Determining Young Modulus for

Various Materials Using a Computer Simulation

Introduction

When given the option to select to

do a laboratory report on whatever I wished, I was drawn to the investigation

of Young Modulus of materials. As an individual particularly interested in

engineering fields, I am very interested in the mechanical properties of solid

materials and how this shapes design in all facets of engineering. After

participating in a job shadowing opportunity in civil engineering, I was drawn

in particular to the mechanical properties of concrete and industrial steel.

Concrete has the ability to resist a tremendous amount of compressive force but

has very low tensile strength whereas steel has a large elastic range under

stress. Therefore, concrete that has been reinforced with rebar is one of the

best composite construction materials in the world. Having discussed Young

Modulus to great depth in my IB Design Technology course, I was curious as to

how the differing slopes of each material at various points on the stress

strain curve could result in the determination of single values for the

ultimate tensile strength and yield strength of each material. This curiosity

prompted me to select the focus of my Internal Assessment to be determining

Young Modulus for various materials using a computer simulation so that I could

compare these found values with the generally accepted values for the selected

materials as seen in textbooks, in order to give insight on the use of

composite materials in engineering and construction.

Background

Youngs

Modulus is a measure of stiffness of an elastic material and is used to

describe the elastic properties of objects when stretched or compressed. When a

load is applied to a solid material, the material is deformed because of the

weight of its load. Materials often undergo deformation due to dead loads and

live loads, both of which must be considered in construction and often in

manufacturing as well. Dead loads are static, permanent forces that materials

undergo. When considering reinforced concrete, as an application of this

investigation, the weight of a concrete building and its permanent fixtures are

dead loads, while consideration of live loads—the forces exerted on a material

with varying magnitude and intensity—take into account the weight of people in

a building and the tensile strength required of the reinforced concrete to bear

wind, seismic shifts, and weather conditions such as snow and ice that

contribute to greater loads on the structure. Youngs Modulus measures the

material’s ability to return to its original shape when these loads are

removed, resulting in the stress-strain curve, and represents the constant, k, in

Hooke’s law, which relates the stress and the strain, though law is only applicable

under the assumption of an elastic and linear response. For most materials, the

stress-strain curve only remains linear under a limited amount of deformation. Hooke’s

law states that the force F needed to extend or compress a spring by some

distance x has a linear relationship with that distance. F=kx, where k is a

constant: the stiffness of the spring, and the value of x is minor compared to

the total possible deformation of the spring. Real materials

eventually fail when placed under a very large force, so this model doesn’t

hold true for large distances.

Hooke’s

law can be expressed in terms of stress and strain where stress is the force on

a unit area of a material as a result of the externally applied force, while

strain is the relative deformation this causes.

Extension Produced ? Load Attached

Stress ? Strain ? k

Research Question

The

primary aim of this investigation is to model Young Modulus for a variety of

materials to compare to established values of said materials using a computer

simulation of Searle’s apparatus.

I

chose to use a computer simulation because this experiment involves specialized

equipment that is not readily available. Furthermore, using a computer

simulation provides better accuracy and precision because it provides a closed

system that lacks confounding variables such as pressure and temperature.

Design and Methodology

The set-up of Searle’s Method is

illustrated in figures 2 and 3 utilizing spherometers.

The

apparatus utilized in Searle’s Method consists of two wires of the same

material, length and diameter, each of which have one end of the wire tightened

in torsion screws above to a fixed support, and the other attached below: the

first to a weighted control mass and the latter to the masses applied to vary

the stress of the load on the wire of material. Thus, the first wire (a) becomes

the auxiliary wire while the second (b) becomes the experimental wire. Each

wire is then loaded equally with 1 kg. The pitch and the least count of the

spherometer are then determined according to the following:

Pitch, P =

Least Count, LC =

The

central screw is then adjusted so that the bubble in the spirit level on the

apparatus is exactly at the center, indicating that the weights are aligned

horizontally to one another. The scale reading of the spherometer is then

calibrated to display zero weight being hung from the experimental wire.

Weights are then added to the experimental wire in half-kg increments, causing

the air bubble in the level to shift away from the center. Between the addition

of each weight, the spherometer screw is adjusted so that the bubble returns to

the initial position and corresponding spherometer readings for each weight

observed. The same procedure of noting the spherometer reading and adjusting

the level is then repeated for unloading the weights in the same half-kg

increments. From these spherometer readings, the extension of the wire material

can be determined and Young’s modulus found.

Simulation Selection

In selecting a simulation for this

investigation, I was searching for a program that would resemble the Searle’s

Method of determination of conducted in the real-world. I required the ability

to hold variables constant, to measure the extension of the wire with

precision, and to exchange the material making up the wire to calculate Youngs

Modulus for multiple metals. This lead me to select a simulation from Amrita

University which had fairly precise measuring instruments and the ability to

keep multiple components of the simulation constant, greatly reducing error

that would cause inconsistencies in my results.

Deriving Equations

If

a wire with length L and radius r has a load of mass M under the acceleration

due to gravity g and the extension produced is denoted by l, then,

Normal Stress = ,

where Force F= mg and Cross-Sectional Area A= , then

Normal Stress = , and

Longitudinal Strain = , then

Longitudinal Strain = .

Thus, as

Young Modulus .

Therefore,

Error = , and

% Error =

The

program I used for this investigation provided me with the ability to keep

variables L, r, and g constant and in the simulation, I set these variables as

follows for all trials:

·

Wire

length, L= 5.0m

·

Wire

radius, r= .0001m

·

Acceleration

due to gravity, g= 9.8m/

Data

Observation

Load on Hanger, M (kg)

Spherometer Screw

Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y (GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

2

0.5

4.83

4.83

4.83

4.41

176.9285

3

1.0

8.24

8.24

8.24

7.82

199.5537

4

1.5

12.65

12.65

12.65

12.23

191.3953

5

2.0

17.06

17.06

17.06

16.64

187.5612

6

2.5

20.47

20.47

20.47

20.05

194.5773

7

3.0

24.88

24.88

24.88

24.46

191.3953

8

3.5

29.29

29.29

29.29

28.87

189.1854

9

4.0

32.71

32.71

32.29

193.3118

18.35

190.4886

Given

Youngs Modulus for Material A: 1.90 x Pa = 190 GPa

= 190.4886

GPa

Slope

= = 8.1443

= 191.6075727 GPa

By

calculation, the Youngs Modulus of the material of the given wire is 190.4886 GPa.

From

the graph, Youngs Modulus of the material of the given wire is 191.6075727 GPa.

Observation

Load on Hanger, M (kg)

Spherometer Screw

Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y

(GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

2

0.5

11.55

11.55

11.55

11.13

70.10375

3

1.0

22.68

22.68

22.68

22.26

70.10375

4

1.5

34.81

34.81

34.81

34.39

68.06526

5

2.0

45.94

45.94

45.94

45.52

68.56369

6

2.5

57.07

57.07

57.07

56.65

68.86626

7

3.0

68.20

68.20

68.20

67.78

69.06947

8

3.5

79.34

79.34

79.34

78.92

69.20658

9

4.0

90.78

90.78

90.36

69.07966

50.88

69.1323

Given

Youngs Modulus for Material B: 6.90 x Pa = 69.0 GPa

= 69.1323

GPa

Slope

= = 22.609

= = 69.02161 GPa

By

calculation, the Youngs Modulus of the material of the given wire is 69.1323 GPa.

From

the graph, Youngs Modulus of the material of the given wire is 69.02161 GPa.

Observation

Load on Hanger, M (kg)

Spherometer Screw

Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y

(GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

2

0.5

7.09

7.09

7.09

6.67

116.9797

3

1.0

13.75

13.75

13.75

13.33

117.0675

4

1.5

20.42

20.42

20.42

20.00

117.0382

5

2.0

27.09

27.09

27.09

26.67

117.0236

6

2.5

33.75

33.75

33.75

33.33

117.0499

7

3.0

40.42

40.42

40.42

40.00

117.0382

8

3.5

47.09

47.09

47.09

46.67

117.0299

9

4.0

53.76

53.76

53.34

117.0236

30.00

117.0313

Given

Youngs Modulus for Material C: 1.17 x Pa = 117 GPa

=

117.0313 GPa

Slope

= = 13.334

= = 117.03236 GPa

By

calculation, the Youngs Modulus of the material of the given wire is 117.0313 GPa.

From

the graph, Youngs Modulus of the material of the given wire is 117.03236 GPa.

To

better demonstrate the accuracy of attributing a linear model—such as that

Youngs Modulus provides—to the relationship between mass of load and extension

of wire material, I calculated and graphed the residuals of the

experimentally-found l and the l? predicted by the least squared

regression line (LSRL) for each of the three materials selected in the

simulation.

Mass (kg)

Extension (mm)

0.5

4.41

4.072

0.338

11.13

11.305

-0.175

6.67

6.667

0.003

1.0

7.82

8.144

-0.324

22.26

22.609

-0.349

13.33

13.334

-0.004

1.5

12.23

12.216

0.014

34.39

33.913

0.477

20.00

20.001

-0.001

2.0

16.64

16.289

0.351

45.52

45.218

0.302

26.67

26.668

0.002

2.5

20.05

20.361

-0.311

56.65

56.523

0.127

33.33

33.335

-0.005

3.0

24.46

24.433

0.027

67.78

67.827

-0.047

40.00

40.002

-0.002

3.5

28.87

28.505

0.365

78.92

79.132

-0.212

46.67

46.669

0.001

4.0

32.29

32.577

-0.287

90.36

90.436

-0.076

53.34

53.336

0.004

0.9991

0.9999

1.0000

Evaluation

The

above demonstrates that LSRLs calculated using Youngs Modulus are good models

for the data because the range of the residuals is minimal in comparison to the

data, the residual plot is scattered, and the value is very

close to 1.00 for all materials within the investigation, meaning that nearly

100% of the variance in wire extension can be attributed to the LSRL of wire

extension and mass of load for each material studied.

According

to my results, the Youngs Modulus for each of the three materials studied was

found to be within ±00.9% of the given Youngs Modulus value for the material,

indicating a confirmation of this model’s validity.

While

within the range of mass loads studied, there is an extremely strong positive

correlation between mass of load and extension of wire with a slope equating to

the Youngs Modulus of the material studied, it is important to note that to

assume this same correlation for all loads of mass M would be to extrapolate

upon the results of this investigation. As noted in the “Background” section of

this investigation, materials fail when placed under large forces, so this

model doesn’t hold true when extension l

is very large due to large Mg.

Limitations

Although

this investigation was conducted with the use of a computer simulation, the

accuracy and precision of the results found were in fact limited chiefly by the

precision of the spherometer utilized to calculate extension, which was accurate

to the nearest millimeter. Errors may have arisen from the limited ability to

measure extension because of this, resulting in the presence of residuals when

comparing experimental and predicted values for the extension of the wire.

However, as a simulation, many of the errors that could have resulted in the

experimental setting were not present. Had this investigation taken the form of

an experiment rather than simulation, instrumental errors could have resulted

from the linear scale zero having not been calibrated to exactly coincide with

the circular scale zero on the apparatus. Additionally, wire outside of a

simulated setting may not be uniform throughout in composition or diameter,

resulting in error when radius of the wire is measured and used to calculate

cross-sectional area. Therefore, error within this investigation was minimized

to a great degree through the utilization of computer simulation.

Conclusion

As

this investigation was performed by a computer simulation, its purpose proved

to primarily act as an exercise in understanding one component of material

indices that allows for the calculation of material performance. This allows

for engineers, manufacturers, those in the construction industry, etc. to

enhance the selection of materials for applications in analysis of component

functions. In calculating Young Modulus using Searle’s Method, I was able to

determine the value of the materials selected quite accurately.

When

considering ways in which I could improve or extend this investigation, I would

like to be able to calculate Youngs Modulus in multiple ways. There are various

methods of finding Young Modulus, including Searle’s Method, the Tabletop

Method, and the Beam Deflection Method. The multitude of options prompted me to

wonder which of these methods of determining Young Modulus proves most accurate

when compared to the generally accepted value, such as that found in a IB

Physics textbook. Additionally, I would like to physically perform this

experiment. The computer simulation utilized for this investigation was

programmed to achieve specific results, and so the accuracy of the values found

for Youngs Modulus is incredibly high. A physical experiment, however, while

perhaps less accurate, would better reflect the conditions of the real world,

outside of a closed system, where other factors must be taken into

consideration in the realm of design and engineering such as temperature and

pressure, which could lead to further investigation of the benefits of cold

rolling, annealing, and other metal-working techniques.