Tension Notes

©2014 R.H. Sturges 1

Tension • Properties of Materials for Manufacturing: Mechanical: σ, e, σt , ε, E, Y, UTS, ν Physical: ρ, cv, k, Ω Tensile testing, Strain hardening, ductility, Toughness, modulus of resilience Find mechanical properties through testing Mechanical properties predict forces, heat, time, cost of cutting, forming and success/failure of a process •Testing helps predict:

Material Use, Energy Use, Waste

History of Analysis in Materials Testing 1. Robert Hooke 1676 F=kx

2. Thomas Young 1807 E 3. Claude-Louis Navier 1822 E=σ/ε

Example Tensile Test 2” long, gauge length

2” wide • .0025” thick = 0.005in2 Area

=> σ = LOAD/.005in2 = 200 P; e=Δ”/2”

UTS = ef = E = NB: went down near fracture, but was Af = A0? Af = => σ actually went up all the time!


Example: Tensile Test data l0 = 50mm lf = 80mm d0 = 4mm df = 3mm Pmax = 10kN

Enter: Poisson’s Ratio, ν

(Baby Gate Model) ν =

€

latelonge

=

For metals ν = .1 to .5 but ν =

€

12

for all PLASTIC flow!

For most materials ν is really negative: the density increases with strain. TRUE STRESS & STRAIN

True stress σt = P / A where A varies. True strain ε =

€

dll

l0

l

∫ = ln ll0

#

$ % &

' (

Also, since volumes are constant until NECKING, ε =

€

ln A0

A"

# $

%

& '

NB: When ε small, ε ≈ e , but not when e is large (>.01) Replot a tensile test:

Notice that the material keeps getting stronger the more it is strained!

Study Fig 2.6 of K; it will be fundamental to us for the next 12 weeks!

As ε↑, so does σy!


Now, we can model this actual curve with an approximate equation, σ = Kεn

NB: This model has some interesting properties:

1. K = σ when ε = 1 Strength Coefficient

2. εu (when σ = UTS) = n Strain Hardening Exponent

Why? Because

€

dPdε

= 0 at necking.

3. But Y is missing! so, we plot on log paper:

…And Call Y the intercept But it is just a model. The REAL curve should be used wherever available! For Certain Materials, like Al 1100-O, an exponential model is poor, and a LINEAR model should use σ = A+Bε

DUCTILITY: Strain, ε, before failure, given by Elongation and/or Reduction in Area MODULUS OF RESILIENCE: Area under the linear elastic part of the σ/ε curve = energy per unit volume when ε ≈ e

MR = σ dε =

€

Yε0

2=

Y2

2E ;

€

in#in3 =

#in2

Examples: Copper = 3 Carbon Steel = 30 “Spring” Steel = 300!

TOUGHNESS = σ dε = area under ENTIRE true σ/ε curve = energy of deformation/unit volume = total specific work to fracture

€

ln∫

€

0

e f

∫


EXAMPLE: Suppose a material has σ = 100,000 e0.5 psi

Find true UTS and Eng’g UTS

First: Plot it!

Why? Because at UTS,

ε = n = 0.5

So, UTSt = 100k(.5).5 =

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12

100ksi = 70.7ksi

Now, find UTSENG’G = Pmax/A0. Since we don’t know anything about the area from this

data, we must use the ratio of areas: A0/An since ln (A0/An) = εn !

Thus,

€

A0

An

= eεn = e.5 OR, An = A0e-.5

At Necking, the Pmax must occur, so σn = Pmax/An.

OR, Pmax = σnAn = σnAe-.5 But what is σn? UTSt!

So, Pmax = UTStA0e-.5 = (70.7k)A0(.606) = 42.8k A0 (#)

and UTSENG’G = Pmax/A0 = 42.8ksi The actual area cancels out.

Q: Why 2 different UTS values?

Q: Can there be 2 different UTSTRUE values?


Hardness, and other tests

Short Quiz on the shapes of σ/ε curves:

HARDNESS TEST 1. Material is constrained all around; not free, so Y appears to go up even more. How Much Apparent Increase in Yield? 2. Tests are Easy to do! 3. Brinell: WC balls on low to medium hardness materials.

HB = Load/Curved area of indention =

€

2P

πD(D− D2 − d2) kg/mm2

where D = indention sphere, d = indention circle 4. BEWARE: Strain hardening is not considered here. Recall that Y changes with cold work! Example Hardness Test: Marble = 16mm (measure) Load = .35 kg (on label) Result = mm dia. spot=> HB =

Convert to SAE (back flap of your book) ÷ 7•10-4 Y =

Other: Rockwell - fixed load, fixed ball dia. , measure depth. Easy to do!


STRAIN RATE: or, suppose V > 0?

Strain rate:

€

˙ e =dedt

=ddt

l− l0l0

#

$ %

&

' ( =

ll0

• dldt

=Vl0

€

˙ ε = dεdt

=ddt

lnll0

#

$ %

&

' ( =

1l• dl

dt=

Vl

Q: Where did l0 go to?

A: Materials are characterized by σ = cεm

m = strain rate sensitivity c = another strength coefficient total NB: Highly temperature-dependent: UTS↓ and m↑ with temp↑. m copper, brass .1 steel .1 - .3 titanium (s.p.) .5 - .8

Q: Liquids 1 ∴ τ ∝ velocity Q: What does ε, m mean to manufacturing? A: CREEP: or, very low speed strain rates. 1. High speed steels @ high temperature

2. Copper and nylon @ room temperature demo solder B = Area = P = ε =

3. See advice on potential creep deformation


HYDROSTATIC PRESSURE: 1. Increases ε at fracture

2. Extends σ/ε curve (same shape)

3. No change in ε or P at necking

Q: What good for manufacturing? A: COMPRESSION TEST: 1. Good for Brittle materials that would ordinarily fracture in tension. 2. Bad since friction affects results and shape. UTSTENS = UTSCOMP for ductile, ≠ for brittle. ideal actual PLANE STRAIN: 1. Only 2 directions are allowed (show with clay and lead)

2. (see text page 72) TORSION: 1. Shear stress, for a thin tube

2. Shear strain, γ = Q: Do we need to find true vs eng’g γ , τ ? A: 3. G = τ /γ where G depends on E and Poisson’s ratio! Q: What good is this knowledge? A:

τ = T2πr2t

Y' = 23

Y = 1.15Y

rφl


BENDING (Flexure):

1. Material fails at the fracture stress (or model of rupture)

σR = moment • y (

€

12

thickness)/ I (area moment of inertia)

2. σR depends on geometry of test!

3. σR depends on isotropy of material! σR ≠ σy OR σU T S !!

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Tension Notes