Fracture Toughness and Fracture Design

Overview:
One of the most important properties of any material for virtually all design applications is fracture toughness. Given the unusual units of MPa(m1/2), fracture toughness is a quantitative way of expressing "a material's resistance to brittle fracture when a crack is present." [5]. If a material has a large value of fracture toughness it will probably undergo ductile fracture. Brittle fracture is very characteristic of materials with a low fracture toughness value.

Differences between Fracture Toughness:
There are actually four different types of fracture toughness, KC, KIC, KIIC, and KIIIC. KC is used to measure a material's fracture toughness in a sample that has a thickness that is less than some critical value, B. When the material's thickness is less than B, and stress is applied, the material is in a state called plane stress. The value of B is given in equation Te-1. A material's thickness is related to its fracture toughness graphically in figure Tf-1. Equation Te-2 shows a material's KC value in relation to the material's width.

Eq. Te-1 [1]. The minimum thickness of material before plane strain behavior occurs.
B = minimum thickness to distinguish between KC and K1C
KC = fracture toughness, when the sample has a thickness less than B
sy = yield stress of material

Figure Tf-1 [1]. Fracture Toughness as a function of material thickness

Eq. Te-2 [1]. The fracture toughness of a material with a thickness less than B.

KC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material

KIC, KIIC, and KIIIC all represent a material's fracture toughness when a sample of material has a thickness greater than B. If a stress is applied to a sample with a thickness greater than B, it is in a state called plane strain. The differences between KIC, KIIC, and KIIIC, however, do not depend on the thickness of the material. Instead, KIC, KIIC, and KIIIC are the fracture toughness of a material under the three different modes of fracture, mode I, mode II, and mode III, respectively. The different modes of fracture I, II, and III are all graphically expressed in figures Tf-2, Tf-3, and Tf-4. Equation, Te-3 shows how KIC can be calculated knowing the material's parameters.

Figure Tf-2 [1]. Mode I Fracture

Figure Tf-3 [1]. Mode II Fracture

Figure Tf-4 [1]. Mode III Fracture

Eq. Te-3 [1]. The fracture toughness of a material with a thickness equal to or greater than B; when it fractures in mode I.

KIC = fracture toughness, when the sample has a thickness greater than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material

KIC values can be used to help determine critical lengths given an applied stress; or a critical stress values can be calculated given a crack length already in the material with equations Te-4 and Te-5.

Eq. Te-4 [1]. Critical applied stress required to cause failure in a material.
sC = critical applied stress that causes the material to fail
KIC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)

Eq. Te-5 [1]. Critical crack length required to cause failure in a material.
a = critical crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material
KIC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry

Callister provides a table, Table Tt-1, of fracture toughness of common engineering materials.

KIC values for Engineering Materials
 Material K1C MPa (m)1/2

 Metals Aluminum alloy 36 Steel alloy 50 Titanium alloy 44-66 Aluminum oxide 14-28

 Ceramic Aluminum oxide 3-5.3 Soda-lime-glass 0.7-0.8 Concrete 0.2-1.4

 Polymers Polymethyl methacrylate 1 Polystyene 0.8-1.1

Table Tt-1 [1].

Students may notice that the ceramic materials have a much lower KIC value than the metals. The low KIC value reflects the fact that ceramic materials are very susceptible to cracks and undergo brittle fracture, whereas the metals undergo ductile fracture.

Example Problem : Fracture Toughness
Plates of a ceramic material called boron carbide are being used to absorb neutrons in a nuclear reactor. Boron carbide has a KIC of approximately 4 MPa(m)1/2 and a high yield strength of 400 MPa. The compressive stresses being applied to the boron carbide plates, only 10-3 m thin, are about 5 MPa; Y = 1.1. What is the critical crack length in the boron carbide plates?

Step 1) Are the ceramic plates thick enough to be in plain strain when a stress is applied to them?
B = 2.5 (KIC / sy)2
B = 2.5 (4 MPa(m)1/2 / 400 MPa)2
B = 2.5*10-4 m
The width of the material is greater than 2.5*10-4 m, therefore plane strain applies to the problem

Step 2) Plug the known and calculated values into the equation given and calculate the critical crack length.
ac = (1 / Pi) (4 MPa(m)1/2 / (5 MPa * 1.1)2
ac = 0.168 m

Fracture Design
Sources

Submitted by Matt Gordon

Virginia Tech Materials Science and Engineering

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

Last updated: 4/25/97