1. Lecture #19 Failure & Fracture

2. Strength Theories
Failure Theories
Fracture Mechanics

3. Failure = no longer able to perform design function
FRACTURE in brittle materials
YIELDING / excessive deformation in ductile materials

4. Stages of Cracking Failure
Behavior of concrete in compression. Discuss the development of cracking as a function of stress level.

5. Static Fatigue
If a constant load is maintained between 75% and 100% of the strength, failure will eventually occur, because the unstable cracks are given sufficient time to propagate catastrophically.

6. Bond and Microcracking
There is more linearity stress-strain between the paste and aggregates in high strength concrete rather than normal strength concrete because of the reduced micro-cracking. The same is true of the light weight concrete.

7. Stress Conditions Mechanical testing under simple stress conditions
Design requires prediction of failure for complex stress conditions
principal stresses (s1 > s2 > s3)
biaxial stress state (s3=0)

8. Strength
Envelope
For Concrete

9. Simple Failure Theories
Rankine
s1=sft
St. Venant
e1= eft
neither agree w/ experimental data
either are rarely used

10. Complex Failure Theories
Max Shear Stress (Tresca)
ductile materials
tmax= ty
s1-s3= sy
s2-s3= sy
s1-s2= sy
sy/2 = max shear stress
at yield
2 = y If 2 > 1 > 0
1- 2 = -y
If 1< 0 and 2 > 0
1 = y If 1 > 2 > 0
2 = -y If 1 < 2 < 0
1- 2 = y
If 1> 0 and 2 < 0
1 = -y If 2 < 1 < 0

11. Complex Failure Theories
Max Distortional Strain Energy (octahedral shear stress, von Mises)
best agreement with experimental data
hydrostatic + distortional principal stresses

12. Failure Theories Mohr’s Strength both yielding & fracture sft  sfc OR

13. Failure Theories
Mohr’s Strength

14. Failure Envelope
Mohr’s Strength
failure envelope

15. Effect of Confinement

16. Comparison of Failure Theories
equivalent to Max Shear Stress
if sft=sfc
ductile and modified
if sft  sfc (brittle)

17. concentrated stress at crack tip (see Fig. 6.7)
Fracture Mechanics
max stress criterion
not sufficient
relationships between applied stress, crack size, and fracture toughness
probability of failure, critical crack size
(size effect, variability of material properties)
focus on linear fracture mechanics, tensile loading, brittle materials
all materials contain flaws, defects, cracks
concentrated stress at crack tip (see Fig. 6.7)

18. Crack Growth

19. Fracture Mechanics Theoretical cohesive strength Griffith Theory
fracture work resisted by energy to create two new crack surfaces
Griffith Theory
flaw / crack size sensitivity

20. Fracture Mechanics stress concentration at crack tip (see Fig 6.9)
for C>>

21. Stress Intensity Factorx
Crack Tip
Stress Distribution

22. Fracture Mechanics Three modes of crack opening
Focus on Mode I for brittle materials

23. Fracture Mechanics

24. Fracture Mechanics

25. Fracture Mechanics KI = stress intensity factor = Fs(pC)1/2
F is a geometry factor for specimens of finite size
KI = KIC OR GI=GIC unstable fracture
KIC= Critical Stress Intensity Factor
= Fracture Toughness
GI=strain energy release rate (GIC=critical)

26. 2 d KI c c
2 a
Alpha = a/d F
Alpha

28. Flexure (Bending) Fracture Yielding similar as in tension
brittle materials
nonlinear s distribution
initiates as tensile failure
flexural strength > tensile strength
Yielding
similar as in tension
ductile materials
extreme fiber
progresses inward
gradual change masks proportional limit
In a brittle material, nonlinearity of the stress distribution contributes to the flexure strength exceeding the tensile strength by approximately 50%.

29. Failure Criterion

30. Linear Fracture Mechanics
Non-Linear Fracture Mechanics

31. a cf
Crack
Process
Zone KI d
Alpha = a/d

32. Fracture specimens

33. Specimen Apparatus

34. Specimen Preparation

35. Test Specimens

36. Determination of Fracture Parameters
sN = cn KIf / [g’(a0)cf + g(a0)d]1/2
sN = cn P/(sr) - split tensile (eq. 5.12)
sN = cn P/(bd) - beam (eq. 5.13)
Linear Regression
Y = AX + B
Y = cn2 / [g’(a0) sN2]
X = g(a0) d / g’(a0)
KIf = 1 / A1/2
cf = B / A

38. Application of Fracture Method Strength Determination
g(a ) = c2nF2(a)
Basic Geometry - split tensile
cn = 2/p ;  = (1) 0.0, (2) , or (3)
(1) F(a) = 0.964; g(a ) = 0.0; g’(a ) =
(2) F(a) = a a a3
F() = , g(a ) = ; g’(a ) =
(3) F(a) = a a a3
F() = , g(a ) = ; g’(a ) =
Basic Geometry - beam
cn = 1.5 s/d ;  = a/d
F(a) = a a a a4

39. Failure Criterion

40. Applications of Fracture Parameters Strength Determination
sN = cn KIf / [g’(a0)cf + g(a0)d]1/2

41. Applications of Fracture Parameters Strength Determination
Size effect on strength
( a0 = 0.2; Bfu = 3.9 MPa = 566 psi; da = 25.4 mm = 1 in)
log (d/da) Specimen or structure size log (sN / Bfu) sN
d (mm or inch) (MPa or psi)
or or 373
or or 312
or or 254