ELASTIC CONSTANTS IN ELASTIC CONSTANTS IN ISOTROPIC MATERIALSISOTROPIC MATERIALS

1. Elasticity Modulus 1. Elasticity Modulus (E)(E)

2. Poisson’s Ratio 2. Poisson’s Ratio (())

3. Shear Modulus 3. Shear Modulus (G)(G)

4. Bulk Modulus 4. Bulk Modulus (K)(K)

1. Modulus of Elasticity, E(Young’s Modulus)

= E

Linear- elastic

E

F

Fsimple tension test

Units:E: [GPa]

Slope of stress strain plot (which is Slope of stress strain plot (which is proportional to the elastic modulus) proportional to the elastic modulus) depends on bond strength of metaldepends on bond strength of metal

Adapted from Fig. 6.7, Callister 7e.

E=

2. 2. Poisson's ratio, Poisson's ratio,

Units:: dimensionless

T

L

F

Fsimple tension test

“” is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force.

Either transverse strain or longitudional strain is negative, ν is positive

T L

T : Transverse Strain

L : Longitudional Strain

Virtually all common materials Virtually all common materials undergo a transverse contraction undergo a transverse contraction when stretched in one direction and a when stretched in one direction and a transverse expansion when transverse expansion when compressed. compressed.

IIn an isotropic material the allowable n an isotropic material the allowable (theoretical) (theoretical) range of Poisson's ratio range of Poisson's ratio is from -1.0 to +0.5, based is from -1.0 to +0.5, based on on the the theory of elasticitytheory of elasticity.. metals: ~ 0.33ceramics: ~ 0.25polymers: ~ 0.40

G

= G

3. Shear Modulus, G3. Shear Modulus, G

simpletorsiontest

M

M

Units:G: [GPa]

4. Bulk Modulus, K4. Bulk Modulus, K

avg = KV

Vo avg

V

KVo

P

P

P

Initial Volume = V0

Volume Change = V

Units:K: [GPa]

σavg is the average of three stresses applied along three principal directions.

= E

= G

avg = KV

Vo

Stresses

Strains

Elastic Constants

Normal

Shear

Volumetric

Example: Example: Uniaxial Loading of a Prismatic Uniaxial Loading of a Prismatic SpecimenSpecimen

AfterBefore

10 cm

10 cm

10 cm

10.4 cm

9.9 cm

9.9 cm

Determine E and

P=1000 kgf

10cm

10cm

Δl/2=0.2cm

Δd/2=0.05cm

1000 kgf

P=1000 kgfP=1000kgf → σ= 10*1

0

1000

= 10kgf/cm2

E=

σε =

100.0

4

= 250 kgf/cm2

εlong

=

Δll0

= =0.04

0.410

εlat=Δdd0

= = -0.01-0.110

ν = --0.010.04

= 0.25

For an isotropic material the stress-For an isotropic material the stress-strain relations are as follows:strain relations are as follows:

RELATION B/W K & ERELATION B/W K & E Consider a cube with a unit volumeConsider a cube with a unit volume

σ

1

1

1

σ

D

C

BA

σ causes an elongation in the direction CD and contraction in the directions AB & BC.

The new dimensions of the cube is :

• CD direction is 1+ε

• BC direction is 1-νε

• AB direction is 1-νε

VV00 = 1 = 1

Final volume Final volume Vf of the cube is now: of the cube is now:

(1+(1+εε)) (1-(1-ννεε)) (1-(1-ννεε)) == (1+(1+εε)) (1-2(1-2ννεε++μμ22εε22))

= 1 - 2= 1 - 2ννεε ++ μμ22εε22 ++ εε-2-2ννεε22 ++ μμ22εε33

= 1 + = 1 + εε -- 22ννεε -- 22ννεε22 ++ μμ22εε22 ++ μμ22εε33

ε is small, ε2 & ε3 are smaller and can be neglected. Vf = 1+ ε - 2νε → ΔV = Vf - V0 = ε (1-2ν)

If equal tensile stresses are applied to each of the other two pairs of faces of the cube than the total change in volume will be :

ΔV = 3ε (1-2ν)

σ

σσ

σ

σσ

Ξ + +

K =(σ+σ+σ)/3

σ=

3ε (1-2ν)

=3ε (1-2ν)

E

3 (1-2ν)

K =E

3 (1-2ν)

ΔV = 3ε (1-2ν) =ε (1-2ν) ε (1-2ν) ε (1-2ν)+ +

=avg

V/V0

Moreover the relation Moreover the relation between G and E is :between G and E is :

G =

E

2 (1+ν)

The relation between The relation between G, E and K is :G, E and K is :

E1 1= 1+

9K 3G

K =E

3 (1-2ν)

The relation between K and E is :The relation between K and E is :

Therefore, out of the four elastic constants only two of them are independent.

For very soft materials such as pastes, gels, For very soft materials such as pastes, gels, putties, K is very largeputties, K is very large

Note that as K Note that as K → ∞ → → ∞ → νν →→ 0.5 & E ≈ 3G 0.5 & E ≈ 3G

If K is very large → If K is very large → ΔΔV/VV/V00 ≈ 0 ≈ 0 **No volume changeNo volume change

For materials like metals, fibers & certain For materials like metals, fibers & certain plastics K must be considered.plastics K must be considered.

Modulus of Elasticity :Modulus of Elasticity :

• High in covalent compounds such as High in covalent compounds such as diamonddiamond

• Lower in metallic and ionic crystalsLower in metallic and ionic crystals• Lowest in molecular amorphous solids Lowest in molecular amorphous solids

such as plastics and rubber.such as plastics and rubber.

Elastic Constants of Some Elastic Constants of Some MaterialsMaterials

E(psi)x10E(psi)x106 6

(GPa)(GPa)G(psi)x10G(psi)x1066 (GPa)(GPa) νν (-)(-)

Cast IronCast Iron 16 16 110110 7.4 7.4 5050 0.170.17

SteelSteel 30 30 205205 11.8 11.8 8080 0.260.26

AluminuAluminumm

10 10 7070 3.6 3.6 2525 0.330.33

ConcreteConcrete 1.5-5.5 1.5-5.5 10-10-4040

0.62-2.30 0.62-2.30 4-4-1515 0.20.2

WoodWood Long 1.81 Long 1.81 1212

Tang 0.10 Tang 0.10 0.70.7

0.11 0.11 0.70.7

0.03 0.03 0.20.2

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