true stress (σt true stress is the stress determined by...
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True Stress (σT)True stress is the stress determined by the instantaneous load acting on the
instantaneous cross-sectional areaTrue stress is related to engineering stress:Assuming material volume remains constant
AA
AP
AA
AP
AP o
oo
oT ** ===σ
ll AA oo =
)1(1 ε+=+δ
=+δ
==oo
o
o
o
AA
ll
l
l
l )1()1( ε+σ=ε+=σo
T AP
True Strain (εT)The rate of instantaneous increase in the instantaneous gauge length.
)ln(
lnln
ln
εε
ε
ε
+=
∆+⇒
∆+=
== ∫
1T
oo
o
o
oT
oT
d
l
l
l
l
l
ll
l
l
l
l
Strain Hardening Parameter (n)
Strain hardening parameter
0<n<1T
T
T
T
dd
nεσ
εσ
=
nTT Kεσ =
True Stress-Strain Curve
)ln( ; o
Ti
T ll
AF
== εσ
o
o
o lll
AF −
== εσ ;
Instability in TensionNecking or localized deformation begins at maximum load, where the increase in stress due to decrease in the cross-sectional area of the specimen becomes greater than the increase in the load-carrying ability of the metal due to strain hardening. This conditions of instability leading to localized deformation is defined by the condition δP = 0.
T
T
AA
σδσδ
=−0=+= TT AAP δσδσδAP Tσ=
From the constancy-of-volume relationship,
TAA
ll δεδδ
=−=iioo lAlAV ==
so that at the point of tensile instability
T
TnT
T
TnTT nKnK
εσ
εδεδσ
εσ === −1 TT
T σδεδσ
= But
Instability occurs when ε = n
The necking criterion can be expressed more explicitly if engineering strain is used. ( ) T
O
o
TT
T
LL
LLLL
σεδεδσ
δεδσ
δδ
δεδσ
δεδε
δεδσ
δεδσ
=+=
=== 1
//
εδεδ
+=
1Tσσ
1+ε
σT
ε1
Fracture BehaviorDuctile material – Significant plastic deformation and energy absorption (toughness) before fracture.
Characteristic feature of ductile material - necking
Brittle material – Little plastic deformation or energy absorption before fracture.
Characteristic feature of brittle materials – fracture surface perpendicular to the stress.
Steel
Before and after fracture
Ductile Fracture (Dislocation Mediated): Extensive plastic deformation. Necking, formation of small cavities, enlargement of cavities, formation of cup-and-cone. Typical fibrous structure with “dimples”.
(a) Necking, (b) Cavity Formation, (c) Cavity coalescence to form a
crack, (d) Crack propagation,(e) Fracture
Crack grows 90o to applied stress
45O - maximum shear stress
Scanning Electron Microscopy: Fractographic studies at high resolution. Spherical “dimples” correspond to micro-cavities that initiate crack formation.
Brittle Fracture (Limited Dislocation Mobility): very little deformation, rapid crack propagation. Direction of crack propagation perpendicular to applied load. Crack often propagates by cleavage - breaking of atomic bonds along specific crystallographic planes (cleavage planes).
Brittle fracture in a mild steel
Intergranular fracture: Crack propagation is along grain boundaries (grain boundaries are weakened or embrittledby impurities segregation etc.)
Transgranular fracture: Cracks pass through grains. Fracture surface has faceted texture because of different orientation of cleavage planes in grains.
Stress-Strain Behavior of CeramicsFlexural Strength: the stress at fracture under the bending tests. It’s
also called Modulus of rupture, fracture strength, or the bend strength
3-point Bending tests
3
223
RLFbdLF
ffs
ffs
πσ
σ
=
=
BrittleDuctile