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## Explanation & Answer

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Objective

The aim of this experiment was to determine the strength and inelastic properties of a

material and to observe the deformation and fracture of material under load.

Introduction and Procedure

This experiment uses a button head round bar specimen of A36 Steel alloy and a Tensile

test machine. Before experiment two reference marks called gage points are created on the

specimen. Initial gage length as well as diameter is measured. As load is applied to the specimen,

changes in gage length is noted to calculate elongation for load. This data is obtained till point of

failure. Schematic diagram of the tensile test setup is shown below:

Data is analyzed to find stress, strain, True stress, true strain and many more important

characteristic values. Stress is defined as:

Where A0 is the initial cross-sectional area. Strain is defined as:

It can be noted that cross sectional areas and gage lengths are continuously changing as applied

load value increases, and thus stress and strain calculated by above formula needs a correction

factor to be as close to actual value as it can. True stress and True strain values are used to

correct the situation. True Stress and True strain are also calculated by using these equations:

Graphs for the data are plotted and various other important values like modulus of elasticity,

ductility. Upper yield strength, lower yield strength, ultimate tensile strength and so on are

calculated for the specimen.

Calculations

1) Results for first few load values

Extensometer

Load Cell Load

Displacement

Engineering

Engineering

True Stress

(Lbs)

(inch)

Stress (psi)

Strain

(psi)

True Strain

15

0.00002

75.02419

1.09637E-05

75.02501

1.09636E-05

337

0.00016

1680.481

8.15624E-05

1680.618

8.15591E-05

547

0.00024

2722.49

0.00011756

2722.81

0.000117553

1016

0.00043

5061.403

0.000212348

5062.477

0.000212325

1699

0.00056

8460.238

0.000278996

8462.598

0.000278957

1654

0.00060

8236.97

0.000298415

8239.428

0.000298371

1774

0.00066

8836.513

0.000327088

8839.403

0.000327034

1926

0.00069

9593.212

0.000341449

9596.487

0.000341391

2017

0.00071

10046.36

0.000352124

10049.9

0.000352062

2098

0.00075

10451.11

0.000372945

10455.01

0.000372875

2196

0.00081

10939.88

0.000401352

10944.27

0.000401271

2) Sample calculations

(P) Max load: 13750 lbs

(L) Gage length:

L0 = 2.0086 in

Lf = 2.749 in

(D) Neck Diameter:

D0 = 0.5056 in

Df = 0.3133 in

(Lf – L0) Elongation: 2.749 in – 2.0086 in = 0.7407 in

(A) Cross-sectional area: A0 = (π/4) D0² = (π/4)(0.5056 in)² = 0.2007724 in²

Af = (π/4) Df² = (π/4)(0.3133 in)² = 0.077092241 in²

(σ) Engineering stress: P/A0 = 13750/0.2007724 = 68485.5 psi

(ε) Engineering strain: (Elongation)/L0 = 0.7407 in / 2.0086 in = 0.3687643

(σT) True Stress: σ(1+ ε) = 68485.5 psi (1+0.3687643) = 93740.51 psi

(εT) True strain: ln(1 + ε) = ln(1 + 0.3687643) = 0.31390836

3) Graphs (from excel data)

Stress Vs Strain till Strain is 0.3%

60000

50000

Stress

40000

y = 1E+07x + 957.87

30000

20000

10000

0

0.00000 0.00050 0.00100 0.00150 0.00200 0.00250 0.00300 0.00350 0.00400 0.00450 0.00500

Strain

Stress Vs Strain

100000

90000

80000

70000

Stress

60000

50000

Engineering Stress Vs Strain

40000

True Stress Vs Strain

30000

20000

10000

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Strain

True Stress Vs True Strain in plastic region

100000

90000

80000

y = 98172x0.1165

70000

Stress

60000

50000

40000

30000

20000

10000

0

0

0.05

0.1

0.15

0.2

Strain

4) Material properties obtained from above graphs:

a) Modulus of elasticity = 10^7 psi

0.25

0.3

0.35

b) Upper yield strength 53693 psi, lower yield strength 50468 psi, yield strength for an offset of

0.2% = 50801 psi

c) Tensile strength (ultimate) = 71348 psi

d) Ductility: Percentage elongation in two inches at fracture = 36.88%, Percentage reduction in

area at fracture = 61.60%

e) Modulus of resilience = 0.5*σy*εy = 76.38241 psi

f) Modulus of toughness = (2/3)*Su*εf = 21476.00236 psi

5) Strain hardening coefficient = m = 0.1165 and σ0 = 98172 psi

Questions and Discussion

1) Necking begins after maximum load has been achieved. The formula written in lab notes

uses the fact that volume remains consistent as necking does not occur.

2) Elastic strain is the strain in a material which is not in plastic region and material will

fully recover as soon as load is removed. Plastic strain is permanent in nature and will

not recover even if the load is removed. When a material’s 0.2% offset yield strength is

σ0 , it’s modulus of elasticity is E and the stress reaching σ0 is equal to σ0 /E, the plastic

strain is equal to 0.002. The total strain is equal to ε(elastic) + ε(plastic) and total strain

is also equal to σ0 /E + 0.002.

3) A36 Steel specimen behaved like a ductile material, which was expected given its

properties. Percentage elongation in the specimen came out to be 36.88% which is

significant ductility. Deformation was clearly observed along with necking. Cup-and cone type fracture occured. Cast iron, however, behaved very differently. Significant

deformation was not observed and fracture was sudden without necking. Ductility could

not be calculated, leading to the result that it behaved like brittle material.

4) True strain can be calculated as follows:

εT = ln(L/L0) = ln(A0/A) = ln[([(π/4 D0 ²)/(π/4 D²)] = ln(D0²/D²)

Conclusion

In conclusion, the experiment was a success. Data was collected for A36 Steel alloy

specimen using the Tensile testing machine. Values for ultimate tensile stress, fracture stress,

percent elongation, percent reduction in area, elastic modulus, modulus of toughness and so on

were calculated. The difference of fracture of brittle and ductile material was observed.

Understanding these differences and properties will allow an Engineer to choose the best

possible alternative for many applications.

stress_strain

NO Load readings

Load Cell Extensometer

0.007568

-0.002069

load slope

ext slope

8662062 243.3301072

Load Cell Extensometer

Voltage

0.007577

0.007780

0.007912

0.008208

0.008638

0.008610

0.008686

0.008781

0.008839

0.008890

0.008952

0.009014

0.009069

0.009134

0.009199

0.009255

0.009321

0.009388

0.009445

0.009513

0.009580

0.009640

0.009708

0.009776

0.009835

0.009904

0.009976

0.010038

0.010111

0.010185

0.010249

0.010328

0.010408

0.010476

0.010558

0.010644

0.010717

0.010806

0.010896

0.010974

Voltage

-0.002068

-0.002065

-0.002064

-0.002059

-0.002056

-0.002056

-0.002054

-0.002054

-0.002053

-0.002052

-0.002051

-0.002051

-0.002050

-0.002049

-0.002048

-0.002047

-0.002047

-0.002045

-0.002045

-0.002044

-0.002043

-0.002042

-0.002042

-0.002041

-0.002039

-0.002039

-0.002038

-0.002037

-0.002036

-0.002035

-0.002034

-0.002033

-0.002032

-0.002032

-0.002030

-0.002028

-0.002028

-0.002026

-0.002025

-0.002024

Initial

Excitation

5.458496

A36 Steel

Cast Iron

Final

A36 Steel

Cast Iron

Excitation

Voltage

5.458505

5.458452

5.458485

5.458534

5.458512

5.458516

5.458541

5.458507

5.458508

5.458464

5.458483

5.458456

5.458539

5.458539

5.458477

5.458488

5.458476

5.458537

5.458556

5.458559

5.458507

5.458559

5.458561

5.458558

5.458559

5.458476

5.458532

5.458487

5.458498

5.458548

5.458516

5.458539

5.458486

5.458480

5.458500

5.458537

5.458555

5.458545

5.458550

5.458512

Load Cell Load

(Lbs)

15

337

547

1016

1699

1654

1774

1926

2017

2098

2196

2295

2382

2486

2589

2678

2782

2888

2979

3088

3194

3289

3397

3505

3598

3708

3822

3920

4036

4153

4256

4380

4507

4616

4746

4881

4998

5138

5282

5406

...