2. Safety Factor for Design
The failure criterion for ductile metals and plastics under tensile or steady-tensile loading is the yield strength
However, the yield strength is not taken as the design or working stress for the material.
It is common for engineers to employ a “factor of safety” to ensure against uncertainties due to variations in the properties of a piece of material.
In designs using yield strength the safety factor is two.
In designs using tensile strength, the safety factor is four.
Yield strength is used for ductile materials like structural steels.
Tensile strength is used for brittle materials such as cast iron.
Since most brittle materials are used under conditions of compression, why then would the tensile strength be used?

3. Safety Factor for Design
So, for the design stress, sdes,
Using the design stress, we can obtain the size or area of the material from the maximum load, Pmax, the structure or component is going to sustain where
This safety factor method oversizes the material, which is fine for static structures, but is a factor for transport structures such as airplanes and freight trucks, i.e. it’s not appropriate.

4. Safety Factor for Design
If the mode of loading is shear, we can approximate the maximum shear strength, tmax, of a material from its yield strength by
And, the design method is the same as for ductile materials like steel.
For brittle ceramics and glasses, the design for the safety factor is more complex and Weibull statistics are used, which we’ll discuss in more detail later.

5. Safety Factor for Design
Using the yield strength and ultimate tensile strength is sufficient for most designs, but it’s not a guarantee that catastrophic failure will not occur, which you’ll be held legally accountable as the professional engineer responsible for the design.
Current design methods for safety and performance use the “fracture toughness” of a material.
The basic premise in using fracture toughness, or fracture mechanics in design is to assume that materials have defects or cracks in them.
The material property that resists the propagation of these cracks is the “fracture toughness”.

6. Design - Fracture Toughness
What is fracture toughness?
The stress intensity at the crack tip is dependent on both the applied stress and the length of the crack.
A mechanical variable, Stress Intensity Factor, KI, is used to describe the relationship:
Where,
f is a dimensionless constant (related to geometry of specimen and flaw)
is the applied stress
a is the crack length or half the length of an internal crack
KI is a variable but NOT a material property
KI has unusual unit of Mpa(m)½ or psi(in)½ .

7. Design - Fracture Toughness
When the stress intensity, KI is increased to a critical value, i.e., the critical stress intensity, KIC , crack propagation will occur, which will lead to fracture.
It is written as:
Where,
KIC is a measure of a materials resistance to crack propagation.
It is a material property.
KIC is dependent on temperature, microstructure, and strain rate.
KIC usually increases with a reduction in grain size.

8. Fracture Toughness Design
Higher fracture toughness materials, such as material B in the graph, tolerate higher stresses and thus larger cracks.
Crack size

9. Fracture Toughness How to use KIC ?
Fracture toughness is most useful in mechanical designs involving materials with limited toughness or ductility.
Usually s < syield/n is good enough for ductile materials, which are statically loaded, e.g., soft aluminum.
The design criterion that should be used is KIC , where
Taking into account KIC , which is a material property, the allowable stress (s) and/or the allowable flaw size (a) can be determined.

10. Fracture Toughness Material Selection:
If the maximum applied stress, smax , and maximum crack length are specified for a certain application, then only the materials with KIC greater than KI can be used:
As a professional engineer, if you are tasked with applying materials for a design requiring structural integrity, you should request from the designer the maximum applied stress and maximum crack length.

11. Fracture Toughness
Allowable stress design (if “amax” and KIC are specified by the application constraints) then the allowed stress is,
Allowable crack size design (if the stress level, smax , and KIC are specified) then, E

12. Critical Stress-Intensity Factor
Variation of K with thickness of the material. As the thickness increases, the stress intensity becomes the material fracture toughness, KIc, and independent of thickness. Thus, objects break independent of their size.
A crack propagates when KIc is attained at the tip of a crack.

13. Fracture Toughness – Crack Surface Energy
When a material has an applied strain, it undergoes an elastic strain related to the modulus of elasticity, E, of the material.
When a crack propagates, this strain energy is released, which reduces the overall energy.
However, two new surfaces are created by the extension of the crack, which increase the energy associated with the surface.
By balancing the strain energy and the surface energy, g, we find that the critical stress required to propagate the crack is given by:
This equation shows that even small cracks can severely limit the strength of a material.
This equation is particularly applicable to ceramics.

14. Fatigue
Fatigue is the lowering of strength or failure of a material due to repetitive stress, which may be above or below the yield strength.
Many engineering materials such as those used in cars, planes, turbine engines, machinery, shoes, etc are subjected constantly to repetitive stresses in the form of tension, compression, bending, vibration, thermal expansion and contraction or other stresses.
At a local size scale, the stress intensity exceeds the yield strength.
For fatigue to occur at least part of the stress in the material has to be tensile.
Fatigue is most common in metals and plastics, whereas ceramics fail catastrophically without fatigue because of their low fracture toughness.

15. Fatigue
Many components fail by fatigue when subjected to cyclic loads, which generate nominal stresses below the static ultimate stress of the material.
Fatigue occurs because each half stress cycle produces minute strains, which are not recoverable.
When these minute strains are added, they produce local plastic strains, which are sufficient to cause submicroscopic cracks.
These small cracks act as stress intensifiers so that the local stress in the region of the crack can exceed the stress to propagate the crack.
The crack grows, often over a long period, until the cross sectional area is lowered below the limit to support a stress that can cause catastrophic fracture (the Griffith relationship)
The presence of a notch or other stress intensifiers can act as a starting point for the process of fatigue.

16. Fatigue – 3 Stages
There are typically three stages to fatigue failure.
First a small crack is initiated or nucleates at the surface and can include scratches, pits, sharp corners due to poor design or manufacture, inclusions, grain boundaries or dislocation concentrations.
Second the crack gradually propagates as the load continues to cycle.
Thirdly, a sudden fracture of the material occurs when the remaining cross-section of the material is too small to support the applied load.

17. Fatigue Fatigue failures are often easy to identify.
The fracture surface near the origin is usually smooth. The surface becomes rougher as the crack increases in size.
Microscopic and macroscopic examination reveal a beach mark pattern and striations.
Beach mark patterns indicate that the load is changed during service or the load is intermittent.
Striations are on a much finer scale and show the position of the crack tip after each cycle.

18. Fatigue
The most important fatigue data for engineering designs are the S-N curves, which is the Stress-Number of Cycles curves.
In a fatigue test, a specimen is subjected to a cyclic stress of a certain form and amplitude and the number of cycles to failure is determined.
In a rotating beam fatigue testing machine, the specimen is bent as it rotates.
The reduced middle section of the specimen alternates between states of tensile and compressive stress.
How is this misalignment compensated in a car’s driveshaft?
This often happens in rotating shafts used in motors.

19. The S-N Curve
Results of fatigue tests are presented as plots of nominal cyclic stress, S, versus number of cycles to failure, N.
At a nominal stress equal to the ultimate stress, the component will fail after the first half cycle.
At a nominal stress below the yield strength, the number of cycles to failure is relatively large but still finite.
In iron-based materials, there is a nominal stress below which fatigue does not occur during “normal” life times, the endurance limit, which is used as a design parameter.

20. The S-N curves for a tool steel and an aluminum alloy showing the number of cycles to failure
Al does not show a fatigue limit but continuously decreases.
For materials, which do not show a endurance limit such as Al, Cu, and Mg (non-ferrous alloys), fatigue strength is specified as the stress level at which failure will occur for a specified number of cycles, where 107 cycles is often used.

21. Fatigue Fatigue Limit:
For some materials such as steels and Ti alloys, the S-N curves become horizontal when the stress amplitude is decreased to a certain level.
This stress level is called the Fatigue Limit, or Endurance Limit, which is typically ~35-60% of the tensile strength for steels.
In some materials, including steels, the endurance limit is approximately half the tensile strength given by:

22. Example of Surface Stress Raiser on S-N Curve
The endurance limit is sensitive to the size of the stress raiser that may exist in the material.
The endurance limit decreases as the size of the stress raiser decreases (radius of crack), which agrees with the increase in the concentrated stress as the crack radius decreases.

23. Fatigue Failures Types of stresses for fatigue tests include,
1) axial (tension – compression)
2) flexural (bending)
3) torsional (twisting)
From these tests the following data is generated.
By convention, tensile stresses are positive and compression stresses are negative.

24. Fatigue Failures
Examples of stress cycles where a) shows the stress in compression and tension, b) shows there’s greater tensile stress than compressive stress and in c) all of the stress is tensile. a b c

25. Fatigue Failures
As the mean stress increases, the stress amplitude must decrease in order for the material to withstand the applied stress. This condition is summarized by the Goodman relationship:
Where sfs is the desired fatique strength for zero mean stress, sm and sTS is the tensile strength of the material.
Example, if an airplane wing is loaded near its yield strength, vibrations of even a small amplitude may cause a fatigue crack to initiate and grow. This is why aircraft have a routine inspection in order to detect the high-stress regions for cracks.

26. Fatigue Failures Crack Growth Rate
To estimate whether a crack will grow, the stress intensity factor (DK), can determine the crack geometry and the stress amplitude to be used.
Below a threshold DK a crack doesn’t grow.
For somewhat higher stress intensities, the cracks grow slowly.
For still higher stress-intensities a crack
grows at a rate given by:
Where C and n are empirical constants that depend on the material.
When DK is high, the cracks grow in a rapid and unstable manner until fracture occurs.

27. Fatigue Failures From the steady state crack growth relationship of
if we integrate between the initial size of a crack and the crack size required for fracture to occur, we find that the number of cycles to failure, N, is given by
where C and n are empirical constants that depend on the material.

28. Stress-Corrosion Failure
Stress corrosion happens when a material reacts with corrosive chemicals in our environment.
A good example is salt on the roads reacting with the steel in cars causing reduced lifetime of the car’s components such as its frame and suspension system.
Another example is the salt in the ocean reacting with boats and their moorings where the corrosion reduces the life of the engine, which is cooled by the salt water, and the structural integrity of the boat, which is jeopardized if salt water sits in the hull or around the drive shaft.

29. Stress-Corrosion Failure
Stress-corrosion will cause failure of materials below their yield strength.
Why? - because the corrosion will cause cracks to form, usually along grain boundaries.
Usually if there is a corrosion product on the surface there is a crack inside the material.
The surface flaws themselves can be nucleation sites for crack growth during service of the material.
Usually materials are coated to reduce or prevent corrosion. The automotive industry has shown excellent results by applying metal coatings (Sn) and polymer coatings on the sheet steel used on the body of cars.

30. Stress-Corrosion Failure
Intergranular cracks near a stress-corrosion fracture in a metal. Note the many branches where the corrosion has eaten into the grain boundaries of the metal.
On the surface, you’ll see a corrosion by-product. The crack inside is typically much larger than the surface by-product.

31. Stress-Corrosion Failure
Corrosion failures are also strongly affected by the alloying additions to metals.
The best alloying addition to many metals such as iron and zirconium is Chromium.
Cr preferentially oxides over Fe and forms a thin stable film, which substantially reduces further oxidation.
We will see in our Stainless Steel Lectures that the Fe-Cr alloy must be of sufficient Cr concentration and properly heat treated in order for the Cr to be effective against corrosion.
Corrosion resistant steels, eg., 316 steel, are used for containing chemicals, such as sulfuric acid and foods, which often contain organic acids, eg. milk.

32. Stress-Corrosion Failure
Chromia (Cr2O3) blocks oxygen diffusion at grain boundaries, dislocations and defects on the surface of the material so that oxidation of the host material is substantially reduced.
We will also see in our discussion of dislocations that the crystal structure plays a big role in determining whether a material will form cracks or a surface layer of rust during corrosion.
So, we haven’t finished with corrosion.

33. Creep
Creep is an important material behaviour at elevated temperature.
At elevated temperatures, (> 0.5 Tm), a material will undergo slow plastic deformation even under a static stress lower than the yield strength of the material. This is called creep.
Creep Test: This is subjecting a specimen to a constant load or stress at a constant temperature and determining the deformation or strain as a function of time.
Important properties from the creep test include:
1) creep rate
2) time to rupture
3) elongation or reduction in area

34. Creep
The rate and extent of creep is very small at temperatures less than 0.5 Tm so creep is often referred to as a high temperature effect.
Creep occurs at room temperature in lead alloys, e.g., the drain pipes in very old (Roman) buildings are fatter at the bottom than at the top since 0.5 Tm for PB is 20 oC.
In the process of creep:
A load is applied which produces a stress less than the yield stress.
The load causes an instantaneous elastic extension as in a tensile test.
The specimen extends plastically over a relatively long period of time
The creep rate, i.e., the strain rate of the plastic deformation, varies with time and temperature.

35. Creep
The form of the plot of creep strain against time is primarily a function of temperature – with respect to 0.5 Tm.
There are three distinct stages of creep observed at intermediate temperature:
The first stage is marked by a relatively rapid initial creep rate, which decays with time.
The second stage is marked by a constant creep rate, referred t as steady state creep.
The third stage is marked by an increase in creep rate, which accelerates as the specimen necks as it approaches rupture.

36. Stages of Creep
Creep-rupture curve showing three stages.

37. Creep
If the temperature is well below 0.5 Tm or the stress is well below the yield point:
- after the first stage, the creep rate decays to a very low constant value and the specimen does not rupture over finite times.
If the temperature is above 0.5 Tm and the stress is a significant fraction of the yield point:
- the first stage is curtailed and a relatively high creep rate is observed in the second stage, which is followed by an accelerated creep rate to rupture.
Low temperature creep is usually ignored.

38. Creep
Creep rates determined in the second stage – under steady state conditions – are used as design parameters, e.g., for Al alloys at room temperature.
High temperature creep is a serious problem as it can significantly shorten the life of a component, which must be used at high temperatures, e.g., for boiler tubes and blades for turbine engines.

39. Creep In general, the creep rate is given by an Arrhenius relationship
Where Qc is the activation energy, R is the gas constant (~ cal/mol/K), T is the absolute temperature A is the pre-exponential constant, which is dependent on the applied stress, s, given by,
where C and n are constants. Qc is related to the activation energy for self-diffusion when dislocation climb is important.

40. Creep - Rupture Time
The rupture time, tr, due to creep also follows an Arrhenius relationship
Where Qr is the activation energy for rupture, and K and m are constants. tr is dependent on the applied stress, sm.

41. Creep – Rupture Time
The stress - time to rupture curves for a heat-resistant alloy. Note the significant decrease in time for rupture with an increase in stress and temperature.

42. Creep
In tensile creep deformation, it can be described by a tensile viscosity, h,
Where s is the tensile stress and e is the tensile strain rate.
Deformation mechanisms involved in creep include:
viscous creep: for amorphous solids
vacancies or atoms : diffusion
dislocations : slip
grain boundaries : grain rotation, grain boundary sliding

43. Viscous Creep
Viscous creep for amorphous solids such as many types of plastics is a diffusion dependent process that is enhanced by increasing the temperature, i.e., thermally activated process, and follows the Arrhenius equation.
Where Q is the activation energy for creep in cal/mol, R is the gas constant, and T is the absolute temperature in K.
As seen before, during creep A depends on the applied stress.

44. Creep Mechanisms
for amorphous solids

45. Creep
In crystalline materials, creep occurs either by diffusional or dislocation creep.
Diffusional creep involves the motion of vacancies and this may occur primarily through the grains or along the grain boundaries.
Vacancy motion through the grains is called the Nabarro-Herring mechanism.
Vacancy motion along the grain boundaries is called the Coble mechanism.

46. Creep Mechanisms
Note that the vacancies and atoms move in opposite directions.

47. Creep These strain rates are given by Nabarro-Herring Coble
Where d is the diameter of the grain, Qv is the activation energy of self or volume diffusion, and Qb is the activation energy for grain boundary diffusion, which is usually half that of self or volume diffusion. A2 is a material constant.

48. Creep
In crystalline materials, dislocation creep involves the motion of dislocations where dislocation climb is an important factor.
Dislocation climb means that the edge of the extra plane of atoms move to another plane parallel to the previous plane that it was before.
This dislocation motion also involves the diffusion of vacancies and thus the strain rate is thermally activated having the form,
Dislocation creep
Where m varies from one material to another and is typically on the order of 5.
Thus creep can become quite complex.
More sophisticated methods are often applied to creep by using the Sherby-Dorn parameter and Larson-Miller parameter.

49. Creep Mechanisms

50. Creep
Only solid solution hardening and precipitation hardening remain effective at elevated temperatures to help prevent creep.
Grain boundary sliding during creep causes a) the creation of voids at an inclusion trapped at the grain boundary and b) the creation of a void at a triple point where 3 grains are in contact.

51. Steady-State Creep
The creep rate at various temperatures for carbon steel used for pressure vessels. Note the logarithic scales, resulting in the exponential dependency of stress on the strain rate.

52. The End (Any questions or comments?)