Propellant grain

In one lineThe grain’s shape sets the thrust curve; its rubbery mechanics decide whether it survives being cold.

Before thisHow a solid rocket motor works · Case sizing

Why this matters

The grain is the motor’s fuel tank, combustion chamber wall and throttle all at once. Its geometry fixes how much burning surface is exposed at every instant, and therefore chamber pressure and thrust for the whole burn. It also fixes how much propellant fits in the case and how high the peak pressure is — which is what the case has to be sized for.

Its structural integrity is a safety issue, not a performance one. A crack is new burning surface. More surface means more gas, higher pressure, and — because pressure amplifies changes by $1/(1-n)$ — potentially a case failure. That is why cold-soak cracking and debonding are taken so seriously.

This page has two halves: geometry and burn-back, then structural integrity.

Part 1 — Geometry and burn-back

The burning surface moves like a grass fire

Over most of a burn, every exposed point of propellant recedes perpendicular to the local surface at the same burn rate $r$ . After burning a distance $y$ (the web burned), the surface is simply every point that is a distance $y$ from the original surface.

Two consequences follow directly:

Port corners that point into the propellant (star tips, slot ends) round off into arcs as they burn.
Propellant ridges that point into the port (between star points) stay sharp and recede quickly, because they burn from both sides. When those ridges burn out, burning area drops sharply.

Words you need

Term	Meaning	Why it matters
Web, $w$	Distance the flame burns before first reaching the case	Sets burn time: $t_b \approx w / \bar r$
Web fraction	$w$ divided by the grain outer radius	High web fraction = long burn, more propellant, but more structural strain
Volumetric loading	Propellant volume ÷ volume available inside the case	How much of the case you actually filled with propellant
Sliver	Propellant left when the flame first touches the case	Burns in a low-pressure tail-off, contributes little useful thrust
$K_n$	Burning area ÷ throat area	Sets chamber pressure
Port ÷ throat area	Flow area down the bore relative to the throat	Too small means fast bore flow and erosive burning

Shape sets the thrust curve

Port shape sets the thrust–time curve

Sketches, not results — proportions matter as much as the family. A star can be tuned progressive, neutral or regressive by its point count, depth and angle. Use the simulator below to see real numbers.

Tube (internal-burning cylinder). The port only gets bigger, so burning area and thrust rise all the way: progressive. Simple and robust, but the peak pressure at the end of the burn sizes the case.
Star. The points’ area shrinks while the circular region grows, which can be balanced to give near-neutral thrust. The price is a sliver left behind the star valleys, unless the design adds fillets or sliver-filling inserts.
Core plus radial slots (wagon wheel, and the slotted section of a finocyl). Very large initial burning area for a high-thrust boost, usually regressive as the slots open up.
End burner. Burns like a cigarette, face-on: perfectly neutral, the longest burn time and the most propellant for the case, but low thrust — and the case wall behind the face sees hot gas for the whole burn.
BATES (stacked short tubes, ends burning). The bore grows while the ends shrink inward, roughly cancelling.

That BATES balance is a good example of the geometry reasoning. For a segment of outer diameter $D$ , port $d$ and length $L$ , with both ends burning, the burning area after web $y$ is:

A_b(y) = \underbrace{\pi (d + 2y)(L - 2y)}_{\text{bore}} + \underbrace{2 \cdot \frac{\pi}{4}\left(D^2 - (d+2y)^2\right)}_{\text{two ends}}

Setting the area at ignition equal to the area at burnout ( $d + 2y = D$ ) gives the classic sizing rule:

L = \frac{3D + d}{2}

With that length the trace starts and ends at the same level, with a gentle hump in between.

From burning area to thrust

Combine the burning area with the pressure balance from the fundamentals page. At every web position $y$ :

p_c(y) = \left(\rho_p\,a\,c^*\,\frac{A_b(y)}{A_t}\right)^{\frac{1}{1-n}} \qquad \frac{dt}{dy} = \frac{1}{a\,p_c^{\,n}} \qquad F = C_F\,p_c\,A_t

March through the web, accumulate time, and you have the pressure and thrust traces. One useful check falls out: because mass flow through the throat is $p_c A_t / c^*$ , the total impulse is

I_t = \int F\,dt = C_F\, c^*\, m_p

whatever the grain shape. Geometry changes when the impulse is delivered, not how much there is.

Simulator

The simulator finds the burn-back by computing, for every point in the cross-section, its distance to the original port outline — the grass-fire rule, done numerically. Try the presets, then change one dimension at a time.

Grain burn-back & thrust profile

Thrust profile: —
Burn time: —
Total impulse: —
Main-burn avg / peak thrust: —
Main-burn avg / peak p_c: —
Propellant mass: —
Volumetric loading: —
Web fraction: —
Sliver: —
A_b max ÷ min: —
Throat Ø · K_n start: —
Port ÷ throat area: —

Burning surface area vs web burned

Chamber pressure and thrust vs time

Quasi-steady 0-D model: uniform burn rate everywhere, no erosive burning, ignition transient, throat erosion or nozzle losses. Star points are sharp (real stars have fillet radii that change the sliver). Good for understanding and early layout; real motors are designed with full internal-ballistics codes and test data.

Things to try:

Tube → star. Same propellant, same case: the star flattens the thrust curve but leaves a sliver. Watch the burn map — the sliver is the propellant left between the star valleys and the case when the flame first touches the wall.
Make the star points deeper (larger tip diameter) and watch the profile tip from neutral into regressive.
BATES: change the segment length away from $(3D+d)/2$ and see the trace tilt.
Shrink the tube port to 50 mm and watch the port-to-throat warning appear.

Worked example — tube grain by hand

Grain outer diameter 150 mm, port 80 mm, length 600 mm, ends inhibited. Propellant as the defaults: $a = 4.05$ mm/s at 1 MPa, $n = 0.35$ , $\rho_p = 1770$ kg/m³, $c^* = 1540$ m/s, $C_F = 1.5$ . Throat sized so peak pressure is 7 MPa.

Burning area. Start: $\pi \times 80 \times 600 = 150{,}800$ mm². End (port reaches 150 mm): $\pi \times 150 \times 600 = 282{,}700$ mm². Ratio 1.875 — progressive.
Throat. Peak pressure is at the end. $K_n = 7^{0.65} \times 10^9 / (1770 \times 4.05 \times 1540) = 320.9$ , so $A_t = 282{,}700 / 320.9 = 881$ mm² and $D_t = 33.5$ mm.
Starting pressure. $p \propto A_b^{1/(1-n)}$ , so $p_0 = 7 \times (80/150)^{1.538} = 2.66$ MPa. A 1.9× swing in area becomes a 2.6× swing in pressure.
Burn time. Burn rate at the end is $4.05 \times 7^{0.35} = 8.00$ mm/s, and $r \propto (\text{port radius})^{n/(1-n)}$ . Integrating $dt = dy/r$ from a port radius of 40 mm to 75 mm gives $t_b = 5.11$ s.
Mass and impulse. $m_p = \frac{\pi}{4}(150^2 - 80^2) \times 600 \times 1770 \times 10^{-9} = 13.4$ kg, and $I_t = 1.5 \times 1540 \times 13.4 = 31.0$ kN·s, so the average thrust is about 6.1 kN.

The simulator’s tube preset reproduces these numbers to within about 1%. The automated tests check both.

Part 2 — Structural integrity

The material: a filled rubber with a memory

Composite propellant is around 85–90% solid particles (oxidiser and aluminium) held in a rubber binder. Mechanically it behaves like a viscoelastic solid: its stiffness and strength depend on how fast it is loaded and how cold it is.

Three facts drive everything below:

Propellant is nearly incompressible. It changes shape easily but barely changes volume (bulk modulus is thousands of times its shear modulus). Wherever its volume wants to change, the geometry decides where the movement goes.
Relaxation lowers stress, not strain. In a case-bonded grain the bore strain is set by geometry and expansion coefficients, not by stiffness. Slow cool-downs relax the stress away, but the strain stays.
Capability depends on rate, temperature, pressure and age. There is no single “strain to failure”. Each load case is compared with the capability measured at its own rate and temperature, and capability falls as the propellant ages.

The load cases

Load case	What happens	Typically governs
Cure cool-down and cold storage	Propellant tries to shrink far more than the case it is bonded to	Bore cracking, bond tension
Temperature cycling	Repeated cool-down strains	Cumulative damage over the service life
Ignition pressurisation	Case dilates; incompressible grain amplifies it at the bore	Bore cracking at cold firing
Flight acceleration, vibration	Inertial loads on a soft grain	Bond shear at the ends, slender grains
Long-term storage (slump)	The grain sags under its own weight for years	Large motors, bore deformation

Cool-down: why the bore cracks

Propellant is cured warm and bonded to the case at that temperature, where it is stress-free. Its thermal expansion coefficient is around $10^{-4}$ /K — roughly eight times steel’s — so when it cools it tries to shrink far more than the case does. The outside is glued to the case and can’t follow. Because it can’t change volume either, the shrinkage has to come out of the bore, which is pulled open.

What the grain feels — deformation exaggerated

The dashed circle is the bore at cure. In both loads the outer surface is held by the case, and because propellant is nearly incompressible, the only place the volume change can go is the bore.

For a long, case-bonded cylindrical grain with bore radius $a$ and outer radius $b$ , write $m = b/a$ . Incompressibility means the area of each ring of propellant can’t change except by thermal expansion, which fixes the displacement field. Applying the case boundary condition gives the bore hoop strain on cooling by $\Delta T$ (negative for cooling):

\varepsilon_{\theta}(a) = \Delta T \left[\frac{3\alpha_p - \alpha_c}{2}\,(1 - m^2) + m^2\,\alpha_c\right] \;\approx\; \tfrac{3}{2}\,(\alpha_p - \alpha_c)\,|\Delta T|\,(m^2 - 1)

The approximate form is within $\alpha_c |\Delta T|$ (about 0.1%) of the full one. The key term is $m^2$ : bore strain grows with the square of the outer-to-bore ratio. A grain with a small bore and a thick web — which is exactly what you want for propellant loading — is the hardest to keep from cracking. That is the central structural trade in grain design.

The same analysis gives the radial tension pulling the grain off the case at the bond, using the propellant’s relaxed modulus $E$ for this slow load:

\sigma_r(b) = E\,(m^2 - 1)\,(\alpha_p - \alpha_c)\,|\Delta T| \;\approx\; \tfrac{2}{3}\,E\,\varepsilon_\theta(a)

A soft propellant keeps bond stress low; it does nothing for bore strain.

Ignition: the case stretches, the bore stretches more

At ignition, pressure pushes the grain against the case and the case dilates by a fraction of a percent. The propellant can’t change volume much, so that small outward movement at radius $b$ has to be matched by a much larger movement of the bore:

\varepsilon_{\theta}(a) = m^2\,\varepsilon_{\theta,\text{case}} + \frac{m^2 - 1}{2}\left(\varepsilon_{z,\text{case}} + \frac{p}{K}\right)

where $K$ is the propellant bulk modulus, and for a thin closed-end case of modulus $E_c$ , Poisson’s ratio $\nu_c$ and wall $h$ :

\varepsilon_{\theta,\text{case}} = \frac{p\,b}{E_c\,h}\left(1 - \frac{\nu_c}{2}\right) \qquad \varepsilon_{z,\text{case}} = \frac{p\,b}{E_c\,h}\left(\frac{1}{2} - \nu_c\right)

This links the grain directly to case sizing: a lighter, more highly stressed case stretches more, and the grain pays for it at the bore. The worst case is usually a cold firing, where the cool-down strain is already there when ignition adds its share.

Calculator

Case-bonded grain — structural screenlong cylinder · incompressible propellant

m = b / a: —
Web fraction: —
Case hoop strain at MEOP: —

Bore hoop strain and margin

Cool-down: —
Pressurisation: —
Cold ignition (sum): —

Case bond, cool-down

Radial tension at bond: —

Bore strain vs web fraction for this propellant, case and temperature. The dot is your design.

Closed-form screening (generalised plane strain, elastic, incompressible except for the bulk-modulus term in pressurisation). Ignores grain ends, stress-relief flaps, gravity slump, acceleration, thermal gradients, ageing and cumulative damage. Default capabilities are an illustrative curve from REF·DB v1, not data for any real propellant.

Worked example — HTPB grain in the D6AC case

The same 300 mm D6AC case as the case-sizing worked example (wall 1.377 mm, MEOP 10 MPa), with a 100 mm circular bore ( $m = 3$ ). Composite propellant with $\alpha_p = 100 \times 10^{-6}$ /K, relaxed modulus 3.6 MPa and bulk modulus 3.5 GPa; steel $\alpha_c = 12 \times 10^{-6}$ /K. Stress-free at 60 °C, coldest firing at −40 °C, so $\Delta T = -100$ K.

Cool-down strain. $100 \times \left[\frac{300 - 12}{2} \times 8 - 9 \times 12\right] \times 10^{-6} = 10.4\%$ .
Case strain at MEOP. $\frac{p b}{E_c h} = \frac{10 \times 150}{205{,}000 \times 1.377} = 0.531\%$ , so hoop $0.452\%$ and axial $0.106\%$ .
Ignition strain. $9 \times 0.452\% + 4 \times (0.106\% + 10/3500) = 4.07\% + 1.57\% = 5.6\%$ .
Cold ignition. $10.4\% + 5.6\% = 16.1\%$ . Against the illustrative 39.6% capability with a safety factor of 1.5: margin $= 39.6 / (1.5 \times 16.1) - 1 = +0.64$ .
Bond tension. $3.6 \times 8 \times 88 \times 10^{-6} \times 100 = 0.25$ MPa.

Now set the concentration factor to 2.5, roughly what a star point might add. The cold-ignition strain becomes 40% and the margin goes negative. The same grain that is comfortable with a circular bore fails with a star — which is why star points get generous fillet radii and their own detailed analysis.

Design levers

Larger bore / smaller web — the most powerful lever, because strain goes as $m^2$ . It costs propellant loading.
Stiffer or thicker case — lowers ignition strain directly. Composite cases are light but stretch more.
Raise the minimum firing temperature — every degree of $\Delta T$ removed is strain removed. Often a system-level negotiation.
Lower the stress-free temperature — curing cooler, or curing under pressure, reduces the cool-down $\Delta T$ .
Stress-relief flaps and boots at the grain ends, where the bond terminates and stresses are far higher than the long-cylinder values above.
Cartridge-loaded grains (made separately and slid into the case) avoid the bonded cool-down problem entirely, at the cost of support and inhibition complexity. Common for double-base grains, which go glassy at low temperature.

Where these formulas stop

Ends and terminations. Bond stresses at the grain ends can be several times the values above.
Non-circular ports. Star points, slot tips and fin roots need FEA. The concentration factor here is a placeholder.
Dewetting. At high strains the binder separates from the solid particles and voids open up. The material is then no longer incompressible, and its response changes.
Cumulative damage. Repeated thermal cycles each use up some capability. Programmes track this with cumulative damage models, and through surveillance testing of aged motors.

Differences from REF·DB v1

On the drawing

Grain geometry — port profile, fillet radii, web, and the tolerances on each. Fillet radii at star points and slot tips are structural features, not cosmetic ones.
Liner, insulation and inhibitor — specifications, thicknesses, and which surfaces are inhibited.
Bond and void acceptance — the NDT method (typically radiography or CT for voids, cracks and debonds) and its acceptance criteria.
Stress-relief features — flap geometry and the release treatment that lets them separate.
Temperature limits — storage and firing temperatures belong in the motor specification, because the structural margin depends on them.

References and further reading

NASA SP-8073, Solid Propellant Grain Structural Integrity Analysis (1973) — the open reference for this half.
NASA SP-8076, Solid Propellant Grain Design and Internal Ballistics (1972).
G. P. Sutton & O. Biblarz, Rocket Propulsion Elements, 9th ed., Wiley (2017) — chapters on solid propellant rocket fundamentals and grain design.
A. Davenas (ed.), Solid Rocket Propulsion Technology, Pergamon (1993).
M. L. Williams, P. J. Blatz & R. A. Schapery, Fundamental Studies Relating to Systems Analysis of Solid Propellants, GALCIT SM 61-5, California Institute of Technology (1961).
openMotor — an open-source solid motor internal-ballistics simulator that uses the same distance-map approach to burn-back.