COL #2 | Mechanical Properties of Materials

Engineering stress (σ)

σ = F/A₀: force divided by ORIGINAL cross-sectional area. Units MPa (1 MPa = 1 N/mm²).

Engineering strain (ε)

ε = Δl/l₀ = (l − l₀)/l₀. Fractional change in length; dimensionless.

Elastic deformation

Temporary deformation: atomic bonds stretch and spring back when the load is removed. Stress ∝ strain.

Plastic deformation

Permanent deformation: dislocations move and atomic planes slip. Does not recover on unloading.

Hooke's law

σ = Eε. Valid ONLY in the elastic region (σ below the yield strength).

Modulus of elasticity (E)

Slope of the linear elastic region = stiffness. Set by bond strength, so alloying/heat treatment barely change it. Units GPa.

Poisson's ratio (ν)

ν = −ε_x/ε_z: lateral (sideways) strain over axial (pulling-direction) strain. Metals ≈ 0.3.

Using Poisson's ratio

ε_z = σ/E (lengthwise), then ε_x = −ν·ε_z (sideways), then Δd = ε_x·d₀. Negative Δd = diameter shrinks in tension.

Yield strength (σy)

Stress at which plastic deformation begins.

0.002 offset method

Draw a line with slope E starting at ε = 0.002; where it crosses the stress–strain curve is the yield strength.

Proportional limit

Stress where the curve first departs from a straight line. Slightly below σy.

Strain hardening (work hardening)

Plastic deformation creates more dislocations that tangle and block each other, so more stress is needed to keep deforming. Material gets stronger, less ductile; σy rises on reloading.

Ultimate tensile strength (UTS / TS)

Maximum engineering stress on the stress–strain curve.

Necking

Localized thinning that begins at the UTS. Engineering stress then falls (uses A₀) even though true stress in the neck keeps rising.

Ductility

Amount of plastic deformation before fracture.

Percent elongation (%EL)

%EL = (l_f − l₀)/l₀ × 100. Uses plastic strain only (elastic part recovers at fracture).

Percent reduction in area (%RA)

%RA = (A₀ − A_f)/A₀ × 100.

Elastic strain recovery

On unloading from the plastic region, the specimen springs back along a line of slope E. Elastic strain σ/E is recovered; plastic strain remains.

Permanent (plastic) strain after unloading

ε_plastic = ε_total − σ/E. Final length = l₀(1 + ε_plastic).

Resilience (concept)

Energy a material absorbs elastically and gives back on unloading = area under the ELASTIC part of the curve. Energy stored in stretched bonds.

Modulus of resilience formula (NOT on formula sheet)

U_r = σy²/(2E). Comes from triangle area ½·σy·ε_y with ε_y = σy/E. Units J/m³ (1 MPa = 1 MJ/m³).

Toughness (concept)

Energy absorbed up to fracture = area under the WHOLE stress–strain curve. Needs both strength and ductility.

Toughness quick estimate

Toughness ≈ TS × ε_f (treat the area as a rectangle).

Toughness by integration (elastic + plastic)

U = Eε_y²/2 + [K/(n+1)]·(ε_f^(n+1) − ε_y^(n+1)). Put E and K in Pa to get J/m³.

True stress (σ_T)

σ_T = F/A_i: force over the INSTANTANEOUS area.

True strain (ε_T)

ε_T = ln(l_i/l₀).

Engineering → true stress (NOT on formula sheet)

σ_T = σ(1 + ε). Valid before necking (constant volume).

Engineering → true strain (NOT on formula sheet)

ε_T = ln(1 + ε). Valid before necking.

Strain-hardening equation

σ_T = K·ε_Tⁿ. K = strength coefficient (MPa); n = strain-hardening exponent (how fast stress rises with strain).

Finding K and n from two data points

Convert both points to true values. n = log(σT2/σT1)/log(εT2/εT1); K = σT1/εT1ⁿ.

Length from a given true stress

ε_T = (σ_T/K)^(1/n), then l_i = l₀·e^(ε_T).

Effect of temperature on E

E decreases as T rises (atoms vibrate and sit farther apart; bonds less stiff).

Effect of temperature on σy, TS, and ductility

σy and TS decrease; ductility increases (thermal energy helps dislocations move).

Monomer

Small starting molecule that links up to form a polymer (e.g. ethylene, C₂H₄).

Polymer

Very large molecule (macromolecule) made of many repeat units covalently bonded in a chain.

Repeat unit (mer)

The structural unit that repeats along a polymer chain, e.g. –CH₂–CH₂– in polyethylene.

Homopolymer

Polymer made of only one type of repeat unit.

Copolymer

Polymer made of two or more different repeat units.

Random copolymer

Two repeat units placed in no particular order along the chain.

Alternating copolymer

Two repeat units alternating regularly: ABABAB.

Block copolymer

Long runs of each repeat unit: AAAA–BBBB.

Graft copolymer

Chains of one repeat unit attached as side branches to a main chain of the other.

Chain (addition) polymerization

Monomers add one at a time to the active end of a growing chain; no by-product. Three stages: initiation, propagation, termination.

Step (condensation) polymerization

Molecules join in steps and release a small by-product such as H₂O (e.g. nylon, polyester).

Polyethylene (PE)

Side atoms: all H. Repeat unit C₂H₄, m = 28.05 g/mol. Simple structure → crystallizes easily.

Polypropylene (PP)

Side group: CH₃. Repeat unit C₃H₆, m = 42.08 g/mol.

Poly(vinyl chloride) (PVC)

Side group: Cl. Repeat unit C₂H₃Cl, m = 62.49 g/mol.

Polytetrafluoroethylene (PTFE, Teflon)

Side atoms: all F. Repeat unit C₂F₄, m = 100.02 g/mol.

Polystyrene (PS)

Side group: bulky benzene (phenyl) ring. Repeat unit C₈H₈, m = 104.14 g/mol. Bulky group → hard to crystallize.

Degree of polymerization (DP)

Average number of repeat units per chain. DP = M̄n/m (m = repeat-unit molecular weight).

Identifying an unknown polymer

Compute m = M̄n/DP, then match m to the repeat-unit weight of a candidate polymer.

Number-average molecular weight (M̄n)

M̄n = Σ xᵢMᵢ, using NUMBER fractions xᵢ and the midpoint Mᵢ of each size range.

Weight-average molecular weight (M̄w)

M̄w = Σ wᵢMᵢ, using WEIGHT fractions wᵢ. Always ≥ M̄n.

Total chain length (L)

L = N·d·sin(θ/2). N = number of C–C bonds (= 2 × DP); d = 0.154 nm; θ = 109°.

Average end-to-end distance (r)

r = d√N. Much smaller than L because the chain is coiled and tangled.

Linear polymer

Separate long chains with no branches. Pack closely; crystallize most easily.

Branched polymer

Side branches off the main chain. Pack poorly, so lower density and crystallinity.

Crosslinked polymer

Neighboring chains joined by covalent bonds at points; chains can't slide past each other.

Network polymer

Fully 3D web of covalent bonds. Requires trifunctional (or higher) monomers.

Bifunctional vs trifunctional monomer

Bifunctional: forms 2 bonds → linear chains only (e.g. PE, PTFE). Trifunctional: forms 3+ bonds → can form networks.

Tacticity

Arrangement of side groups (R) along the chain. Only applies when there is a distinct R group.

Isotactic

All R side groups on the same side of the chain.

Syndiotactic

R side groups alternate sides along the chain.

Atactic

R side groups placed randomly along the chain. Irregular → harder to crystallize.

Head-to-tail

R groups on every other backbone carbon (the usual arrangement).

Head-to-head

R groups on adjacent backbone carbons.

Structures NOT possible for PTFE

No tacticity and no head-to-head/head-to-tail (all side atoms are F, no distinct R); no network (bifunctional); no 'random' (that describes a copolymer). Possible: linear, branched.

Crosslinking

Covalent bonds between neighboring chains. Stops chains sliding: stiffer, no viscous flow, springs back (e.g. vulcanized rubber).

Glass transition temperature (T_g) — polymer

Below T_g: rigid, brittle glass (chain segments frozen). Above T_g: rubbery and flexible (segments can rotate).

Melting temperature (T_m) — polymer

Temperature at which crystalline regions melt. T_g is lower than T_m.

Crystallinity

Fraction of the polymer where chains are neatly folded and packed in order. Higher → denser, stronger, stiffer.

Amorphous (polymer)

Regions where chains are tangled randomly with no order.

What helps a polymer crystallize

Linear chains, small/simple side groups, regular tacticity (iso or syndio), alternating copolymer. Hurts: branching, crosslinks, bulky groups, atactic, random.

Thermoplastic

Linear/branched polymer that softens when heated and can be remolded (PE, PP, PVC, PS).

Thermoset

Crosslinked/network polymer that does not soften on heating (epoxy, phenolic, vulcanized rubber).

Relaxation modulus E_r

Stress remaining after 10 s at fixed strain, divided by that strain. High = stiff; low = relaxed/flowing.

Relaxation modulus vs T: the regions

Glassy (high, flat) → leathery (drop at T_g) → rubbery plateau → rubbery flow → viscous flow.

Effect of molecular weight on the relaxation curve

Higher MW → more entanglement → rubbery plateau extends to higher T before flow.

Effect of crosslinking on the relaxation curve

No viscous-flow region; plateau continues and sits higher with more crosslinking.

Cation vs anion

Cation: positive ion, usually small (metal). Anion: negative ion, usually large (e.g. O²⁻).

Coordination number (CN)

Number of anions touching each cation. Set by the radius ratio.

Radius ratio

r_cation/r_anion (cation on top). Table on formula sheet: 0.414–0.732 → CN 6; 0.732–1.0 → CN 8.

Rock salt (NaCl) structure

CN 6. FCC anions, cations in octahedral holes (cube edges + center). n′ = 4.

Rock salt contact equation

Ions touch along the cube edge: a = 2(r_C + r_A).

Cesium chloride (CsCl) structure

CN 8. Anions at cube corners, one cation at the body center. n′ = 1.

CsCl contact equation

Ions touch along the body diagonal: √3·a = 2(r_C + r_A).

Antifluorite structure (e.g. Li₂Se)

FCC anions, cations in all 8 tetrahedral holes. n′ = 4.

Antifluorite / zinc blende contact equation

Ions touch along ¼ of the body diagonal: (√3/4)·a = r_C + r_A.

n′ (formula units per cell)

Rock salt 4 · CsCl 1 · antifluorite 4 · zinc blende 4.

Theoretical density of a ceramic

ρ = n′(ΣA_C + ΣA_A)/(V_C·N_A), V_C = a³. Convert a from nm to cm (×10⁻⁷) before cubing.

Largest ion in an empty octahedral hole

Along the cube edge: a = 2r_A + 2r_i, so r_i = (a − 2r_A)/2.