Department of Mechanical and Industrial Engineering

ME 2356: Laboratory for Mechanics of Materials

Fall 2015

Lab #2: Torsion Test

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IntroductionObjectiveThe core objective of this laboratory exercise was to establish the material characteristics of cylindrical bars of aluminum and brass that are subjected to torsional forces, that is, the twisting moments and shear stress and strain that a cylindrical bar experiences under torsion.

Background Test components or materials under torsion tests are twisted to a given extent, with a given force quantity, or up to the failure of the sample in torsion. Under this test, the sample is subjected to the force needed to twist it by anchoring one of its end in order to restrict it from any sort of rota-tion and/or movement while moment is applied to the other end, a procedure that results in speci-men rotating about its axis. In other cases, both ends of the specimen may be subjected to the moment of rotation. In such scenarios, the rotating moments are (always) applied in opposite di-rections. An analogous scenario of this kind of mechanical test is that of a string whose one end is tightly held, say in a hand, whereas the other hand twists its other end.

According to torsional theory, the shearing stress ( ), polar moment of inertia (J), specimen length (L) and radius (r), twisting moment or torque (T), angle of twist ( ), and modulus of rigidity (G) are related as in equation [1] below.

[1]

Consider the bar with a circular cross-section as depicted in figure 1 below:

Figure 1 Cylindrical bar with circular cross-sectionThe rate of twist of the specimen is given as:

[2]At the outer surface, the experienced shear strain ( ) is given as:

[3]When the material is under pure torsion, the rate at which it is twisted is constant and is given by equation [4] below.

[4]

Where is angle of twist in radiansIn addition, an analysis of the Hooke’s law reveal equation [5] below.

[5]That is, shear stress divided by shear strain gives the shear modulus or modulus of rigidity.Thus, the shear stress at the very outer surface of the material can be expressed as:

[6]Another vital interrelationship exist among Poison’s ratio, Shear modulus, G and Young’s mod-ulus, E. This is represented in equation 7 below.

[7]Where is the Poisson’s ratio, E is the Young’s modulus, and G is the shear modulus.

Torsion tests aim at establishing the performance of a given material when subjected to torsional forces, especially the moments that result in shear stress about its very axis. A number of vital as well as critical values determined through this test, and which are crucial in manufacturing in-clude: ultimate shear strength, modulus of elasticity in shear, torsional fatigue life, ductility, modulus of rapture in shear, and yield shear strength. It is worth noting that, inasmuch as these values are similar to those determined through tensile test, they are not the same as they particu-larly focuses on shearing stress or forces. They are vital in that the helping simulating service conditions of a material so as to establish its quality (Czichos, 2006).

A number of materials in various applications experience torsional forces (torque) and thus re-quires to undergo this kind of test to gauge their capabilities. Biomedical, structural, and automo-tive applications are the common areas where materials used experience substantial torque.

The gradient or slope in the linear region of the torque-angle of twist curve gives the modulus of rigidity, Gof a given material.

Methodology Equipment and Apparatus

1. The vernier calipers 2. The steel rule3. The Instron Torsion Machine

Experimental Layout

Figure 2. Torsion testing machine

Figure 3. Cylindrical Specimen Mounted in Torsion Tester (Top View)

Procedure1. The work length as well as diameter of each and every specimen (brass and aluminum)

was measured steel rule and verniar calipers respectively. These parameters were recorded as in table 1 below. Three readings for diameter were taken and only the average recorded.

2. The specimen was then installed in the chucks, ensuring that its work section coincided with the free surface of every chuck.

3. Initial adjustments on the testing machine were then made as follows: the level was ze-roed, followed by zeroing the coarse angular displacement scale, and finally the torque meter.

4. The fine scale was as well zeroed followed by the counter. The zeroing process was done manually.

5. A torque was then applied to the specimen by twisting it in angular steps of 1o (fine scale). The magnitude of the torque as well as the angle readings from the radian meter were noted while torque was applied for every 1o angular step. This was carried out until

the yield point was observed. The readings in all of these steps were recorded as in table 2 below. Afterwards, the angle increments were gradually increased up to the failure point.

6. Another specimen was introduced in the testing machine, its work section made to prop-erly coincide with the free surface of the chuck, and steps 3 through 5 repeated. Results were recorded as in table 2 below.

Experimental ResultsAfter the above outlined procedure was keenly followed, the results in the following tables were obtained. Even though it might be early, it is worth noting that the stated results were not the same as the theoretical ones due to a number of errors that will be later outlined under the section of experimental sources of error.

Table 1. The work length and diameter of the experimental specimens.Specimen Work length, L (mm) Work diameter, r (mm)

Aluminum 77.76 6.13

Brass 77.76 6.23

Table 2. The magnitude of the torque and the angle reading for the specimensAluminum (preload torque = 0.8 Nm) Brass (preload torque = 1.4 Nm)

Angle, ϕ (o) Torque (Nm) Angle, ϕ (o) Torque (Nm)

1 1.6 1 2.8

2 2.4 2 4.4

3 3.2 3 6.3

4 4.1 4 8.0

5 4.9 5 9.7

6 5.8 6 11.0

7 6.7 7 12.2

8 7.5 8 13.2

9 8.2 9 13.9

10 8.8 10 14.4

11 9.2 11 14.8

12 9.6 12 15.2

13 9.8 13 15.5

14 10.0 14 15.7

15 10.2 17 16.2

16 10.3 20 16.5

17 10.4 23 16.7

20 10.7 26 16.9

23 10.8 29 17.2

26 11.0 41 17.2

29 11.0 47 17.5

32 11.1 59 17.5

35 11.1 71 17.8

41 11.1 83 17.9

47 11.2 107 18.0

53 11.3 131 18.2

65 11.4 155 18.3

77 11.5 179 18.5

89 11.6 203 18.6

113 11.8 227 18.6

137 11.9 275 18.9

161 12.0 323 19.0

185 12.1 371 19.1

233 12.2 398 Fracture (breaking point)

281 12.4

319 Fracture (breaking point)

Note: Preload values of torque are those values taken at an angle of 0o

The data in table 2 above was utilized in constructing one graph for each and every sample. These curves are as shown in figures 4 and 5 below.

Figure 4. A graph of torque in Nm versus angle of twist in degrees for aluminum.

Figure 5. A graph of torque in Nm versus angle of twist in degrees for brass.

Observations from the graphsAs clearly depicted in figures 4 and 5 above, it can be seen that:

i. The fully plastic torque of brass is more than that of aluminumii. The torque needed in order for brass specimen to begin undergoing yielding is more

than that needed in the aluminum specimen case.iii. The torque remains fundamentally constant towards the specimens’ fully plastic states.iv. Modulus of rigidity, G of the specimens can be determined by determining the slope

in the linear region of the curves in figures 4 and 5 above.

Discussion Experimental valuesFrom the experimental values and hence the constructed graphs, the values of shear stress, τY and shear strain, γYat the yield point were obtained as below. Correspondingly, the value of modulus of rigidity or shear modulus was determined from these values.Shear stress, shear strain, and shear modulus for aluminum

From equation [1],

Thus

But 1.386 × 10-10m4

Hence, = 203.45

Also, using equation 6 for the maximum shear stress, it can as well be obtained as follows:

Similarly, shear strain for aluminum can be obtained as in equation 3 and 4 as:

From equation 5, modulus of rigidity or shear modulus for aluminum can now be determined as:

Hence G = Shear stress, shear strain, and shear modulus for brass

From equation [1],

But 1.479 × 10-10 m4

Hence, = 278.01

Also, using equation 6 for the maximum shear stress, it can as well be obtained as follows:

Similarly, shear strain for brass can be obtained as in equation 3 and 4 as:

From equation 5, modulus of rigidity or shear modulus for brass can now be determined as:

Hence G =

Comparison with the theoretical values Table 3. Comparison between the experimental and theoretical shear modulus values

Specimen Theoretical Shear Modulus (GPa)

Experimental shear mod-ulus (GPa)

Aluminum 26.0 26.88

Brass 37.0 36.15

Clearly, there is a significant discrepancy between theoretical and experimental values of shear modulus. These deviations are due to a number of experimental error sources that are outlined below under the section of errors.Young’s modulus computationAccording to equation [7], the inter-linkage among Poison’s ratio, shear modulus, and Young’s modulus is represented and hence E can be determined as follows:

For aluminum:E = G × 2(1+ ) = 26.88 × 2(1 + 0.3)

= 69.89 GPaFor brass:E = G × 2(1+ ) = 36.15 × 2(1 + 0.33)

= 96.16 GPaComparison with theoretical values Table 4. Comparison between the experimental and theoretical Young’s modulus values

Specimen Theoretical Young’s Modulus (GPa)

Experimental Young’s modulus (GPa)

Aluminum 70.0 69.89

Brass 100.0 96.16

As well, there is a notable discrepancy between the two values. Some of the experimental errors attribute to this scenario.

The experimental data, as clearly seen from tables 3 and 4, perfectly compares with the nominal one. Also, the limitation of this experiment is almost insignificant as there are minimal caution-ary measures to be considered. Sample limitation is simply getting the specimen properly held in the chucks of the testing machine. As such, it is imperative to assert that these results can easily be reproduced since the limitations are minimal.

Sources of errorAs per table 3 above, it is clear that there exists a significant difference between the values ob-tained during execution of the experiment and the theoretical ones. Various aspects of the experi-ment attributed to this significant discrepancy. Key of them is discussed in the following para-graphs.

Chiefly, the specimen orientation in the chucks of the Instron machine proved challenging. Inap-propriate fixing of the specimen in the chucks of the machine, or any slightest misalignment of the specimen will cause errors to the results. In addition, the machine’s grips would sometimes fail to tightly hold the specimen. As such, there was an occurrence of strain outside the consid-ered length which reflected in the wrong results tabulated.

By the same token, there are a number of assumptions made when calculating theoretical values. This is to say that, inasmuch as the experimental values are always the ones said to be wrong, the manner in which the theoretical values are arrived at ought to be assessed. There is the assump-tion based on a single grain and very few samples for accuracy’s sake. However, this experiment utilized reasonably bigger samples (≈66x6.5x1.5mm), certainly made up of numerous grains as opposed to just fewer grains samples used in obtaining accurate theoretical values. Thus the dis-crepancies in the tabulated results.

In addition, the aging effect of the specimen/samples and the test apparatus is also one of the key causes of errors. As equipment age, they lose their capability of operating appropriately so as to give true results. As for the samples, they will fracture earlier than expected thus give wrong re-sults.

Lastly but not least, there is an issue of approximation error. As it can be seen clearly, the experi-mental values of the shear modulus for the given specimens above have been obtained through approximations in almost every computation step. As values and figures are approximated, there is a significant percentage error that is set up in the final results.

ConclusionThe goal of this experiment was to determine the material characteristics of cylindrical bars of aluminum and brass that are subjected to torsional forces. The Shear modulus, a property com-monly utilized in categorizing mechanical properties of materials is obtained from the slope of the linear regions of the torque-angle of twist curves. The selected samples underwent torsion test carried out using the Instron testing apparatus. Each sample’s data was recorded as in tables

1 and 2 above and utilized in constructing torque-angle of twist curves in figures 4 and 5 above. The Young’s modulus was then determined from the relationship shown in equation 7. As depicted, it can be asserted that the experimental values of both Shear and Young’s modulus of the metals generally differ with their nominal ones. However, the estimated values of the sam-ples differed with the true ones due to the explained reasons under error sources section. None-theless, it can be affirmed that the set objective was achieved as the deviations in the typical and practical values are very minimal.

ReferenceCzichos, H. (2006). Springer Handbook of Materials Measurement Methods. Berlin: Springer.