Eats 2470

Introduction

The purpose of this lab was to verify the Hooke's law sigma = ε * E

Equipments:

Weights, Elastic material, rules/tape measure, digital caliper.

Procedure

The length, the width and the thickness of the objected to be experimented on, in this case a rubber thread loaded with 2kg is measured. Then after a 1.5kg weight is loaded on the object, the measurements are taken again. The process is repeated for 1kg and 0.5 kg weights.

The strain is calculated using the equation

ε11, ε22 and ε33 are also calculated from the observed values. The values observed and the values calculated are noted in the chart below.

No Mass (kg) Length

(inch)

(x3) Width

(mm) (x1) Thickness

(mm) (x2) Normal Stress σ_33=mg/(x_1.x_2 ) (Pa) Extension

ε33 Contraction

- ε11 Contraction

- ε22

1 0.5 21 1/8 18.35 7.08 0.377 ×105 0 0 0

2 1.0 21 6/16 18.32 7.02 0.762 ×105 0.0118 -1.635x 10-3 -8.475x 10-3

3 1.5 21 8/16 18.29 6.98 1.152×105 0.01774 -3.269 x 10-3 -.01412

4 2.0 21 9/16 18.14 6.95 1.555 ×105 0.01942 -11.44 x 10-3 -18.36 10-3

Table 1: The experimental value of the parameter of a rubber sample from trail 0 to 4. Trial 0 represents prior to the load of a weight on the sample.

Calculations:

Length (inch to mm):

21 9/16 = 547mm

21 8/16 = 546.1 mm

21 6/16 = 542.93 mm

21 1/8 = 536.58 mm

ε11 = (x10 - x1) / x10

ε22 = (x20 - x2) / x20

ε33 = (x30 – x3) / x30

If the strains in x and y directions are not equal, and then take the average value between the two.

Average strain at trail 1:

ε=(ε1+ε2)/2= -5.055 x 10-3

Average strain at trail 2: ε = (ε1+ε2)/2= -1.6415x 10-3

Average strain at trail 3:

ε = (ε1+ε2)/2= -8.694x 10-3

Normal Stress vs Extension ε33

Then apply these computed average strains to determine the Young's modulus (E) and Poisson's ratio (v).

Manipulating the equation 1 gives you the formula for Young's Modulus and Poisson's ratio as followings.

E= σ/ε ,v=- ε1/ε3= -ε2/ε3 , where ε1, ε2, ε3 indicate the strain tensors in x, y, z directions, respectively.

Computation of Poisson's ratio

When 1 kg of weight loaded

v= ε11/ε33= -0.1386

When 1.5 kg of weight loaded

v= ε11/ε33= -0.1843

When 2 kg of weight loaded

v= ε11/ε33= -0.589

Calculation for Young's modulus

When 1kg of weight is loaded

E1= σ/ε=