osgood

model, which uses Equation E.1 as a base equation for relating the predicted torque to the (θ/L)

response.

~ε=~σE

+( ~σH )

1n (Equation E.1)

Just like the Power-Hardening model (Appendix C) the coefficients H and n are the strength

coefficient and the strain hardening exponent, respectively. The values for H and n are: H= 407

MPa, and n=0.0490. As discussed in Appendix C, the use of effective stress and effective strain

changes Equation E.1 into:

~γ xy=~τ xyG

+√3(√3~τ xyH )

1n

(Equation E.2)

Where:

G= E2(1+ν) (Equation E.3)

The modulus of elasticity, E, for Aluminum 6061-T6 is 68.5 MPa, and the Poisson’s ration,ν, is

0.34 . Also, the equation becomes less daunting when a “shear” strength coefficient is defined,

H τ , as shown below:

H τ=H

3(n+1 )

2(Equation E.4)

This changes Equation E.2 into:

~γ xy=~τ xyG

+( ~τ xyH τ

)1n

(Equation E.5)

With all of these equations an equation for torque is made as shown below in Equation E.6.

T=2π c3 ~τmax [ 14+

2n+13n+1

β τ+n+2

2n+2β τ

2+1

n+3βτ

3

(1+βτ)3 ] (Equation E.6)

21

Where~τ max=τ c, meaning that the max shear stress occurs where the radius is the largest. Also:

β τ=~γ pc~γ ec

(Equation E.7)

~γ pc=( τ cH τ)

1n

(Equation E.8)

~γec=τ cG

(Equation E.9)

~γc=~γec+~γ pc (Equation E.10)

Equations E.6 through Equation E.10 are used to predict the torque vs. (θ/L) response based on the Ramberg-Osgood model, with (θ/L) being equal to:

θL=γcc

(Equation E.11)

Just like the Power-Hardening model the Ramberg-Osgood model was only fit to the Aluminum 6061-T6 data set.

22