What is
fracture toughness K_{c}?
Fracture toughness, K_{c}, 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:
K_{c} = 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.
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 throughthickness surface crack as shown in Figure 2
Y = 0.73 For a halfcircular surface crack as shown in Figure 3
Fracture toughness, K_{c}, has the English customary units
of psi in^{1/2}, and the SI units of MPA m^{1/2}.
What is the
plane strain fracture toughness, K_{Ic}?
For thin samples, the value K_{c} decreases with increasing sample thickness, b, as shown by Figure 4.
Ultimately, K_{c} becomes independent of b, at
this point the sample is said to be under the conditions of
plane strain. This fixed value of K_{c} becomes known as the
plane strain fracture toughness, K_{Ic}. K_{Ic}
is mathematically defined by:
K_{Ic} = 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 K_{c}. The I subscript for K_{Ic}, stands for mode I, or tensile mode, crack displacement as shown in Figure 5(a).
In general, K_{Ic} is low for brittle materials and high
for ductile materials. This trend is supported by the K_{Ic}
values in Table 1 (3,4).
 
 
 

Source: Materials Science and Engineering: An Introduction,
3rd Edition, John Wiley & Sons, Inc. New York.
How is K_{Ic} experimentally developed?
There are many different experiments which can be run in order
to obtain a value for K_{Ic}. 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 K_{Ic} involves a sample,
with known dimensions, similar to the one shown in Figure 6 and
a screwdriven universal testing machine. This testing machine
loads the specimen, at a constant strain rate, while a Load vs.
Displacement curve is plotted by a XY recorder. From this plot,
a possible value for can be calculated. With this value for ,
K_{Ic} 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 K_{Ic}?
As stated in the previous sections, many factors figure into the value of K_{Ic} for a given material. By definition (Equation 2), K_{Ic} varies with crack size and location, because crack size and location both factor into the value for a. As shown by Figure 4, K_{Ic} does not change with sample thickness. This nonvariance is due to the condition of plane strain that the sample must be under to have a value for K_{Ic}. K_{Ic} does vary with temperature and strain rate.
It can be shown, by Figures 7 and 8, K_{Ic} increases with decreasing strain rate and increasing temperature. General strengthening methods, such as solidsolution hardening and strain hardening, tend to increase the samples yield stress, but these procedures commonly lead to a decrease in K_{Ic}. Additionally, K_{Ic} 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:
K_{Ic} = Y σ SQRT{π a}
50.0 MPa SQRT{m^{1/2}} =
1.00 σ SQRT{π{5*10^{3}}}
(2)
σ = 564 MPa
From Table 1, K_{Ic} 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.
193195.
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.
194196.
4. Mechanical Behavior of Materials: Engineering Methods for
Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice
Hall, 1993, p. 287291.
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.
193195.
7. Mechanical Behavior of Materials: Engineering Methods for
Deformation, Fracture, and Fatigue, Norman E. Dowling, Prentice
Hall, 1993, p. 302310.
8. Physical Metallurgy Principles, 3rd Edition, Robert
E. ReedHill and Reza Abbaschian, PWS Publishing Company, 1994,
p. 737738.
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.
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