Mechanical Properties - Stresses & Strains

Types of Deformation : Elasic Plastic Anelastic Elastic deformation is defined as instantaneous recoverable deformation

Hooke's law : For tensile loading, σ = E ε where σ is stress defined as the load per unit area : σ = P/Ao, N/m2, Pa

and strain is given by the change in length per unit length ε = ∆llo , %

The proportionality constant E is the Young's modulus or modulus of Elasticity :

E ~ 10x106 psi [68.9 GPa] for metals [varying from 10x106 psi for Al, 30x106 for Fe

and 59x106 for W].

Poisson's Ratio [ν] : ratio of lateral contraction to longitudinal elongation ν = - εx / εz = - εy / εz [for isotropic materials]; in general, ν ~ 0.3

Thus the total contractile strains is less than the expansion along the tensile axis thereby resulting in a slight increase in the volume of the material under stress - this is known as Elastic Dilation .

Modulus Of Rigidity or Shear Modulus [G] : G = τ / γ ; G is the shear modulus and is related to E andν , G = E / 2(1+ν ).

Bulk Modulus [κ] : the change in volume to the original volume is proportional to the hydrostatic pressure [σhyd] : ∆V/V = β σhyd , where β is the compressibility .

The inverse of the compressibility is the bulk modulus [κ] : κ = 1/β .

κ is also related to E and ν : κ = E / 3(1-2ν ) = 2G(1+ν ) / 3(1-2ν ). • • • Thus one can evaluate the various elastic moduli from one or more experimentally evaluated constants. Note that the elastic moduli are related to the interatomic bonding and thus decrease [slightly] with increasing temperature. Any change in the crystal structure, for example following a phase change [polymorphism], one notes a distinct change in the elastic moduli

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Stress - Strain Curve

Definitions

Nominal (engineering) vs True

S = P

Ao , e =

∆llo σ =

PA , ε = ln (

llo )

Proportional limit (PL) Yield strength (Sy) 0.2% offset; (SLY)

Tensile strength (TS or UTS or SUTS) Fracture strength (SF) Uniform elongation (eu)

Total elongation (ductility) (et or ef in 2” )

Necking strain (en = et - eu)

Reduction in area (ductility) (RA) Volume increases (Elastic Dilation)

σ = S (1+e) & ε = ln (1+e) true Yield stress (σy) 0.2% offset

true Tensile strength (TS or UTS or σUTS)

true Fracture strength (σF)

true Uniform strain (εu)

true Total elongation (ductility) (εt or ef in 2” )

true Necking strain (εn = εt - εu)

Volume is conserved Aolo = Al

Energy to fracture

Elastic (σ = E ε ) Plastic (σ = K εn )

Resilience (Uel = σ2

2E ) units (J/m3) Toughness (J = K εn+1

n+1 )

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σ - ε curves smooth (SSs & fcc) with yield point (steels & bcc)

Rate Effects

Plastic deformation is rate dependent (generally at high temperatures) : σ ≡ f( ε& ) = A ε& m, m = SRS = )

lnd(

ε&lnd σ

T,ε

m ~ 0 at low temperatures m|max = 1

m↑ et↑ vs n↑ eu↑

Group Work :

left of the instructor : (1) Derive relation between σ and S :

right of the instructor : (2) Derive relation between ε and e :

all (3) Show that εu= n

KL Murty

from Ashby and Jones

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⇒ ⇒

0dedS n

TSu =

⇔ε=

1nKU

1npl

+ε

=+

σ = Kεn where n is the work-hardening parameter and K is the strength coefficient.

n = 0 perfect plastic solid and n = 1 perfect elastic solid

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Hardness Def. Resistance to surface penetration

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Charpy Impact Test

Fracture Surfaces

T, C

Effect of C-content on CV for steel

(note decreased energy and increased DBTT

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