ME 2733 exam 3 lol

engineering stress

how much force is being applied relative to the original size of the material; σ=F/A0​

engineering strain

how much the material stretches or compresses compared to its original length; ϵ=ΔL​/L0

units of stress

Pa (N/m²) or MPa

units of strain

dimensionless

stress formula for tension/compression

σ=F/A

stress formula for shear

τ=F/A

true stress

accounts for the fact that as a material stretches, its area decreases; its more accurate bc it reflects the changing geometry of the material; σt​=F/Ainstant​

true strain

measures deformation incrementally and continuously rather than only comparing start and end states; ϵt​=ln(L/L0​)

convert to true stress

σt​=σ(1+ϵ)

convert to true strain

ϵt​=ln(1+ϵ)

resolved shear stress (slip systems)

stress acting along a slip plane/direction; a portion of an applied force that actually causes atoms to slide along specific planes inside the material; τR​=σcosϕcosλ; (ϕ: angle between normal to plane and force; λ: angle between slip direction and force)

universal testing machine (UTM)

machine measures mechanical properties such as load vs displacement (strain stress curve); applies controlled forces and measures how a material responds to loading, producing a stress strain curve (tells us how a material behaves from initial loading all the way to failure

elastic deformation

reversible; removable load; returns to original shape; applied stress below yield strength

plastic deformation

permanent deformation; doesnt return to original shape; applied stress is above yield strength

engineering curve

uses original area; doesnt keep rising after necking; ignores geometry changes; appears to drop

true curve

uses instantaneous area; keeps rising after necking; accounts for shrinking area; continues increasing

hookes law

a linear relationship between stress and strain in the elastic region; material behaves like a perfect spring (for a bit); σ=Eϵ

modulus of elasticity (E)

slope of elastic region (stiffness); high resists deformation; low easily stretches;

poisson’s ratio

how material contracts sideways when stretched; ν=−ϵlateral​​/ϵaxial​

Relationship between E, G and v

G=E/2(1+ν)

lateral strain

stretching in one direction causes shrinking in another; ϵlat​=−νϵaxial​

yield strength

stress where plastic deformation begins; taken as the lower yield point

yield strain

amount of deformation that occurs at the onset of yielding; transition from elastic to plastic behavior

yield point phenomenon

sudden drop after upper yield point

tensile strength (UTS)

maximum stress before necking and weakening

tensile strain

the amount of elongation a material experiences under tensile loading; usually measured relative to its original length

ductility

ability to plastically deform before fracture; allows materials to deform without sudden failure

percent elongation

%EL=((Lf​−L0)/L0)​​×100

percent reduction in area

%RA=((​A0​−Af​​)/A0)×100

Resilience

energy absorbed elastically

modulus of resilience

Ur​=σy​ϵy​/2

toughness

total energy absorbed before fracture (area under curve); elastic + plastic

what happens when load is removed after plastic deformation

elastic portion recovers; plastic portion remains

hardness

a material’s resistance to localized plastic deformation, such as indentation scratching or suface penetration

Rockwell’s hardness test

measures how deep an indenter penetrates into a material under a specific load, giving a direct hardness number based on depth

Brinell hardness test

presses a hard spherical ball into the material and measures the diameter of the indentation

slip

the process where layers of atoms slide past each other along specific planes and directions, causing plastic deformation

general rule of slip

occurs on the densest plane in the densest direction bc it requires the least energy

slip system

combo of a slip plane and slip direction where dislocation motion occurs; the more of these, the more ductile;

why do materials deform by slip instead of all atoms moving at once?

moving one row at a time requires much less energy than shifting the entire plane simultaneously

FCC slip system

Planes: {111}; Directions: <110>; Total: 12 slip systems; very ductile

BCC slip system

planes: {110}, {211}, {321}; directions: <111>; slip systems: 12, 12, 24

HCP slip system

planes: {0001}, {10-10}, {10-11}; directions: <11-20>; slip systems: 3, 3, 6

resolved shear stress

portion of applied stress that actually causes slip along a specific system

critical resolved shear tress (CRSS)

minimum shear stress required to initiate slip; τR=σcos⁡ϕcos⁡λ

yielding begins

when τR​≥τCRSS​

why do we strengthen metals

to make dislocation motion harder, increasing strength

3 main strengthening mechanisms

solid solution strengthening; grain size reduction; strain hardening

solid solution strengthening

adding impurity atoms creates lattice distortions which block dislocation motion

grain size reduction

grain boundaries act as barriers to dislocation movement

strain hardening

increasing strength by plastic deformation, which increases dislocation density

percent cold work

%CW=((​A0​−Af​​)/A0)×100

how do strenghtening methods affect properties?

yield strenght increases; tensile strength increases; ductility decreases

recovery

internal stress is reduced; dislocations rearrange; no major strength change; low temp

recrystallization

new strain-free grains form; strenght decreases; ductility increases; medium temp

grain growth

grains become longer; strength decreases further; ductility slightly increases; high temp

fracture

failure by complete separation of a material into two or more pieces due to stress

fatigue

failure due to repeated cyclic loading, often occuring at stresses below yield strength

creep

time dependent deformation under constant stress, usually at high temperature

5 steps ductile fracture

1 necking begins; 2 void formation; 3 void growth; 4 void coalescence; 5 final fracture

ductile fracture surface

rough; dimpled; cup and cone shape; significant plastic deformation

3 steps brittle fracture

crack initiation; crack propagation; sudden fracture

brittle fracture surface

flat; smooth; little/no plastic deformation; fast failure; chevron fan shaped

stress concentration

localized increas in stress due to geometric discontinuities (holes, cracks, notches)

how does crack size affect stress

larger crack - higher stress concentration - easier failure

how does crack tip radius affect stress

sharper tip (smaller radius) - higher stress concentration

max stress at crack tip

σmax​=σ(1+2(sqrt(a/ρ​​))); a is crack length; p is tip radius

stress intensity factor (K)

a parameter that describes the stress field near a crack tip; K=Yσsqrt(πa​); Y is a geometry factor; o is applied stress; a is crack length;

crack propogation condition

crack grows at K≥Kc​

fracture toughness (Kc)

material’s resistance to crack propagation

plain strain fracture toughness (K_IC)

a conservative value of fracture toughness under thick, constrained conditions; the higher the Kic, the tougher and more crack resistant

toughness

total energy to fracture

fracture toughness

resistance to crack growth

design parameters

KIC​: material resistance; a: crack size; o: applied stress

crack size

length of an existing flaw or crack in the material;

design goal

K<KIC​

ductile to brittle transition

a change where material behavior shifts from ductile to brittle as temp decreases; BCC does, FCC does not, HCP sometimes

what happens at low temperature

less atomic movement; reduced slip; more brittle behavior

alloying

can lower transition temperature and improve toughness

stress field

distribution of stress throughout the material; how stress varies at every point not just worst spot

what does S represent in an S-N curve

stress amplitude ( how intense each load is)

what does N represent in an S-N curve

number of cycles to failure (fatigue life)

what does the S-N curve show

relationship between applied cyclic stress and how long the material survives

s-n curve type 1 (with fatigue limit)

curve levels off; material can survive infinite cycles below a certain stress (common in steels)

s-n curve type 2 ( no fatigue limit)

curve keeps dropping; eventual failure always occurs

fatigue limit

maximum stress below which the material will not fail, no matter how many cycles

fatigue strength

stress level at which material fails after a specific number of cycles

fatigue life

number of cycles a material can endure before failure at a given stress

safe region

region where applied stress is below the fatigue limit; no failure expected

factors affecting fatigue life

high stress; surface defects; stress concentrations; corrosion; high temperature;

shot peening

process that introduces compressive surface stresses, slowing crack initiation

case hardening

hardens the surface layer to resist crack formation and wear

thermal fatigue

failure due to repeated temperature changes causing expansion/contraction

corrosion fatigue

fatigue accelerated by chemical/environmental attack

creep

slow, time dependent deformation under constant stress, usually at high temperature; occurs above 0.4-0.5 of melting temp in kelvin

main creep stages

primary (transient) - decreasing rate; secondary ( steady-state) constant rate; tertiary (acceleration) - failure;

steady state creep rate

constant deformation rate during secondary creep

rupture time

time until final fracture under creep conditions

design use steady state creep and rupture time

turbines and short term high risk applications

stress and creep

higher stress means faster creep and shorter life