Chap 2

Elasticity • Elastic strain • Hooke’s law – Stress is proportional to strain.

Longitudinal stress and strain dl l = ln 1 lo l lo

True strain

dl dε = l

εt = ∫

Engineering strain

∆l l1 − lo εe = = lo lo

εt = ln (1 + εe )

True stress

F σt = A

Engineering stress

σe =

Hooke’s law

σ E= ε

l1

F Ao E: Young’s modulus

Shear stress and strain τ =FA

γ = dl l = tan θ ≈ θ G= τ

γ

G: shear modulus or rigidity

Poisson’s ratio A body, upon being pulled, tends to contract laterally. Poisson’s ratio, υ = -(lateral strain)/(longitudinal strain) The stress σ33 generates ε11, ε22, ε33. υ = -ε11/ε33 = -ε22/ε33

Generalized Hooke’s law Normal stresses generate only normal strains. Strains produced by σ11: ε11 = σ11/E; ε22 = ε33 = - υσ11/E Shear stresses generate only shear strains. 1 ε 11 = [σ 11 − υ ( σ 22 + σ 33 ) ] E 1 ε 22 = [σ 22 − υ ( σ 11 + σ 33 ) ] E 1 ε 33 = [σ 33 − υ ( σ 22 + σ 11 ) ] E

σ 12 G σ 13 γ 13 = G σ γ 23 = 23 G γ 12 =

Plane stress: in sheets or plates (one dimension can be neglected with respect to the other two.) Plane strain: one of the dimensions is infinite with respect to the other two.

Elastic properties of materials

Mohr circle

Positive shear stresses produce counterclockwise rotation. Negative shear stresses produce clockwise rotation.

Principal stresses and maximum shear stresses Maximum shear stress orientation 45o

Principal stress orientation

Principal stress orientation

Pure shear

γ 2 = AC AO

Pure shear D

O

− ε 11 = AB OD γ 2 = AC AO

Simple shear

Relationship between G and E For pure shear

1 ( σ 1 − υσ 2 ) = σ 1 (1 + υ ) E E τ = −σ 1

ε 11 =

τ = Gγ Gγ (1 + υ ) E γ ε 11 = − 2

ε 11 = −

G is related to E by means of Poisson’s ratio.

G=

E 2(1 + υ )

Anisotropic effects  σ 11 σ 12 σ 13   σ 11 σ 12 σ 13       σ 21 σ 22 σ 23  ≡  σ 12 σ 22 σ 23  σ  σ  σ σ σ σ 31 32 33 13 23 33    

11→1;

22→2;

33→3;

σij = σji; εij = ε ji

23→4; 13→5;

12→6

 σ 11 σ 12 σ 13   σ 1 σ 6 σ 5       σ 21 σ 22 σ 23  ≡  σ 6 σ 2 σ 4  σ     31 σ 32 σ 33   σ 5 σ 4 σ 3   ε 11 ε 12 ε 13   ε 1 ε 6 2 ε 5 2       ε 21 ε 22 ε 23  ≡  ε 6 2 ε 2 ε 4 2  ε     31 ε 32 ε 33   ε 5 2 ε 4 2 ε 3 

ε 4 = 2ε 23 = γ 23 ε 5 = 2ε 13 = γ 13 ε 6 = 2ε 12 = γ 12

Hooke’s law  σ 1   C11     σ 2   C21 σ   C  3  =  31  σ 4   C41 σ   C  5   51 σ   C  6   61 σi = Cij ε j

εi = Sij σj

C12 C22 C32 C42 C52 C62

C13 C14 C23 C24 C33 C34 C43 C44 C53 C54 C63 C64

C15 C25 C35 C45 C55 C65

C16  ε 1    C26  ε 2  C36  ε 3    C46  ε 4  C56  ε 5  C66  ε 6 

C: stiffness S: compliance

orthorrhomb ic

tetragonal

cubic

C11 C12 C  12 C22 C13 C23  0  0  0 0  0  0 C11 C  12 C13   0  0  C16

C13

0

C23 0 C33 0 0 C44 0 0 0 0

0 0 0 0 C55 0

0  0  0   0  0   C66 

C12 C11 C13 0

C13 C13 C33 0

0 0 0 C44

0 0 0 0

0 − C16

0 0

0 0

C44 0

C16  − C16  0   0  0   C66 

C11 C12 C12 0 C  12 C11 C12 0 C12 C12 C11 0  0 0 C44  0  0 0 0 0  0 0 0  0

0 0 0 0 C44 0

0  0  0   0  0   C44 

isotropi c

C11 C12 C12 0 C  12 C11 C12 0 C12 C12 C11 0  0 0 C44  0  0 0 0 0  0 0 0  0

C44 =  S11 S  12  S12  0 0   0

0 0 0 0 C44 0

     0  0   C44  0 0 0

C11 − C12 2 S12 S11 S12

S12 S12 S11

0 0 0

0 0 0

0 0 0

0 0 0

S 44 0 0

0 S 44 0

S 44 = 2( S11 − S12 )

     0  0   S 44  0 0 0

•

The stiffness and compliance matrices are symmetric, and the 36 components are reduced to 21. • The number of independent elastic constants depends on the symmetry of the crystals. • For cubic system, the number is three. C11, C12, C44 •

For isotropic system, C44 = (C11 – C12)/2. – two independent constants – Lame’s constants: µ = C44 = 1/S44 = G λ = C12 – Young’s modulus: E = 1/S11 – Rigidity or shear modulus: G = 1/2(S11 – S12) = 1/S44 – Poisson’s ratio: υ = -S12/S11 – Compressibility (B) and bulk modulus (K): ε 11 + ε 22 + ε 33 1 B= = K 1 (σ + σ + σ ) 11 22 33 3

Relationships between stresses and strains for isotropic materials 1 [σ 1 − υ ( σ 2 + σ 3 ) ] E 1 ε 2 = S12σ 1 + S11σ 2 + S12σ 3 = [σ 2 − υ ( σ 1 + σ 3 ) ] E 1 ε 3 = S12σ 1 + S12σ 2 + S11σ 3 = [σ 3 − υ ( σ 2 + σ 1 ) ] E

ε 1 = S11σ 1 + S12σ 2 + S12σ 3 =

σ 1 = C11ε 1 + C12ε 2 + C12ε 3 = ( 2µ + λ ) ε 1 + λε 2 + λε 3 σ 2 = C12ε 1 + C11ε 2 + C12ε 3 = λε 1 + ( 2µ + λ ) ε 2 + λε 3 σ 3 = C12ε 1 + C12ε 2 + C11ε 3 = λε 1 + λε 2 + ( 2µ + λ ) ε 3

σ4 G σ ε 5 = 2( S11 − S12 )σ 5 = 5 G σ6 ε 6 = 2( S11 − S12 )σ 6 = G ε 4 = 2( S11 − S12 )σ 4 =

1 ( C11 − C12 ) ε 4 = µε 4 2 1 σ 5 = ( C11 − C12 ) ε 5 = µε 5 2 1 σ 6 = ( C11 − C12 ) ε 6 = µε 6 2

σ4 =

Relations among elastic constants for isotropic materials

Orientation dependence of elastic moduli for monocrystals In a cubic material, the elastic moduli can be determined along any direction, from the elastic constants and the direction cosines of the direction [i j k]

(

)(

)

(

)(

)

1 = S11 − 2 S11 − S12 − 1 S 44 li21l 2j 2 + l 2j 2lk23 + li21lk23 2 Eijk

1 = S 44 + 4 S11 − S12 − 1 S 44 li21l 2j 2 + l 2j 2lk23 + li21lk23 2 Gijk

Orientataion dependence of Young’s modulus of single crystal

Elastic properties of polycrystalline metals