The estimate of Frenkel considers two rows of atoms that shear past one another. The spacing between rows is a
r
and the
spacing between atoms in the slip direction is a
0
. The shear stress is and is considered to be sinusoidal. The well-known
result is:
= B/A( /2 ) SIN(2 X/B)
(EQ 8)
where μ= shear modulus. The maximum value, which is also the point at which the lattice is mechanically unstable and
slip occurs, is σ= b/a(μ/2π). Because a b, the theoretical shear strength is σ
theo
/2π, which is several orders of
magnitude greater than the value usually observed for soft crystals. There have been numerous variations and
improvements to Eq 8 in an effort to improve predictions of material strength, but the result remains essentially the same.
Unfortunately, the strength of a given material predicted by theoretical calculations is much larger than the observed
strength. The question is "why?" Certainly it is important that slip in crystals occurs well below the ultimate stress and
that slip occurs by the movement of dislocations, as postulated by Taylor (Ref 12), Orowan (Ref 13), and Polanyi (Ref
14). But these observations do not completely answer the question, and we are led to search for other reasons for
weakness.
In looking for points of weakness, we begin by noting that pure metals by definition contain no alloying constituents (and
may be single crystals or polycrystalline), while structurally useful materials generally contain alloying constituents for
strengthening and may be precipitation hardening, such as many of the aluminum alloys, but may also contain larger
second-phase particles. Structural metals may also contain multiple phases, such as the ferrous alloys do, and have grain
boundary phases as well as phases within the grain interior. A method that has been used to classify materials as to their
mode of failure is that of structure. Shown below are some material properties and their effect on fracture behavior (Ref
15):
Physical property Increasing tendency for brittle fracture
Electron bond Metallic Ionic Covalent
Crystal structure Close-packed crystals Low-symmetry crystals
Degree of order Random solid Short-range order Long-range order solution
For the different classes of materials, crystal structure is of fundamental importance because it influences or determines
the competition between flow and fracture. For example, polycrystals of copper are invariably ductile, while magnesium
polycrystals are relatively brittle. Magnesium has a close-packed hexagonal crystal structure, with parameters of a =
3.202 , c = 5.199 , and c/a = 1.624 (which is very close to the ratio of 1.633 obtained by piling spheres in the same
arrangement). This structure is basic to much of the physical metallurgy of magnesium and magnesium alloys. At room
temperature, slip occurs mainly on (0001) (<11 0>), with a small amount sometimes seen on pyramidal planes such as
(10 1) <11 0>. As the temperature is raised, pyramidal slip becomes easier and more prevalent. However, note that the
slip directions, whether associated with basal or the pyramidal planes, are coplanar with (0001), a general observation for
all observed slip in magnesium and magnesium alloys. Therefore, it is impossible for a polycrystalline piece of
magnesium to deform without cracking unless deformation mechanisms other than slip are available. These mechanisms
are twinning, banding, and grain-boundary deformation.
At the microstructural level, fracture in engineering alloys can occur by a transgranular (through the grains) or an
intergranular (along the grain boundaries) fracture path. However, regardless of the fracture path, there are essentially
only four principal fracture modes:
• Ductile fracture from microvoid coalescence
• Brittle fracture from cleavage, intergranular fracture, and crazing (in the case of polymers)
• Fatigue
• Decohesive rupture
These basic fracture modes are discussed in more detail in the article "Micromechanisms of Monotonic and Cyclic Crack
Growth" in this Volume (with somewhat more emphasis on cleavage than in this article). Cleavage is perhaps more
related to the rapidity of fracturing, as suggested by Irwin's classic paper (Ref 7).
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