Is E a material property?

“Is Young’s Modulus (E) a material property?”

Thomas Young 1773-1829

,P EA

,M yy EI

Hooks Experiment

• “Is Young’s Modulus (E) a material property?”Material E (kg/mm2)

Aluminum Bronze 12237Aluminum 7036

Brass 12746Bronze 12237

Carbon Fiber Reinforced Plastic 15296

Concrete 1733Copper 11931Glass 92692Gold 7546

Grey Cast Iron 13256Iron 21414

Nickel 17335

Steel, Structural ASTM-A36 20394

Titanium Alloy 12237

• P = applied load (measured in test)• = strain (measured in test)• A = Area of the un-deformed cross section

• • σ and are to be in elastic region to determine the

Young’s modulus

( )P stress a purely calculated valueA

' modulus

E

Young s

The Deformed and Un-deformed shapes and sizes of a uniform rectangular bar under uniform axial loading

• We want to find the answer to this question through testing• Young’s modulus is the slope of stress-strain curve in the linear

elastic range as shown schematically in fig.1. Young’s modulus can also be expressed as:

1

1

E

1

1

E

Fig.1. Typical stress-strain curve for a metallic material

“Is Young’s Modulus (E) a material property?”

If this curve is a material property it must be independent of the type of testing, for eg: • Tension•Compression•Bending• Others

• The value of E for a given material is usually determined by conducting a uni-axial tension/compression test on a specimen using a UTM (Universal Testing Machine)

• A schematic of a specimen (bar) under uni-axial tension is shown in fig-2 here

• Such a test specimen mounted in an UTM is shown in the photograph-1

• A strain gage is bonded on the specimen for the purpose of measuring the strain

• In order to generate the stress-strain data required to plot the stress-strain curve, we apply the load in steps (say in an increment of 100 kg) and at each step hold the load constant and

1. Measure the strain (record the strain-gage reading)2. Calculate stress by using the formula P

• We continue this σ, recording right up to the failure of the specimen

• The measured strain and calculated stress (for the same external load) when plotted in a σ- graph may appear as shown schematically in fig.3

Fig.3: Experimental data points

• Then we fit a straight line to the initial data points which appear to show a linear trend and obtain the slope of this linear fit and declare it as the Young’s modulus value of E for that material

Straight line fit

• Observe that in order to determine the value of E we measured the strain but calculated the value of stress by holding the load constant

• The external load acting uni-axially on the bar (of given material) has created strain and stress in two entirely independent ways

• We calculate stress using the formula and therefore this calculation is independent of material

• This formula for stress does not recognize the material of the bar

P

• For example if we take a bar of aluminum and a bar of steel and both have the same cross sectional area ‘A’ we will calculate the same value of stress, for any given load ‘P’, for both the materials

P

• However, the measured strains, for the same load level, will be different in these two materials

• Only after experimentally establishing the value of E for a given material we can say that the normal stress and normal strain in the direction of normal external load are related through E that is σ = E

Straight line fit

• We have to first establish the value of E before we can use it elsewhere

Straight line fit

• We have described a hypothetical tensile test to determine the value of E

Straight line fit

• Can this value of E relate the bending stress to bending strain in a beam bending test?

• What is normally done in an engineering stress analysis problem under elastic conditions is to use the relation σ = E for any other structure of the same material in order to calculate strain if we know the normal stress value or Vice-versa

• It does not matter whether the normal stresses in the various members have been created due to axial load or bending

• This is our current understanding

• In short, we say that we can calculate stress from strain or strain from stress in the linear elastic range of loading by the relation σ = E

• Note that this relation can be used only after determining the value of E by conducting a test where strain is measured and stress is calculated (totally independent of magnitude of strain)

• We tend to forget the test through which the value of E has been established while calculating stress from strain in our engineering stress analysis problem

Because• We are given the impression that E is a material property

• As an example of the use of Young’s modulus in engineering stress analysis consider a beam bending problem

• Let us define the problem as • “Calculate the bending moment for a strain measured at the

section shown in the fig.4 of a cantilever beam of aluminum and of rectangular cross section subjected to a tip load and compare it with the bending moment due to the tip load”

Strain gage

X

int

int

Young's modulus under axial loadingIs ?

a Ext

a

ext

IE M M PxC

EM M

• Let us imagine that for a certain load = P we have measured the strain to be 1500 microstrain ( =1500 x 10-6)

Strain gageint

int


a ext

a

ext

IE M M PxC

EM M

• We can straightaway calculate stress value by using the E value of aluminum

• σ = E and σ = 7200 x 1500 x 10-6 kg/mm2

• Since the strain is 1500 microstrain, we are well within the linear elastic range of the aluminum alloy material

• Therefore, using the calculated value of σ we can calculate the internal bending moment as:

int( )CM IC

• Therefore, using the calculated value of σ we can calculate the internal bending moment as:

int( )CM IC

int( )M C IC

• From the known cross sectional dimensions (b x 2C) as shown above, we can calculate I and C and using the above calculated value of σ we can calculate the value of Mint

Exploded view of cross section

σ(C) = E(C) and σ(C) = 7200 x 1500 x 10-6 kg/mm2

int( )CM IC

int( )M C IC

31 (2 )12

I b C

Using the value of σ(C), I & C, we can calculate the value of Mint using the following formula

• From the test we know the magnitude of the tip load which caused that strain of 1500 microstrain

• Therefore, the external moment = P.X and this value we can calculate

• Now we have the numerical value of external bending moment Mext and we have the numerical value of internal bending moment which we have calculated from the measured strain value using the Young’s modulus value of the material obtained under axial loading

• Mext = P.X, , σ(C) = 7200 x 1500 x10-6

• Here we have used the relation σ(C) = E(C) where we have used E = 7200 and (C) = 1500 microstrain

int( )M C IC

• We will find that our calculated Mint is not equal to the external moment Mext

int

int


a ext

a

ext

E M M PxIC

EM M

• This violets the fundamental requirement of moment equilibrium in the beam bending problem

int extM M

• We realize that this violates the requirement of moment equilibrium. This implies that our calculation of Mint is in error

• In this expression in the right hand side (C) is the measured strain, I & C are geometrical quantities

• We have made no error in (C), C and I. Then the source of error can be only in the value of E that we have used

int( ) ( )M C E CI IC C

• We have used the value of E in axial loading to relate the bending stress to bending strain

• It appears that we have made an error in using this value of E

• We can now look at the question “IS YOUNG’S MODULUS A MATERIAL PROPERTY?”• The above discussion indicates that E is not a

material property• We will now try to find the answer to the

question purely through an experimental investigation

• By determining the value of E through a uni-axial tensile test and a four point beam bending test

• Let us consider two different types of tests to establish the young’s modulus of the material. One test is under a uniform axial loading and the other test is under constant bending moment as shown schematically in figure below

• The formula to calculate the axial stress from the Pext is

• Similarly the formula to calculate bending stress from Mext is

extPA

( ) extb

MC C

I

• This bending stress formula is thought to be valid only in the linear elastic range. (This is not true. But we will not talk about this aspect here)

• We can carry these two types of tests in the universal testing machine

• We hold the load constant, measure the strain and calculate the stress for a number of load values in the elastic range

• We can now plot the stress-strain data under axial loading and determine Ea which we denote as Young’s modulus under axial loading

• We can actually use the same specimen and carryout a 4-point bend test which creates a constant bending moment at the strain gage location

• At various value of Mext we measure the bending strain and calculate the bending stress using the formula

• Plot the σb vs b experimental data points and obtain Eb which we denote as Young’s modulus under bending

( )bMC CI

We can now check is Ea equal to Eb?

• These two tests are described in what follows• The specimen which was used in both tests is

shown in the photograph-1

• The dimensions are indicated there along with the values of cross sectional area A, area moment of inertia I and distance ‘C’ from the neutral axis to the outermost fiber (strain gage location)

B = 48.5 mm

2C = 6 mm A = 291 mm2 I = 878.4 mm4

C = 3 mm

• The specimen mounted on the UTM for the axial load test is shown in photograph-2. The bend test set up is shown in photograph-3

• A schematic of the 4-point bend test specimen and loading is shown in figure below

• The constant bending moment at the strain gage location =

4extPLM

Strain gage

These two tests were carried out successfully in the following way:

• Test were conducted to generate strains with increment of 100 microstrains starting from 400 upto 2200 microstrains. The test results are given in table-1

Column-1 Column-2 Column-3 Column-4 Column-5 Column-6 Column-7 Column-8

Micro strains Load (axial)

Load (2P) (bending) Moment (M)

Axial stress σa

Bending stress σb

Axial Young's Ea

Bending Youngs Eb

400 800 41 820 2.75 2.80 6873 7001

Test data under uni-axial loading and constant bending moment

2

20.5 3

40 878.44

20.5 40 8204

820 3 2.8 /878.4

ext

ext

P kg C mmL mm I

PLM kg mm

M C kg mmI

2

2

800 291800 2.75 /291

P kg A mmP kg mmA

=400 ue=400 ue

=400 ue






Axial stress σa

Bending stress σb

Axial Young's Ea

Bending Youngs Eb

400 800 41 820 2.75 2.80 6873 7001


22.8 /b kg mm 22.75 /a kg mm

=400 ue=400 ue

=400 ue

26

26

2.75 6873 /400 10

2.8 7001 /400 10

aa

a

bb

b

E kg mm

E kg mm






Axial stress σa

Bending stress σb

Axial Young's Ea

Bending Youngs Eb

400 800 41 820 2.75 2.80 6873 7001500 994 51 1020 3.42 3.48 6832 6967600 1189 60 1200 4.09 4.10 6810 6831700 1390 70 1400 4.78 4.78 6824 6831800 1585 80 1600 5.45 5.46 6808 6831900 1777 90 1800 6.11 6.15 6785 6831

1000 1976 100 2000 6.79 6.83 6790 68311100 2171 110 2200 7.46 7.51 6782 68311200 2359 120 2400 8.11 8.20 6755 68311300 2553 130 2600 8.77 8.88 6749 68311400 2745 140 2800 9.43 9.56 6738 68311500 2940 150 3000 10.10 10.25 6735 68311600 3136 161 3220 10.78 11.00 6735 68731700 3325 171 3420 11.43 11.68 6721 68711800 3515 181 3620 12.08 12.36 6711 68691900 3705 192 3840 12.73 13.11 6701 69032000 3901 202 4040 13.41 13.80 6703 68992100 4088 213 4260 14.05 14.55 6690 69282200 4280 223 4460 14.71 15.23 6685 6924


Column-1: shows the common axial strain values created under both the loads

Column-1 Micro strains

400500600700800900

1000110012001300140015001600170018001900200021002200

Column-2 : shows the axial load required to achieve the specified strain

Column-1 Column-2 Micro strains Load (axial)

400 800500 994600 1189700 1390800 1585900 1777

1000 19761100 21711200 23591300 25531400 27451500 29401600 31361700 33251800 35151900 37052000 39012100 40882200 4280

Column-3: shows value of load 2P in the four point bend test required for the specified strain

Column-1 Column-2 Column-3 Micro strains Load (axial)

Load (bending)

400 800 41500 994 51600 1189 60700 1390 70800 1585 80900 1777 90

1000 1976 1001100 2171 1101200 2359 1201300 2553 1301400 2745 1401500 2940 1501600 3136 1611700 3325 1711800 3515 1811900 3705 1922000 3901 2022100 4088 2132200 4280 223

Column-4: shows the constant bending moment acting at the strain gage location

Column-1 Column-2 Column-3 Column-4 Micro strains Load (axial)

Load (bending) Moment (M)

400 800 41 820500 994 51 1020600 1189 60 1200700 1390 70 1400800 1585 80 1600900 1777 90 1800

1000 1976 100 20001100 2171 110 22001200 2359 120 24001300 2553 130 26001400 2745 140 28001500 2940 150 30001600 3136 161 32201700 3325 171 34201800 3515 181 36201900 3705 192 38402000 3901 202 40402100 4088 213 42602200 4280 223 4460

2

20.5 3

40 878.44

20.5 40 8204

820 3 2.8 /878.4

ext

ext

P kg C mmL mm I

PLM kg mm

MC kg mm

I

Column-5: shows the calculated axial stress under uni-axial loading

Column-1 Column-2 Column-3 Column-4 Column-5



Axial stress σa

400 800 41 820 2.75500 994 51 1020 3.42600 1189 60 1200 4.09700 1390 70 1400 4.78800 1585 80 1600 5.45900 1777 90 1800 6.11

1000 1976 100 2000 6.791100 2171 110 2200 7.461200 2359 120 2400 8.111300 2553 130 2600 8.771400 2745 140 2800 9.431500 2940 150 3000 10.101600 3136 161 3220 10.781700 3325 171 3420 11.431800 3515 181 3620 12.081900 3705 192 3840 12.732000 3901 202 4040 13.412100 4088 213 4260 14.052200 4280 223 4460 14.71

2

2

800 291800 2.75 /291

P kg A mmP kg mmA

Column-6: shows the calculated bending stress in the four point bend test

Column-1 Column-4 Column-6

Micro strains Moment (M)

Bending stress σb

400 820 2.80500 1020 3.48600 1200 4.10700 1400 4.78800 1600 5.46900 1800 6.15

1000 2000 6.831100 2200 7.511200 2400 8.201300 2600 8.881400 2800 9.561500 3000 10.251600 3220 11.001700 3420 11.681800 3620 12.361900 3840 13.112000 4040 13.802100 4260 14.552200 4460 15.23

2

20.5 3

40 878.44

20.5 40 8204

820 3 2.8 /878.4

ext

ext

P kg C mmL mm I

PLM kg mm

MC kg mm

I

Column-7: shows the calculated value of Ea from uni-axial loading stress-strain data

Column-1 Column-2 Column-5 Column-7


Axial stress σa

Axial Young's Ea

400 800 2.75 6873500 994 3.42 6832600 1189 4.09 6810700 1390 4.78 6824800 1585 5.45 6808900 1777 6.11 6785

1000 1976 6.79 67901100 2171 7.46 67821200 2359 8.11 67551300 2553 8.77 67491400 2745 9.43 67381500 2940 10.10 67351600 3136 10.78 67351700 3325 11.43 67211800 3515 12.08 67111900 3705 12.73 67012000 3901 13.41 67032100 4088 14.05 66902200 4280 14.71 6685

26

2.75 6873 /400 10

aa

a

E kg mm

Column-8: shows the calculated value of Eb from the stress-strain data from the bending test

Column-1 Column-6 Column-8

Micro strains

Bending stress σb

Bending Youngs Eb

400 2.80 7001500 3.48 6967600 4.10 6831700 4.78 6831800 5.46 6831900 6.15 6831

1000 6.83 68311100 7.51 68311200 8.20 68311300 8.88 68311400 9.56 68311500 10.25 68311600 11.00 68731700 11.68 68711800 12.36 68691900 13.11 69032000 13.80 68992100 14.55 69282200 15.23 6924

26

2.8 7001 /400 10

bb

b

E kg mm

• We can observe from the values of Ea in column-7 and values of Eb in column-8 that at each measured strain value, Eb is greater than EaColumn-7 Column-8

Axial Young's Ea

Bending Youngs Eb

6873 70016832 69676810 68316824 68316808 68316785 68316790 68316782 68316755 68316749 68316738 68316735 68316735 68736721 68716711 68696701 69036703 68996690 69286685 6924

• The difference between Ea and Eb is quite small• But they are indeed different




Axial stress σa

Bending stress σb

Axial Young's Ea

Bending Youngs Eb

400 800 41 820 2.75 2.80 6873 7001500 994 51 1020 3.42 3.48 6832 6967600 1189 60 1200 4.09 4.10 6810 6831700 1390 70 1400 4.78 4.78 6824 6831800 1585 80 1600 5.45 5.46 6808 6831900 1777 90 1800 6.11 6.15 6785 6831

1000 1976 100 2000 6.79 6.83 6790 68311100 2171 110 2200 7.46 7.51 6782 68311200 2359 120 2400 8.11 8.20 6755 68311300 2553 130 2600 8.77 8.88 6749 68311400 2745 140 2800 9.43 9.56 6738 68311500 2940 150 3000 10.10 10.25 6735 68311600 3136 161 3220 10.78 11.00 6735 68731700 3325 171 3420 11.43 11.68 6721 68711800 3515 181 3620 12.08 12.36 6711 68691900 3705 192 3840 12.73 13.11 6701 69032000 3901 202 4040 13.41 13.80 6703 68992100 4088 213 4260 14.05 14.55 6690 69282200 4280 223 4460 14.71 15.23 6685 6924

a bE E

• The test data points along with straight line fits are shown in this figure

Conclusions:

• Young’s modulus (E) is not a material property

• For a given material ‘E’ is a loading-action-Property • Under axial loading it has one value and under

bending loading it has a different value (Eb > Ea)

• However, the difference is small but nevertheless consistent

Eb = 6916

Ea = 6643

• However, the differences are small but nevertheless consistent

• Eb will always be greater than Ea

Eb = 6916

Ea = 6643

Axial Young's Ea

Bending Young’s Eb

6873 70016832 69676810 68316824 68316808 68316785 68316790 68316782 68316755 68316749 68316738 68316735 68316735 68736721 68716711 68696701 69036703 68996690 69286685 6924

Conclusions:• All fibers of the bar under axis loading deform

independent of each other• All fibers of the beam under bending load

deform in a coordinated fashion• A rotation requires coordinated motion

Conclusions:• In bending, the

deformation of the outermost fiber [ (C )] is restrained by the smaller deformations of all fibers beneath it. Therefore, it will require a higher stress compared to that under axial loading (compare the values of column-5 and column-6)

Column-5 Column-6Axial stress

σa

Bending stress σb

2.75 2.803.42 3.484.09 4.104.78 4.785.45 5.466.11 6.156.79 6.837.46 7.518.11 8.208.77 8.889.43 9.56

10.10 10.2510.78 11.0011.43 11.6812.08 12.3612.73 13.1113.41 13.8014.05 14.5514.71 15.23

Conclusions:In bending plane section remain plane by rotating about its neutral axisThis leads to linear variation of strain across the neutral axis of the cross section (this is an experimentally verifiable result)This is the result of a coordinated motion of all fibersFibers do not deform independent of each other in bendingUnder axial loading they do

Column-5 Column-6Axial stress

σa

Bending stress σb

2.75 2.803.42 3.484.09 4.104.78 4.785.45 5.466.11 6.156.79 6.837.46 7.518.11 8.208.77 8.889.43 9.56

10.10 10.2510.78 11.0011.43 11.6812.08 12.3612.73 13.1113.41 13.8014.05 14.5514.71 15.23

Food for thought:• In an engineering stress analysis of beam bending

problem the value of Young’s modulus established through axial loading can always be used since difference between Ea and Eb is really very small

• But the answer to the question• Is Young’s modulus a material property?

is…….NOThe value of E has to be established by calculating stress from equilibrium condition and measuring strain in a test (be it axial or bending)

Mext

Strain gage

Mext

Strain gage

C

• The constant bending moment at the strain gage location = 4ext

PLM

2P

P P

P P4L

4L

2L

Strain gage

PP

P P

• The value of E for a given material is usually determined by conducting a uni-axial tension/compression test on a specimen using a UTM (Universal Testing Machine)

• A schematic of a specimen (bar) under uni-axial tension is shown in fig-2 here

Pext

Pext

Strain gage

Fig.2. Typical test specimen to determine Young’s modulus

Column-1: shows the common axial strain values created under both the loads




Axial stress σa

Bending stress σb

Axial Young's Ea

Bending Youngs Eb

400 800 41 820 2.75 2.80 6873 7001500 994 51 1020 3.42 3.48 6832 6967600 1189 60 1200 4.09 4.10 6810 6831700 1390 70 1400 4.78 4.78 6824 6831800 1585 80 1600 5.45 5.46 6808 6831900 1777 90 1800 6.11 6.15 6785 6831

1000 1976 100 2000 6.79 6.83 6790 68311100 2171 110 2200 7.46 7.51 6782 68311200 2359 120 2400 8.11 8.20 6755 68311300 2553 130 2600 8.77 8.88 6749 68311400 2745 140 2800 9.43 9.56 6738 68311500 2940 150 3000 10.10 10.25 6735 68311600 3136 161 3220 10.78 11.00 6735 68731700 3325 171 3420 11.43 11.68 6721 68711800 3515 181 3620 12.08 12.36 6711 68691900 3705 192 3840 12.73 13.11 6701 69032000 3901 202 4040 13.41 13.80 6703 68992100 4088 213 4260 14.05 14.55 6690 69282200 4280 223 4460 14.71 15.23 6685 6924

Is E a material property?

Engineering

Transcript of Is E a material property?