Structural analysis of solid rocket motors is challenging for several reasons, but the most important of these is the complex behavior of the propellant. The mechanical response of a solid propellant is time and temperature dependent. The complexity of the mathematical analysis of the propellant depends on the loading conditions, but for some loading situations, the linear viscoelasticity assumption is reasonable. In particular, linear viscoelasticity is perhaps the most appropriate material behavior description for use in the simulations of stresses related to storage conditions. Typically, simulations use a viscoelastic model in the form of a Prony series and a Williams–Landel–Ferry (WLF) equation. The parameters in these models are derived from stress relaxation experiments, making the stress relaxation experiment a key viscoelastic test, analogous to the tensile test for linear elastic materials.

A typical set of stress relaxation tests is performed at several discrete temperatures that cover a range of temperatures anticipated by the fielded motor. At each of the selected temperatures, the specimen is deformed with approximately a single step in strain, which is then held constant for the duration of the test. While held at this constant strain, the stress decays over time due to relaxation of the rubbery elastomer. During this portion of the test, the stresses are measured, and the ratio of stress to applied strain is determined. This ratio is termed the stress relaxation modulus ER. Using time–temperature superposition, the set of curves at the various temperatures can be shifted horizontally relative to each other to form a master curve. The translation of the curves takes a specific mathematical form, viz., the WLF equation. From this master curve, the Prony series at any given temperature can be calculated, and the calculation can be incorporated into finite element analyses along with the WLF equation, making linear viscoelastic analysis of rocket motors possible.

The Prony series is a common method of approximating the behavior of a viscoelastic material. Various algebraic representations are available, but one of the simplest forms is given by:

Here, ⊺ refers to reduced time, meaning the test time divided by the horizontal shift factor a**T.** The variable E**R** is the relaxation modulus, E_{∞} is the long-term relaxation modulus, and the *n* exponential terms each have a coefficient α**i** and an exponential constant τ_{i}. The relaxation modulus is therefore represented as a sum of a series of exponential terms, each with its own time constant, so that the entire spectrum of relaxation times and temperatures is well represented.

To determine the unknown parameters E_{∞}, α_{i}, and τ_{i} (i=1, 2,…, n), the horizontal shift function is first determined using the WLF equation. This is necessary to obtain the reduced time ⊺ then, numerical methods are employed to derive the Prony series parameters. Typically, E_{∞} is estimated from the data, and then the time constants τ_{i} are chosen arbitrarily, but each a decade apart in time. Next, a least-squares fit of the master curve data to the above equation determines the unknown values for the coefficients α_{i}. These parameters are then incorporated into a finite element program for subsequent analysis of solid rocket motor grains.

The horizontal shift factor a_{T} is determined by the Williams–Landel–Ferry equation:

Here, the horizontal shift factor a_{T} is determined from the shifting of the individual stress relaxation curves relative to a reference curve (in this case, the curve for 20°C), T is the test temperature, T_{ref} is the temperature of the reference curve, and C_{1} and C_{2} are parameters determined from a least-squares fit of the horizontal shift data.

Stress relaxation tests of propellant are currently performed on standard tensile testing machines, and the specimens are subjected to uniaxial stress conditions. Because the propellant is very compliant, the applied strain has to be large; otherwise, the loads would be too small to measure accurately, unless an atypical load cell were used (i.e., an atypical load cell that is not very compatible with commercial tensile testing hardware). Typically, the applied strains will be in the 1–10% range. At these strains, damage is likely occurring due to dewetting of the particulate matter. Therefore, specimens are only used at a single temperature. Because of these considerations, generating a typical master curve requires 18–30 specimens (assuming three to five specimens per temperature and six temperatures).

Specimens were also tested in a dynamic mechanical analyzer, using a dual cantilever beam mode. While the results were similar, the dynamic mechanical analyzer required less material, resulted in reduced variability, and was not sensitive to the applied strain. The quantity of material required was on the order of grams, so that results were obtained with small amounts of propellant, as compared to the conventional uniaxial tension test that requires material quantities on the order of kilograms.

*This work was done by Timothy C. Miller, C. S. Wojnar, and J. A. Louke for the Air Force Research Laboratory. AFRL-0253*

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###### Measuring Propellant Stress Relaxation Modulus Using Dynamic Mechanical Analyzer

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