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Fracture Mechanics: An Introduction

Emmanuel E. Gdoutos

Chapter 5

Critical Stress Intensity Factor Fracture Criterion - all with Video Answers

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Chapter Questions

Problem 1

A three-point bend specimen was tested according to the ASTM E399 procedure. The 0.2 percent offset yield stress of the material is $\sigma_Y=1 \mathrm{GPa}$ and the modulus of elasticity is $E=210 \mathrm{GPa}$. The specimen was tested at a loading rate of $60 \mathrm{kN} / \mathrm{min}$. A chevron starter notch was machined and the specimen was subjected to 20,000 cycles at $P_{\text {max }}=20 \mathrm{kN}$ and $P_{\text {min }}=0$. The final stage of fatigue crack growth was conducted for 40,000 cycles at $P_{\max }=15 \mathrm{kN}$ and $P_{\min }=0$. The specimen dimensions were measured as
$$
\begin{aligned}
& S=40 \mathrm{~cm}, W=10 \mathrm{~cm}, B=5 \mathrm{~cm} \\
& a_1=4.993 \mathrm{~cm} \\
& a_2=5.008 \mathrm{~cm} \\
& a_3=5.999 \mathrm{~cm} \\
& a \text { (surface) }=4.925 \mathrm{~cm} \\
& a \text { (surface) }=4.916 \mathrm{~cm} .
\end{aligned}
$$

The maximum load and the secant load of the test record were measured as $P_{\text {max }}=108 \mathrm{kN}, P_Q=100 \mathrm{kN}$. Calculate $K_Q$ and comment on the validity of the test.

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Problem 2

A compact tension specimen was tested according to the ASTM E399 procedure. The $0.2 \%$ offset yield stress of the material is $\sigma_Y=900 \mathrm{MPa}$ and the modulus of elasticity is $E=210 \mathrm{GPa}$. The specimen dimensions were measured as $W=10 \mathrm{~cm}, B=5 \mathrm{~cm}$, while the crack lengths at equal locations across the crack front were measured as $4.90,4.93,5.05,4.95,4.85 \mathrm{~cm}$. The maximum load and the secant load of the test record were measured as $P_{\text {max }}=70 \mathrm{kN}$ and $P_Q=65 \mathrm{kN}$. Calculate $K_Q$ and comment on the validity of the test.

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Problem 3

The load-displacement test record of the compact tension specimen of Problem 5.2 is shown in Fig. 5.18. Calculate $K_Q$ and comment on the validity of the test.
Figure can't copy

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Problem 4

A three-point bend specimen with $S=40 \mathrm{~cm}, W=10 \mathrm{~cm}, a=5 \mathrm{~cm}$ and $B=5 \mathrm{~cm}$ was tested according to the ASTM E399 procedure. The 0.2 offset yield stress of the material is 400 MPa . The secant load is $P_Q=60 \mathrm{kN}$ and the maximum load is $P_{\text {max }}=65 \mathrm{kN}$. Calculate $K_{\mathrm{Ic}}$ and comment on its validity.

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Problem 5

A compact tension specimen of a steel with $W=20 \mathrm{~cm}, a=9 \mathrm{~cm}$ and $B=8 \mathrm{~cm}$ was tested according to the ASTM E399 procedure. The 0.2 offset yield stress of the material is $\sigma_Y=900 \mathrm{MPa}$ and the secant load $P_Q$ was measured to be 300 kN . Calculate $K_{\mathrm{Ic}}$ and comment on its validity.

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Problem 6

Determine the maximum $K_{\mathrm{lc}}$ value that may be determined according to ASTM standards on a 30 mm thick plate with 0.2 offset yield stress (a) 400 MPa ,
(b) 2000 MPa .

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Problem 7

Assume that the extension $\Delta a$ of a crack of initial length ao is equal to the plane strain plastic zone radius calculated according to the Irwin model. For plane strain show that

$$
\frac{\Delta a}{a_0} \leq 0.0212 \simeq 0.02 .
$$

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02:58

Problem 8

Use the result of the previous problem to justify the determination of $P_Q$ using a 5\% secant offset line on the load-displacement record according to ASTM E399 procedure. For this reason use the result that when the load-displacement $(P-u)$ record takes the form

$$
\frac{u E B}{P}=f(a / W)
$$

where $B$ and $W$ are the specimen thickness and width, and $E$ the modulus of elasticity, the quantity

$$
H=\frac{a_0}{W} \frac{1}{f} \frac{\mathrm{~d} f}{\mathrm{~d}\left(a_0 / W\right)}, \quad f=f(a / W)
$$

takes an average value of 2.5 for the recommended range of values $a_0 / W$ lying between 0.45 and 0.55 .

Anand Jangid
Anand Jangid
Numerade Educator

Problem 9

The crack growth resistance curve of a certain material at a thickness 2 mm is expressed by

$$
R=\frac{K_{\mathrm{lc}}^2}{E}+\frac{1}{2}(\Delta a)^{0.5} \mathrm{MJ} / \mathrm{m}^2, K_{\mathrm{lc}}=95 \mathrm{MPa} \sqrt{\mathrm{~m}}, E=210 \times 10^3 \mathrm{MPa}
$$

Consider a center cracked plate of width 10 cm and thickness 2 mm with a crack of length 1 cm . Calculate the length of stable crack growth, the critical crack length, and the critical stress at instability.

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Problem 10

As in Problem 5.9 for a center cracked plate of width 5 cm .

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Problem 11

Consider a double cantilever beam (DCB, Fig. 4.14) with dimensions: $B=2 \mathrm{~cm}, a=50 \mathrm{~mm}, h=10 \mathrm{~mm}$. For the material of Problem 5.9 calculate the length of stable crack growth, the critical length, and the critical load at instability.

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Problem 12

Determine the stability condition for the double cantilever beam (DCB) of Problem 4.7 tested in a soft (load-controlled) or a hard (displacementcontrolled) testing machine.

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Problem 13

As in Problem 5.12 for the strip of Problem 4.10.

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Problem 14

As in Problem 5.12 for the DCB of Problem 4.24.

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Problem 15

As in Problem 5.12 for a center cracked plate.

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Problem 16

Calculate the load required for crack growth of a double cantilever beam (Fig. 4.14) with $a=20 \mathrm{~cm}, h=4 \mathrm{~cm}, B=1 \mathrm{~cm}, E=210 \mathrm{GPa}$ and $G_c=200 \mathrm{~kJ} / \mathrm{m}^2$.

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Problem 17

Calculate the critical stress for growth of a crack of length 2 mm in a large thin plate of steel, loaded by a uniform stress perpendicular to the crack plane. Take $E=210 \mathrm{GPa}, \gamma=2 \mathrm{Jm}^{-2}, \gamma_p=2 \times 104 \mathrm{Jm}^{-2}$.

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01:05

Problem 18

A sheet of glass with width 50 cm and thickness 1 mm is subjected to a tensile stress of 10 MPa . Determine the minimum crack length that would lead to fracture. Take the following properties of glass: $E=60 \mathrm{GPa}$, $v=0.25, \gamma=0.5 \mathrm{~J} / \mathrm{m}^2, \sigma_f=150 \mathrm{MPa}$.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 19

A center cracked plate of steel with width 10 cm is subjected to a uniform tension 200 MPa perpendicular to the crack plane. Calculate the maximum crack length the plate can withstand without failure. $K_{\mathrm{Ic}}=55 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 20

Consider an edge cracked plate of steel with width 5 cm subjected to a uniform tension 200 MPa perpendicular to the crack plane. Calculate the maximum crack length the plate can withstand without failure. $K_{\mathrm{Ic}}=55 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 21

A crack of length 2 mm emanates from a circular hole of radius 50 mm in a large plate of steel subjected to a uniform stress perpendicular to the crack plane. Calculate the maximum stress the plate can withstand without failure. $\sigma_Y=860 \mathrm{MPa}, K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 22

A spherical vessel of steel with radius 1 m and thickness 40 mm contains a through crack of length 1 mm . Calculate the maximum internal pressure the vessel can withstand without failure. $\sigma_Y=860 \mathrm{MPa}, K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 23

A crack of length 2 mm emanates from an elliptical hole with axes 100 and 50 mm in a large plate of steel subjected to a uniform stress perpendicular to the crack plane. Calculate the maximum stress the plate can withstand without failure. $\sigma_Y=860 \mathrm{MPa}, K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 24

A center cracked plate of width 10 cm contains a crack of length 4 cm . The plate is subjected to a uniform stress perpendicular to the crack plane. Calculate the maximum stress the plate can withstand without failure. $\sigma_Y=$ 450 MPa and $K_{\mathrm{Ic}}=25 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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Problem 25

Calculate the maximum load the three-point bend specimen of Problem 2.32 (Fig. 2.30) can withstand. $\sigma_Y=1500 \mathrm{MPa}, K_{\mathrm{lc}}=60 \mathrm{MPa} \sqrt{\mathrm{m}}$. The factor of safety for failure by yielding is 3 and by fracture is 2 .

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02:28

Problem 26

Calculate the maximum radial interference the shaft of Problem 2.36 (Fig. 2.33) can withstand. $\mathrm{c}=5 \mathrm{~cm}, b=2 \mathrm{~cm}, a=1 \mathrm{~mm} ; E=210 \mathrm{GPa}$, $v=0.3, \sigma_Y=900 \mathrm{MPa}, K_{\mathrm{Ic}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$. The factor of safety for failure by yielding and fracture is 2 .

Chai Santi
Chai Santi
Numerade Educator

Problem 27

Calculate the maximum crack length the cylindrical pipe of Problem 2.37 can withstand when it is subjected to a temperature difference $\Delta T=200^{\circ} \mathrm{F}$, across the wall. $\mathrm{c}=15 \mathrm{~cm}, b=10 \mathrm{~cm} ; E=210 \mathrm{GPa}, v=0.3, \alpha=6.6 \times 10^{-6}{ }^{\circ} \mathrm{F}$, $\sigma_Y=900 \mathrm{MPa}, K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$. The factor of safety for failure by yielding and fracture is 2 .

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01:02

Problem 28

Plot the allowable stress versus crack length curves for a large plate with a through crack subjected to a uniform stress perpendicular to the crack plane for the three steels of Example 5.5.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 29

Consider a circular disk of radius $\mathrm{c}=20 \mathrm{~cm}$ with a hole of radius $b=10 \mathrm{~cm}$ rotating at an angular velocity $\omega=2 \pi N / 60$, where $N$ is the number of revolutions per minute. The disk contains an initial crack of length $a=1 \mathrm{~cm}$ emanating from its hole. The material of the disk has mass density $\rho=7800 \mathrm{~kg} / \mathrm{m}^3$, yield stress $\sigma_Y=1000 \mathrm{MPa}$, Poisson's ratio $v=0.3$, and fracture toughness $K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$. Determine the maximum number of revolutions per minute $N_{\text {max }}$ that the disk can rotate without failure, when the factor of safety is $\mathrm{S}=2$ and $S=3$ against failure by yielding and fracture, respectively.

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Problem 30

The circular disk of Problem 5.29 may be constructed from the three steels of Example 5.5. For each steel:
(a) Plot the maximum permissible number of revolutions per minute $N_{\text {max }}$ versus crack length;
(b) Calculate the maximum permissible crack length for $N=5000 \mathrm{rpm}$;
(c) Calculate $N_{\text {max }}$ for a minimum detectable crack $a=0.05 \mathrm{~mm}$.

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Problem 31

The spherical vessel of Problem 5.22 may be constructed from the three steels of Example 5.5. The factor of safety against failure by yielding is 3 and by fracture is 2 . For each steel:
(a) Plot the maximum permissible pressure $p_{\text {max }}$ versus crack length $a$;
(b) Calculate the maximum permissible crack length $a_{c r}$ for an operating pressure $p_c=10 \mathrm{MPa}$;
(c) Calculate $p_{\text {max }}$ for a minimum detectable crack $a=0.05 \mathrm{~mm}$.

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Problem 32

The cylindrical pipe of Example 5.6 may be constructed from the three steels of Example 5.5. For each steel:
(a) Plot the maximum permissible temperature $T_{c r}$ versus crack length;
(b) Calculate the maximum permissible crack length $a_{c r}$ for an operating temperature difference $\Delta T=200^{\circ} \mathrm{F}$;
(c) Calculate the critical temperature difference $(\Delta T)_{c r}$ for a minimum detectable crack length $a=0.05 \mathrm{~mm}$.

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Problem 33

A rectangular panel of width 1 m is subjected to a tensile load of 100 MN perpendicular to the width. The smallest crack size that can be detected is 1 mm . Two steels with the following values of yield stress and fracture toughness are available for constructing the panel: A. $\sigma_Y=860 \mathrm{MPa}, K_{\mathrm{Ic}}= 100 \mathrm{MPa} \sqrt{\mathrm{m}}$ and B. $\sigma_Y=1550 \mathrm{MPa}, K_{\mathrm{lc}}=55 \mathrm{MPa} \sqrt{\mathrm{m}}$. Find which material gives the smallest thickness of the panel.

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Problem 34

Calculate the maximum stress the plate of Problem 5.19 can withstand without failure when it has a thickness of 5 mm .

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Problem 35

Calculate the maximum stress the plate of Problem 5.24 can withstand without failure when it has a thickness of 20 mm .

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Problem 36

A long steel plate of width 20 cm and thickness 2 cm is subjected to a load 1.2 MN . Calculate the maximum crack length of a central or edge crack the plate can withstand without failure. $\sigma_Y=860 \mathrm{MPa}, K_{\mathrm{lc}}=100 \mathrm{MPa} \sqrt{\mathrm{m}}$.

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