Exercises

1.1 In fracture mechanics manuals, it is customary to use for crjv the maximum tensile stress computed elastically for an unnotchcd specimen. Express on for a beam in terms of the maximum bending moment A/, the beam depth D, and the central moment of inertia of the cross-section [. (Answer: on — M D/2I)

1.2 Determine on as in the previous exercise for a hollow cylindrical bar of outer diameter U and inner diameter aD (a < 1) subjected to a torsional moment Mr. (Answer: on 16Л/т/[тгІ)‘,(1 —a4)])

1.3 . With the same criteria as inthe previous exercises, determine o. n for a circular bar of diameter D subjected to simultaneous tension and torsion; let F be the tensile force and Mt — PPD the torque, where p is some dimensionless constant. Give the coefficient cn corresponding to Eq. (1.4.1). Hint: use Mohr’s circle to find the maximum tensile stress.

1.4 Results from the literature were analyzed by the authors using a characteristic specimen dimension D and a nominal stress оn defined with c, v -= 1. The results for the best fit of Bf’t and Do were 1.15 MPa and 322 mm, respectively. To compare with other results, you want to use a nominal stress defined using the same characteristic size D, but a constant c. v = 2.5. What would the values of the best-fit constants (say В f[ and Do) in this case. (Answer: Вf’t = 2.88 MPa, Do – 322 mm.)

1.5 Generalize the previous exercise and prove that if, for a particular selection of c. v and D, the size effect parameters arc В ft and Do, for a different selection c. v and D (where D/D — constant), their value is Bfl •= (cN/cN)Bfi and Do = (J0/D)D0.

1.6 Find the relationship between hj and cj that make identical Eqs. (1.4.12) and (1.4.14).

Series

Material

Specimen

type"

ao/D

S/D

b

(mm)

CN

Reference

АІ-А6

concrete

SEN-TPB

1/3

4

76.2

6

Walsh 1972

B1

concrete

SEN-TPB

1/6

2.5

38.1

3.75

Bazant and Pfeiffer 1987

B2

concrete

DEN-F. C

1/6

38.1

1

Ibid.

B3

concrete

DEN-T

1/6

19.1

1

Ibid.

B4

concrete

DEN-S

1/6

38.1

1

Bazant and Pfeiffer 1986

Cl

mortar

SEN-TPB

1/6

2.5

38.1

3.75

Bazant and Pfeiffer 1987

C2

mortar

DEN-F. C

1/6

38.1

1

Ibid.

C3

mortar

DEN-T

1/6

. ———

19.1

1

Ibid.

C4

mortar

DEN-S

1/6

38.1

1

Bazant and Pfeiffer 1986

D1

HSC

SEN-TPB

1/3

2.5

38.1

3.75

Gettu, Bazant and Karr 1990

El

marble

SEN-TPB

.0.5

4

30

6

Fatby 1992

E2

granite

SEN-TPB

0.5

4

30

6

Ibid.

FI

limestone

SEN-TPB

0.4

4 –

13

6

Bazant, Gettu and Kazemi 1991

G1

Si02 –

SEN-TPB

0.2b

4

= D

6

McKinney and Rice 1981°

G2

SiCCN-137

SEN-TPB

0.2

4

= D

6

Ibid.

G3

SiC CN-163

SEN-TPB

0.2

4

= D

6

Ibid.

Hl-2

concrete

DP

■—

0.4

Marti 1989

11

microcon.

BPO

—– —–

4/тг

Bazant and §ener 1988

J1

mortar

UPT

0.75

Bazant, §ener and Prat 1988

J2

R. mortar

RPT

—– —–

0.75

Ibid.

K1

R. mortar

LRB-UB

0.5

Bazant and Kazemi 1991

K2

R. mortar

LRB-AB

0.5

Ibid.

LI

microcon.

PS

1 / 7Г

Bazant and Cao 1987

“See specimen types in Fig. 1.5.1.

^Variable. Results have been reduced to a fixed ao/D — 0.2. cSee also Bazant and Kazetni 1990b.