Fracture Toughness

What is fracture toughness Kc?

Fracture toughness, Kc, is the resistance of a material to failure from fracture starting from a preexisting crack. This definition can be mathematically expressed by the following expression:

Kc = Y σ SQRT{π a}

Where Y is a dimensionless factor dependent on: the geometry of the crack and material, the loading configuration (i.e. if the sample is subject to tension or bending), and the ratio of crack length to specimen width, b. is the amount of load (stress) applied to the specimen, and a is the crack length.


Figure 1 A specimen with an interior crack. Note that the entire crack length is equal to 2a.


Figure 2 A specimen with a through-thickness crack.


Figure 3 A specimen with a half circle surface crack.

Source Figures 1,2,3: Materials Science and Engineering: An Introduction, 3rd Edition, John Wiley & Sons, Inc. New York.

Figure 1 shows that a is not always the total length of the crack, but is sometimes half the crack length, as for an interior crack. The values for Y vary with respect to the shape and location of the crack. Some useful values of Y for short cracks subjected to a tension load are as follows:

Y = 1.00 For an interior crack similar to the crack shown in Figure 1

Y = 1.12 For a through-thickness surface crack as shown in Figure 2

Y = 0.73 For a half-circular surface crack as shown in Figure 3

Fracture toughness, Kc, has the English customary units of psi in1/2, and the SI units of MPA m1/2.

What is the plane strain fracture toughness, KIc?

For thin samples, the value Kc decreases with increasing sample thickness, b, as shown by Figure 4.


A fracture toughness vs. thickness graph. Note the location of KIc. Source: Materials Science and Engineering: An Introduction, 3rd Edition, John Wiley & Sons, Inc. New York.

Ultimately, Kc becomes independent of b, at this point the sample is said to be under the conditions of plane strain. This fixed value of Kc becomes known as the plane strain fracture toughness, KIc. KIc is mathematically defined by:

KIc = Y σ SQRT{π a}

This value for the fracture toughness is the value normally specified because it is never greater than or equal to Kc. The I subscript for KIc, stands for mode I, or tensile mode, crack displacement as shown in Figure 5(a).


The three modes of crack surface displacement.(a) Mode I, tensile mode; (b) mode II, sliding mode; and (c )mode III, tearing mode. Source: Materials Science and Engineering: An Introduction, 3rd Edition, John Wiley & Sons, Inc. New York.

In general, KIc is low for brittle materials and high for ductile materials. This trend is supported by the KIc values in Table 1 (3,4).

Table 1 Room-Temperature Plane Strain Fracture Toughness Values



Material
KIc
MPA m1/2
psi in1/2
Metals
2024-T351 Aluminum
36
33,000
4340 Steel

(tempered @ 260 C)
50.0
45,800
Titanium Alloy

(Ti-6Al-4V)
44-66
40,000-60,000
Ceramics
Aluminum Oxide
3.0-5.3
2,700-4,800
Soda-lime glass
0.7-0.8
640-730
Concrete
0.2-1.4
180-1,270
Polymers
Polymethyl methacrylate

(PMMA)
1.0
900
Polystyrene

(PS)
0.8-1.1
730-1,000

Source: Materials Science and Engineering: An Introduction, 3rd Edition, John Wiley & Sons, Inc. New York.

How is KIc experimentally developed?


Two ASTM standard compact specimen of different b sizes. Source: Mechanical Behaviors of Materials, Prentice-Hall, Inc., New Jersey.

There are many different experiments which can be run in order to obtain a value for KIc. Almost any size and shape sample can be used, as long as it is consistent with mode I crack displacement. A possible, an rather simple, experiment that can be performed to find a value for KIc involves a sample, with known dimensions, similar to the one shown in Figure 6 and a screw-driven universal testing machine. This testing machine loads the specimen, at a constant strain rate, while a Load vs. Displacement curve is plotted by a X-Y recorder. From this plot, a possible value for can be calculated. With this value for , KIc can be calculated. For a more in depth explanation of this specific experiment, please refer to Experiment 6 in the Laboratory Manual for ESM 3060 (Prof. Duke) (5).


What is the effect of microstructure, temperature, thickness, and crack size and location on KIc?

As stated in the previous sections, many factors figure into the value of KIc for a given material. By definition (Equation 2), KIc varies with crack size and location, because crack size and location both factor into the value for a. As shown by Figure 4, KIc does not change with sample thickness. This non-variance is due to the condition of plane strain that the sample must be under to have a value for KIc. KIc does vary with temperature and strain rate.


Fracture toughness vs. temperature for various steels. Source: Mechanical Behaviors of Materials, Prentice-Hall Inc., New Jersey.


Fracture toughness vs. temperature for various strain rates applied to A572 steel. Source: Mechanical Behaviors of Materials, Prentice-Hall Inc., New Jersey.

It can be shown, by Figures 7 and 8, KIc increases with decreasing strain rate and increasing temperature. General strengthening methods, such as solid-solution hardening and strain hardening, tend to increase the samples yield stress, but these procedures commonly lead to a decrease in KIc. Additionally, KIc typically increases with decreasing grain size as composition and other microstructural variables are held constant (6,7,8).

Example Problem

1. If a support beam of 4340 Steel (tempered at 260 C) has an interior crack of length 5 mm, how much stress, σ, can be applied to it before it is expected to fracture?

Solution:

KIc = Y σ SQRT{π a}
50.0 MPa SQRT{m1/2} = 1.00 σ SQRT{π{5*10-3}}         (2)
σ = 564 MPa

From Table 1, KIc for 4340 Steel (tempered at 260 C) is 50.0 MPA m. Then by knowing that Y = 1.00 for an interior crack, solve Equation 2 for σ.

Sources

1. Material Science and Engineering: An Introduction, 3rd Edition, William Calister, John Wiley & Sons, Inc. 1994, p. 193-195.

2. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 329 - 330.

3. Material Science and Engineering: An Introduction, 3rd Edition, William Calister, John Wiley & Sons, Inc. 1994, p. 194-196.

4. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 287-291.

5. ESM 3060 Laboratory Manual, Professor Duke, Virginia Polytechnic Institute and State University Copy Center.

6. Material Science and Engineering: An Introduction, 3rd Edition, William Calister, John Wiley & Sons, Inc. 1994, p. 193-195.

7. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 302-310.

8. Physical Metallurgy Principles, 3rd Edition, Robert E. Reed-Hill and Reza Abbaschian, PWS Publishing Company, 1994, p. 737-738.



Sources for the Figures

Figure 1. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 324.

Figure 2. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 301.

Figure 3. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 324.

Figure 4. Material Science and Engineering: An Introduction, 3rd Edition, William Calister, John Wiley & Sons, Inc. 1994, p. 194.

Figure 5. Material Science and Engineering: An Introduction, 3rd Edition, William Calister, John Wiley & Sons, Inc. 1994, p. 191.

Figure 6. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 304.

Figure 7. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 309.

Figure 8. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice Hall, 1993, p. 311.



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Submitted by Matt McMurtry

Virginia Tech Materials Science and Engineering

http://www.eng.vt.edu/eng/materials/classes/MSE2094_NoteBook/97ClassProj/exper/mcmurtry/www/matt.html

Last updated: 4/25/97