Propellant Type | Elastic Modulus (psi) |
PVC/AP/Al | 250 |
Polysulfide/AP/Al | 3000 |
PBAN/AP/Al | 3300 |
HTPB/AP/Al | 2000 |
Polyurethane/AP/Al | 20 000 |
The cast "sugar" based propellants (KN-Sucrose, KN-Dextrose and KN-Sorbitol) all fit into the category of being high-modulus.
Propellant Type | Elastic Modulus (psi) |
KN/Sorbitol | 850 000 |
KN/Dextrose | TBD (similar) |
KN/Sucrose | TBD (similar) |
* Elastic Modulus (E) is defined as E = s / e, where s is stress (force per unit area), and e is strain (change in length divided by original length).
Both failure modes would likely result in catastrophic motor failure due to overpressurization resulting from unexpected increase in burning area.
The likelihood of disbonding failure can be reduced by utilizing a full length grain combined with reinforced casing closures to locally minimize radial expansion due to pressurization. (see Figure 2b).
To prevent radial cracking of the propellant grain due to pressure induced strain (mode 2 failure), it is necessary to limit the strain by using a sufficiently stiff casing. This would normally be achieved by using a casing material of high elastic modulus (such as steel, rather than aluminum) and/or utilizing a sufficiently thick casing wall. But how thick must the casing wall be to sufficiently limit the strain? The change (delta) in casing diameter due to chamber pressure is given by The casing strain,
Consider the following analysis:
Note that the term PD/2t represents casing hoop stress, sc. Considering this, and the fact that the grain must experience the same strain as the casing (as it is bonded to the casing), gives
The poisson ratio for the grain material, ug, is not known. However, for most materials, the value is between 0.25 and 0.33. Assuming that it is similar to that of the casing leads allows this term to drop out of the equation, giving:
This equation reveals that the ratio of the hoop stresses are proportional to the ratio of the elastic moduli.
From this, the maximum allowable casing hoop stress may be calculated, knowing the maximum allowable hoop (tensile) stress of the grain material, as well as the elastic modulus of the grain and casing. Subsequently, the required casing wall thickness may be calculated.
Example:
Consider a proposed case-bonded KN-Sorbitol grain, of 3.0 inch diameter. MEOP of the motor is 800 psi. Determine how thick the casing wall must be to limit the grain tensile stress to 20% of the grain ultimate tensile strength. Both steel and aluminum are being considered for the casing material. Assume that cohesive failure at the bondline will not occur.
For this propellant,
850 000 lb/in2 Elastic modulus
1050 lb/in2 Ultimate tensile stress
0.20 (1050) = 210 lb/in2 Allowable tensile stress
1.Steel casing
Ec = 29 000 000 lb/in2
Maximum allowable casing hoop stress is given by
Therefore, required casing wall thickness is
Such a casing would be rather heavy, having a mass of 0.47 lbs per inch of length.
2. Aluminum casing
Ec = 10 000 000 lb/in2
Maximum allowable casing hoop stress is given by
Therefore, required casing wall thickness is
This casing would also be rather heavy, having a mass of 0.38 lbs per inch of length.
In order to prevent radial cracking of the grain resulting from expansion under pressure, a sufficiently rigid casing must be used, preferably steel. As the examples indicate, the weight penalty of such a thick walled casing is severe.
Care is also essential during casting the grain. Once cast, it is not possible to detect flaws, such as voids or bubbles, which could locally weaken the grain or act as stress raisers.
As such, it would appear that case-bonding of a high-modulus propellant is of questionable practicality.
It is interesting to consider that the objectives sought through case-bonding may well be achieved through other means. For example, to inhibit burning along the outer surface of the grain, the grain may be alternatively cast into a burn-resistant or ablative liner, such as a cardboard tube, of a diameter slightly less than the casing inside diameter. Protection of the casing walls from hot combustion gases may be achieved by a suitable ablative coating (such as polyester or epoxy resin) or a snug-fitting multilayered paper or cardboard liner. With such an arrangement, the grain can be free-standing, and thus subjected to benign compressive stresses only.