Failure

Fracture

Fracture is the separation of a component into two or more parts under an applied stress.
Fracture occurs in two stages:
- Crack initiation
- Crack propagation
Fracture modes depend on:
- Material Type
- Applied Load
- Stress State
- Strain Rate
- Temperature
- Examples:
  - Brittle fracture
  - Fatigue fracture
  - Ductile fracture
  - Creep fracture

Ductile vs. Brittle Failure

Ductile Failure:
- One piece
- Large deformation
- Significant plastic deformation
Brittle Failure:
- Many pieces
- Small deformations
- Little or no plastic deformation
- Catastrophic
Ductility:
- Material's ability to be drawn or stretched under tension and permanently deformed without rupture.
- A form of plasticity deformation
Brittleness:
- Material breaks without significant plastic deformation when subjected to stress.
- Brittle materials absorb little energy prior to fracture.
Ductile fracture is usually more desirable than brittle fracture because it provides warning before failure.
Classification:
- Ductile: Warning before fracture.
- Brittle: No warning.
Moderately Ductile Failure:
- Failure Stages:
  - Necking
  - Void nucleation
  - Void growth and coalescence
  - Shearing at surface
  - Fracture

Linear-Elastic Fracture Mechanics

Applied stress is amplified at the tip of a small incision or notch.
Fracture mechanics was developed by A.A. Griffith during World War I to explain brittle material failure.
Griffith's Motivation:
- The stress needed to fracture glass is around $100$ MPa.
- The theoretical stress needed for breaking atomic bonds in glass is approximately $10,000$ MPa.
Griffith suggested that microscopic flaws in the material explain the low experimental fracture strength and size-dependence of strength.
Experiment:
- Griffith introduced an artificial flaw (surface crack) in glass specimens.
- Result: $σ_f \sqrt{a} ≈ C$
  - $σ_f$ is the fracture stress.
  - $a$ is the flaw length.
  - $C$ is a constant.

The Fracture Process

Under high load, a crack propagates causing separation of atomic layers.
Crack growth requires energy, the so-called surface energy $γ [J/m^2]$ .
Crack growth also releases elastic energy.
Energy Balance:
- Energy supplied by external loading (increased stress) vs. Energy absorbed by crack growth (surface energy).
Crack cannot grow of the energy supplied by external loading is less than the energy absorbed by crack growth.
Crack is unstable and will grow rapidly if the energy supplied by external loading is greater than the energy absorbed by crack growth.

Stress Concentrations

Holes, slots, threads, and geometry changes cause localized stress changes.
Local stresses exceed the nominal background stress.
Higher local stresses promote failure from these locations.
Stress Concentration Factor (SCF) is the ratio of maximum local stress to background stress and must be considered in design.
Design Considerations to Avoid Fatigue Failure:
- Minimize stress concentrations:
  - Avoid abrupt changes in profile.
  - Use chamfers or fillets.
  - Use the largest fillet radius possible.
- Design stresses:
  - Keep maximum stress below the required fatigue stress $σ_f$ .

Stress Concentration

Flaws are stress concentrators.
Griffith crack criterion:
- $σm = 2σ \sqrt{\frac{a}{ρt}}$ , where:
 - $ρ_t$ = radius of curvature of the flaw
 - $σ$ = applied stress
 - $σ_m$ = peak stress at the crack tip
 - $a$ = half crack length
 - $K$ = stress concentration factor
Avoid Sharp Corners!

Crack Propagation

Energy stored in material as it is elastically deformed.
This energy is released when the crack propagates.
Creation of new surfaces requires energy.
Cracks having sharp tips propagate easier than cracks having blunt tips.
A plastic material deforms at a crack tip, which “blunts” the crack.

Mathematical Expression of the Failure Criterion

$K_c = Yσ \sqrt{πa}$
- $K_c$ : Fracture toughness (material property, measured experimentally)
- $Y$ : Geometrical factor
- $a$ : Half critical crack length
- $σ$ : Critical stress/Design stress
Cracks propagate when the fracture toughness of the material is exceeded for a combination of applied stress and crack length.
Interior and edge cracks in a plate of infinite width.

Three Modes of Crack Surface Displacement

Mode I: Tensile loading
Mode II: Shear loading
Mode III: Tearing
Fracture mechanics calculations are usually concerned with Mode I.
Fracture criterion:
- $K_{IC} = Yσ \sqrt{πa}$
Crack growth condition: Largest, most highly stressed cracks grow first!
- $K ≥ K_c = Yσ \sqrt{πa}$
- Scenario 1: Max. flaw size dictates design stress
 - If σ < \frac{Kc}{Y \sqrt{πa{max}}} then no fracture
- Scenario 2: Design stress dictates max. flaw size
 - If a < \frac{1}{π} (\frac{Kc}{Yσ{design}})^2 then no fracture

Linear Elastic Fracture Mechanics

Example 1:
- $K_{IC} = 26$ MPa.m $^{1/2}$
- Maximum flaw size $a = 9$ mm, fails at fracture stress of $112$ MPa.
- Find stress at fracture if maximum flaw size is $4$ mm.
- Solution: $σ = 168$ MPa
Example 2:
- Aluminum alloy 7075-T6 with width of $50$ mm contains an internal crack with length $l = 3.0$ mm.
- Crack propagates at $σ = 364$ MPa.
- Find fracture toughness (assume $Y = 1$ ).
- Solution: $K_{IC} = 25$ MPa.m $^{1/2}$
Exam Example
- $K_{Ic} = Yσ \sqrt{πa}$

Impact Testing

Impact loading: severe testing case, makes material more brittle, and decreases toughness.
Charpy Impact Testing:
- Standard test for measuring impact energy.
- Gives an indication of material characteristics during fracture.
- An arm is swung down in a pendulum motion to impact with the test material.
- The energy required to fracture the sample is recorded. Qualitative measure of toughness.
- Can’t be used to measure $K{Ic}$ or $Gc$ .
- Relative measure of energy absorbed in impact fracture.
$U{before} = mgh1$
$U{after} = mgh2$
Ductile-to-Brittle Transition Temperature (DBTT)…
- BCC metals (e.g., iron at T < 914ºC)
- High strength materials (σ_y > E/150)
- Polymers
FCC metals (e.g., Cu, Ni) – Student self study

Fatigue

Fatigue is failure due to dynamic and fluctuating stresses.
Fatigue occurs at stress levels below $σs$ and below $σy$ .
Characteristics of Fatigue:
- Fatigue refers to the failure of metals caused by fluctuating or dynamic loading.
- Fatigue results in brittle type fracture, even in ductile materials.
- Cracks originate from stress concentrations: at the component surface, or at large internal defects.
- Cracks grow with each loading cycle to leave concentric ‘beach marks on the fracture surface.
- Crack growth continues until a critical size is reached, followed by fast, catastrophic fracture (this happens when the material's fracture toughness is exceeded).
Fatigue refers to crack initiation and propagation under fluctuating or cyclic loads.
To characterise the fatigue behaviour of a material:
Subject the sample to cyclic or fluctuating loads, or subject the sample to cyclic or fluctuating displacements.

Fatigue Loading

Main classifications for load cycles:
- Fully reversed loading:
  - Rotating axle of rail car
- Repeated:
  - Batch pressure vessel
- Full spectrum loading:
  - Ship or oil platform
Fatigue loading may be:
- Axially applied loads (tension-compression or tension-tension)
- Bending (3-point bend test)
- Cantilever + Rotation

Fatigue Loading Definitions

Stress range: $\Deltaσ = σ{max} - σ{min}$
Stress amplitude: $σa = \frac{σ{max} - σ_{min}}{2}$
Mean stress: $σm = \frac{σ{max} + σ_{min}}{2}$
Stress ratio: $R = \frac{σ{min}}{σ{max}}$
If $R = -1$ then $σ_m = 0$

Fatigue S-N Curves

The sample is subjected to a stress cycle with a given stress amplitude σ or stress range $Δσ$ and the number of cycles to failure N is recorded.
Plot stress (σ or $Δσ$ ) against cycles to failure N (log scale).
Two types of fatigue behaviour observed:
- A. Fatigue failure does not occur below a certain stress level.
  - Endurance limit $σ_e$ (also called fatigue limit)
  - Ferrous alloys (carbon steels, stainless steels, alloy steels)
  - Titanium alloys
  - Some polymers
- B. Fatigue failure occurs at all levels of applied stress.
  - No endurance limit
  - Define a fatigue strength $σ_f$ at a certain number of cycles
  - Aluminum alloys
  - Copper alloys
  - Magnesium alloys
  - Nickel alloys

Using S-N Curves

Loading conditions below the curve = safe.
Loading conditions above the curve = unsafe.
Data in S-N curves is for a specific loading type under certain atmospheric conditions.
Should be given on the curve.
Usually data is for:
- Material and condition
- Stress ratio
- Atmosphere
- Fully reversed loading.
- Mean stress ( $σ_m = 0$ ).
- Stress ratio ( $R = -1$ ).
Where $σ_m ≠ 0$ we use Goodman's rule:

$\frac{\Delta σo}{\Delta σ{TS}} = (1 - \frac{σm}{σ{ts}})$

Where $σm ≠ 0$ we use Goodman's rule: $\frac{\Delta σo}{\Delta σ{TS}} = (1 - \frac{σm}{σ_{ts}})$
While the above equation is in terms of stress range it clearly holds for stress amplitude (as $Δσ = 2σ_a$ for all cases).

S-N Curve: Exam Example

High strength low alloy steel.
$σ_{ts} = 855$ MPa
Subjected to cyclic tensile stress.
Stress Range: $\Deltaσ = σ{max} - σ{min}$
Stress Amplitude: $σa = \frac{\Deltaσ}{2} = \frac{σ{max} - σ_{min}}{2}$
Mean Stress: $σm = \frac{σ{max} + σ_{min}}{2}$
σmin = 120 MPa
σmax = 780 MPa
σm = 450 MPa
σα = 330 MPa
$σ_a = \frac{\Deltaσ}{2}$
Stress amplitude (= Δσ/2) with zero mean stress that gives equivalent fatigue behaviour:

Fatigue of Cracked Components

Engineering structures and components are rarely defect-free.
Usually contain internal flaws and cracks.
These defects may be so small they go undetected.
Cyclic loading will cause the cracks to grow.
At some point the crack size may reach the critical crack length for fast fracture.
Need to know the number of cycles (or time) until these cracks reach the critical length.
Consider the stress intensity factor:
- $K = Yσ \sqrt{πa}$
For fatigue loadings we have the fatigue stress intensity factor:
- $ΔK = K{max} - K{min} = YΔσ \sqrt{πa}$

S-N curve: Example 4.3

A steel shaft operates at continuously with rotational speed of 600 rpm.
Using the S-N curve provided determine the maximum continuous life for a stress amplitude of:
- a) 450 MPa
- b) 380 MPa
- c) 310 MPa

Summary

Engineering materials not as strong as predicted by theory
Flaws act as stress concentrators that cause failure at stresses lower than theoretical values.
Failure type depends on T and σ:
- For simple fracture (noncyclic σ and T < 0.4Tm), failure stress decreases with:
 - increased maximum flaw size,
 - decreased T,
 - increased rate of loading.
- For fatigue (cyclic σ):
 - cycles to fail decreases as ∆σ increases.
Sharp corners produce large stress concentrations and premature failure.