Ioan D. Marinescu
2.1. Deformation
When a load is applied to a material, deformation occurs because of a slight change in the atomic spacing. The load is defined as stress (s), and it is typically measured in units of pounds per square inch (psi) or megapascals (Mpa). The deformation is defined as strain (): measured in inches (or centimeters) of deformation per inches (or centimeters) of the initial length or in percent.
Figure 2.1 shows the typical ceramic materials fracture in a brittle mode with only elastic deformation prior to fracture. Typical metals fracture in a ductile mode with initial elastic deformation followed by plastic deformation.
Figure 2.1Types of stress-strain behavior:
(a) brittle fracture typical of ceramics.
(b) plastic deformation with no distinct yield point.
(c) plastic deformation with yield point
[1] 2.2. Dislocation
The mechanism of plastic deformation involves movement of dislocations. A dislocation is a defect in the way planes of atoms are stacked in a crystal structure. There are two types of dislocation: Edge Dislocation and Screw Dislocation.
Edge Dislocation
Edge Dislocation: a partial plane of atoms terminated within the crystal structure.
Screw Dislocation
Screw dislocation: produces a line of discontinuity in the crystal structure.
Under an applied load, these types of dislocations can form and multiply leading eventually to the fracture of the ceramic material.
The criteria for plastic deformation are the same as for metals, as follows:
• The presence of dislocations;
• The mechanism of generation of new dislocations under an applied load; and
• A path along which the dislocations can move.
Ceramics are known as very brittle materials. The phenomena of plastic deformation due to the dislocation activity are limited to very high temperatures especially for polycrystalline ceramics. At room temperatures, the random orientation of the grains severely inhibits dislocation motion (which terminates at the grain boundaries). The two mechanisms for plastic deformation are slip and twinning.
2.3. Slip mechanism
Usually slip in ceramic materials occurs on two different slip systems. In most cubic metals, the crystallographically equivalent slip systems are sufficiently numerous to permit complete flexibility. This means that glide of dislocations on slip planes can produce all the strain components and thus produce any change of shape. In ceramics, a minimum of five independent systems is needed to permit an arbitrary change of shape.
Figure 2.2Simple schematic illustrating an edge dislocation and showing that the displacement b (Burgers vector) is equal to one unit cell edge [1] Figure 2.3Simple schematic illustrating a screw dislocation [1] In ceramics with ionic bonding, the slip systems depend not only on the crystal structure but also on the ionic positions in the dislocation core. Ceramic materials do not exhibit dislocation mobility at room temperatures. But at high temperatures, limited plasticity is possible, and this may have several origins including dislocation motions, grain boundary sliding, or softening of minor faces. Kronberg (1957) has discussed dislocation motion for alumina in detail. The unit cell for the alumina lattice is made up of six layers of ions parallel to the basal planes that are comprised alternately of aluminum and oxygen ions. All sites are filled on the oxygen planes; in the aluminum planes only two thirds of the sites are occupied. The unoccupied sites (or holes) are arranged regularly but differently on each of the three aluminum planes in the unit cell.
Figure 2.4 shows two of the adjacent basal planes, a completely filled oxygen plane, and partially filled aluminum ions laying in the interstices between the oxygen ions. Deformation of single alumina crystals, at temperatures of 1300 °C or higher, shows that slip can occur on the {0001} basal plane. This is the closest packed plane of alumina, and this represents the expected slip plane. The detected slip direction is <1120>, and it is at 30° to the close-packed direction of the oxygen ions (see
Figure 2.4). These geometrical patterns of both aluminum atoms and holes must be restored after shear. Reference to the holes in
Figure 2.4 shows that for the <1010> direction, the repeat distance for holes is times greater than that in the <1120> direction.
Figure 2.4Oxygen ions (large open circles), aluminum ions (small solid circles), and unoccupied octahedral interstitial sites = holes (small open circles) [11] Grooves and Kelly (1963) developed a procedure for determining the number of independent slip systems.
Slip occurs most easily on the basal slip system when the temperature of operation is the lowest and the tensile flow stress the smallest of the three possible slip systems. To activate slip on the second and third (prismatic and pyramidal) slip system, the temperature must rise significantly. The basal slip system can be activated most easily, followed by the prismatic and pyramidal ones. For instance, at 1500 °C the stresses needed to activate the prismatic and pyramidal slip systems exceed the stress to activate the basal slip system by a factor of 8 and 16 respectively. Thus, it can be concluded that alumina oxide is a ceramic material characterized by a strong deformation anisotropy. To achieve manoscopic deformation of polycrystalline body by slip, five independent slip systems must operate.
2.4. Twinning mechanism
Twinning is an important mode of plastic deformation, and it has been observed both in single crystal and polycrystalline ceramics. As compared to plastic deformation by dislocation glide, twinning occurs at low temperatures and high strain rates. The experiments on polycrystalline alumina between room temperature and about 500 °C have shown twinning to be the predominant mechanism
[2].
Figure 2.5 illustrates a strain-rate temperature map with three important deformation modes:
• At elevated temperatures and small strain rates (0001) slip prevails;
• At low temperatures and high strain rates, rhombohedral twinning predominates; and
• For a very narrow band of experimental parameters there is a range of (0001) twinning.
Figure 2.5Occurrence of twinning and slip as a function of strain rate and temperature [2] 2.5. Fracture of ceramic materials
It is very important to understand the strength of real polycrystalline ceramics. For this, it is necessary to emphasize the behavior of microcrack under stress. A lot of scientists have studied the linear elastic fracture mechanism (LEFM); this refers to a crack in a continuous body without taking into consideration what happens on an atomic scale. Among them are Kelly and MacMillan (1986), B. Lawn (1993), Hasselman, Munz, Sakai and Sherchenko (1992).
Figure 2.6Plate containing elliptical cavity, semi-axes b, c, subjected to uniform applied tension s. C denotes notch-tip [3] Figure 2.7The location of microcracks with respect to the residual stress on the grain facets [1] Inglish (1993) says that, considering a sample under a tensile stress s, if an elliptical flaw is introduced, the tensile stress will change especially near the end of the flaw.
The flaw has the length 2c in the X direction and 2b in the Y direction. The equation of the ellipse is:
2c2+y2b2=1
(2.1)
and ? - the radius of the curvature is:
=b2c
(2.2)
The stress distribution around the flaw is very...
