1. Mechanical Properties
11/13/08

2. Stress:
force F
 = sigma = A
area

3. Stress at a single contact point
Stress at a single contact point. The closer the spacing between fringes the higher the stress.
highest stress
* Note
* Fringes are revealed when clear plastics are viewed in
polarized light.

4. Stress and Force: 50 N 50 N 1.0 mm 2.25 mm stress = F / A
= 50 / 2.25
= 50 / 1.0
= 22.2 MPa
= 50 MPa
50 N
50 N

5. Types of deformation:
tensile forces
compressive forces
shear forces

6. Stress concentration at a constriction:F F
higher density of lines
indicates higher stress

7. Stresses:
compressive
stresses
tensile
stresses

8. Strain:  = Lf Li strain = Lf - L i L i by tradition strain is denoted
 =
by tradition strain is denoted
by a Greek epsilon

9. Stress-strain diagrams
0.01
stress (MPa)
0.03
0.05
250
200
150
100 50 A B
strain

10. The ratio of stress to strain. E =  / 
Elastic Modulus:
The ratio of stress to strain.
E =  / 
The slope of the linear portion of a stress – strain diagram
Indicates stiffness, rigidity

11. Elastic modulus Modulus of Elasticity Young’s Modulus Elastic Modulus:
Alternative terminology:
Elastic modulus
Modulus of Elasticity
Young’s Modulus

12. Elastic behavior:  

13. Plastic behavior: 

14. 300 200 100 0.2 0.4 0.6 strain (%) plastic deformation elastic

15. Deformation that is recovered when the stress is removed.
Elastic Deformation:
Deformation that is recovered when the stress is removed.

16. Deformation that is not recovered when the stress is removed.
Plastic Deformation:
Deformation that is not recovered when the stress is removed.

17. 300 200 100 0.2 0.4 0.6 strain (%) yield strength yield point
proportional
limit
100
0.2
0.4
0.6
strain (%)

18. Elastic Limit:
The largest stress that a material can sustain and still return to its original dimensions when the stress is removed.

19. Proportional Limit:
The largest stress that a material can sustain while stress remains proportional to strain.

20. Yield stress:
The most commonly used measurement of the stress at which plastic deformation begins. It is defined at a fixed amount of strain.

21. 300 200 100 0.2 0.4 0.6 strain (%) yield strength yield point
proportional
limit
100
0.2
0.4
0.6
strain (%)

22.  Stress  Strain Energy in purely elastic deformation
Modulus of Resilience
 Strain

23. Modulus of Resilience:
Want high proportional limit
Want low elastic modulus

24. Toughness  Strain  Stress Area under curve is Energy of elastic +
Plastic deformation
Toughness
 Stress
 Strain

25. Ductile vs. brittle fracture:

26. a. b. c. d. e. f. stress stress stress strain strain strain stress

27. An Introduction to Fracture Mechanics

28. Stress concentrations at sharp angles:

29. low load
Note that stresses are
higher at grooves and
corners.
higher load

30. Fracture mechanics: basic idea
The strength of a material depends on its resistance to stress concentrations produced by defects
Cracks are the most important type of defect in materials. (Many materials contain 1000s of micro-scopic cracks).
One wants to develop criteria that indicate how susceptible a material is to the presence of crack. This is what fracture mechanics is all about.

31. Fracture mechanics: Griffith’s work
In 1920 Griffith developed two equations, that help us understand the effect of cracks of stress concentration and on fracture strength.

32. stress concentration factor:
stress in the presence
of the defect c
K = o
stress if the defect had not
been present

33. Stress at an elliptical crack (1):
In 1920, Griffith calculated the stress concentration caused by a crack in a brittle material. 2b 2a

34. Stress at an elliptical crack (2):
max =  b
where 2a and 2b are the lengths of
the major and minor axes of the ellipse
A.A. Griffith, 1920

35. Stress at an elliptical crack (3):
If  is the radius of curvature of the crack
ρ = b2 a
ρ is small when a is large and b is small. When ρ is small, the crack is sharp.
max =  a ρ
1/2
A.A. Griffith, 1920

36. Stress at an elliptical crack (4):
If  << a
1/2 a
max =
2  
A.A. Griffith, 1920

37. Stress at an elliptical crack (5):
max a
K = = 2  
1/2
Stress concentration
Notice that the stress concen-tration increases with the length of the crack and the sharpness of the crack.
A.A. Griffith, 1920

38. Fracture Mechanics: Griffith 1st equation
Unfortunately, although Griffith’s first equation is scientifically important, it is NOT practically important. We do not know:
how long the longest crack is
how sharp the longest crack is
how the material itself acts on the crack; e.g., are some materials more sensitive to the presence of cracks than others?

39. Fracture Mechanics: Griffith’s second equation
Fortunately, Griffith developed a second, more useful, equation.
This equation can be used to determine a material’s susceptibility to weakening by cracks.
This second equation is call the Griffith Equation.
We will go through some of the steps in arriving a the Griffith equation in the following slides.

40. p s a = - Internal energy E 2 2
Griffith developed a second, more useful equation:
Griffith developed a second, more useful equation:
2 2 a
Internal energy E
p s
= -
A crack with a length 2a will increase the internal
energy of the system proportional to the square
of the crack’s length and the square of the
stress. Note that E is the elastic modulus.

41. 4a Crack surface energy g = + s
But energy is required to create new surface.
Here s is the surface energy. Note the crack
length is 2a and two surfaces are being created,
hence the 4.

42. = decrease in internal energy per increase in crack length
increase in crack surface energy
per increase in crack length =
The crack will not grow as long as an increase
in crack length does not decrease the internal
energy more than the creation of new surfaces
increases the surface energy. The last moment
when this is true is when change in the two
energies are equal.

43. ( ) 2 4 f s a E
p s g =
This equation is the differential equation that is equivalent to the previous word equation. Cracks do not extend as long as the change in internal energy (left) is equal to the change in surface energy (right). Fracture occurs when the stress makes the left side larger than the right. Note we have substituted the facture stress, , for the stress, The stress at which this equation holds is the fracture stress. σf σ

44. a E p = s g 4 2 f Differentiating this equation, we obtain:
& solving for σf , we find:

45. Griffith eq. for elastic materials:1 2 s f E a g p =
Note that the fracture strength
decreases as the size of the
largest crack in the material
increases.

46. Stress concentrations at cracks in brittle and ductile materials:
If the material is plastic, the high stress at the crack tip will be reduced by plastic flow at the tip, which will blunt the tip.
If the material is plastic, the high stress at the crack tip will be reduced by plastic flow at the tip, which will blunt the tip.

47. Modifying the Griffith eq. for plastic materials:
We add a term to account for
the increase in plastic energy
with increase in crack length.

48. Modifying the Griffith eq. for plastic materials:
Plastic deformation
energy per unit
length of crack
Plastic deformation
energy per unit
length of crack
We differentiate and solve for σf :
We differentiate and solve for σf :

49. Griffith eq. for plastic materials:
Once again, the fracture strength is
a function of the largest crack in
the material.
Unfortunately, we typically
do NOT know either p , the
plastic energy or a , ½ the
crack length.

50. Fracture mechanics: Determining crack susceptibility
The Griffith Equation could predict fracture strength if we knew a , s , and p . Unfortunately, although we have pretty good values s , in real world situations it is difficult to know a and we generally do not know p .
Fortunately, we are able to determine a material’s susceptibility to the presence of cracks experimentally.

51. Fracture mechanics: Determining crack susceptibility
Engineers are able to calculate the stress intensity factor at crack tips, provided the material is relatively brittle, that the crack length is known, and that the material is formed into specimens with specific dimensions.

52. Fracture toughness – compact tensile specimen:
Fracture toughness specimens are fabricated with precise dimensions. A crack with a sharp point is placed in the specimen. The length of the crack is measured during testing, so that the length of the crack at the moment of fracture is determined experimentally.

53. Fracture toughness – compact tensile specimen:
KI - stress intensity factor is a measure of stress at the crack tip
The load is applied so that the specimen opens like a hinge. This type of loading produces a type of stress intensity factor that is denoted KI. F
There are other ways to open cracks. The stress intensity factor for a crack opened by shear is denoted KII.
The stress intensity factor for a crack opened by twisting is denoted KIII. F

54. K I - the stress intensity factor
Y is a dimensionless constant
whose value is fixed by the
dimensions of a fracture
toughness test specimen.

55. Stress intensity factor – KI : units
The units of stress intensity are a bit unusual. All of the above are equivalent.

56. Critical Stress Intensity
KIC, the critical stress intensity, is the
stress intensity at which fracture
occurs.

57. Not a property of the material A property of the material
stress
ultimate strength
stress intensity
fracture toughness

58. The End