17 Waqas Ahmed

Figure 1. Representation of π/n section geometry of truncated star

with elliptical tip.

FEA Based Stress/Strain Modeling & Simulation of Ballistic Optimized Truncated Star Geometry: Comparison of Circular and Elliptical Star Tip

Waqas Ahmed, Qamar Nawaz and Farooq Nizam Centres of Excellence in Science and Applied Technologies (CESAT), Islamabad, Pakistan

Abstract—Structural integrity of the propellant grain is exceptionally significant for the successful operation of the solid rocket motor. It is well accepted that the elliptical tip of the star grain configuration is an excellent substitute of the circular tip as it ameliorates the structural integrity by minimizing the stresses and strains that occur in the propellant grain. In this paper, an FEA based stress/strain modeling and simulation is employed for the comparison of the ballistic optimized star geometry with circular and equivalent elliptical tip. First, the techniques of calculus and trigonometry are employed to derive analytical expressions for burn perimeter and port area for different burning zones of the truncated star with elliptical tip. Then, ballistic performance of truncated star elliptical tip is compared with that of circular tip. Finite element Analysis (FEA) employing ABAQUS-CAE standard modeling and analysis software was used to assess the viscoelastic response of star grain geometry subjected to ignition pressurization. Various elliptical ratios for star tip were assessed to find out the optimum one. Results were compared with the analytically calculated stress concentration factors for the same geometries.

Keywords: Solid Rocket Motor, Finite element analysis, Structural Integrity, Visco-elasticity, Star Grain Geometry.

I. INTRODUCTION The quest of optimized grain design prerequisites the

evaluation of large numbers of grain designs. The availability of analytical expressions for burn perimeter and port area as a function of burn distance makes this search much more efficient as it obviate the long computation time. Hartfield et al [1] have provided a review for a range of common solid rocket motor grains. The comprehensive text available on solid rocket propulsion [2, 3] is devoid of analytical expression of burn perimeter and port area of propellant grain. Very few geometric details are presented in NASA monographs [4, 5].

The method for determination of burn perimeter and port area of truncated star with classical circular tip is found in literature [1]. This work focuses on the development of analytical expressions for burn perimeter and port area as a function of burn distance for truncated star with elliptical tip. The introduction of elliptical tip instead of classical circular tip in truncated star tends to minimize the stresses and strain that occur in the propellant grain [6]. Hence, it can be deduced that the propellant grain with reduced stresses and strain would have a long storage life and comparatively easy handling and transportation. The use of truncated star with circular

tip or elliptical tip in the fin portion of the finocyl grain [7] further enhances the significance of the work presented here.

II. MATHEMATICAL EXPRESSIONS FOR ELLIPTICAL STAR GEOMETRY

Fig. 1 depicts the star geometry. The parameters used to define the truncated star with elliptical tip are:

• Symmetry number, n • Grain outer radius, R • Web, w • Inner radius, r • Half star valley angle, • Ellipse major axis, a • Ellipse minor axis, b

Based on the burning sequence, there are at least four

types of truncated star [1]. The type described here has the following burning sequence. The first zone ends when the innermost arc vanishes. The diminution of line segment to zero finishes the second zone whereas the third zone lasts till the web. The fourth zone is the sliver region. The relations for other three types can be developed using the method adopted here.

A. Initial Burn Perimeter and Port Area: For initial perimeter, an angle ‘ϕ’ is defined as,

Proceedings of International Bhurban Conference on Applied Sciences & Technology,Islamabad, Pakistan, 10 - 13 January, 2011 244

Using the formula for length of arc [8], the length of portion of ellipse AB with center at O1 (Fig. 2) is given by,

where

Initial burning perimeter of entire cross-section is ‘2n’

times the sum of the lengths of an elliptical arc, a line segment and a circular arc, i.e,

Initial port area is ‘2n’ times is the sum of area under an ellipse with center at O1, a rectangle, a triangle and a sector,

Thus total initial port area is:

B. Various Web Burn Zones: Zone-I:

Figure 2. π/n cross section geometry of truncated star with elliptical

tip: web burning zone-I.

As the web burns, the alteration in the position of point of intersection of straight portion of fin and circular arc induces change in an angle ‘ϕ’. Now this angle is defined as (Fig. 2),

When , the above expression defines the

interface of zone I & II, i.e.

Thus the variable web ‘wx’ for zone I is:

The burning perimeter is defined by:

where

Port area of zone-I is given by,

Zone-II:

Zone II persists till the straight portion of fin completely vanish, thus the variable web for zone II is,


tip: web burning zone-II. Where,

Burning perimeter consists of increasing elliptical arc

and diminishing straight segment,


The port area for zone II is constructed from an ellipse,

a rectangle and a triangle,

where ‘y’ is determined using equation (12). Zone-III: The zone-III continues till the web burn out, i.e,

The perimeter for zone III is defined by the portion of ellipse till x = xB3,

where ‘y’ for ellipse with center at O1,


tip: web burning zone-III.

The point B3(xB3,yB3) lies at the intersection of line segment and elliptical arc A3B3. The ‘y’ values on line segment are determined by,

where

and ‘y’ values on elliptical arc from origin ‘O’ are given by relation,

The value of ‘x’ for which gives us the point

B3(xB3,yB3). The area of a triangle subtracted from the area under the elliptical portion gives us the port area for zone III.

Where

and

Zone-IV:

Zone-IV is the sliver region. The web through sliver

‘ws’ is determined by equating ‘y’ values of line segment


tip: web burning zone-IV.

and elliptical arc A5B5 on point ‘F’, i.e,

The perimeter of zone IV is determined by,


where B4(xB4,yB4) is determined as B3(xB3,yB3). The

point A4(xA4,yA4) lies at the intersection of circular arc GF and elliptical arc A4B4. The ‘y’ values on the circular arc are given by,

where

and ‘y’ values on elliptical arc are given by equation (24). The value of ‘x’ for which gives us the point A4(xA4,yA4).

The port area is represented by the areas under a circular arc and elliptical arc minus area of triangle.

where

and

and

C. Authentication of Developed Equations: A MATLAB code was written to assess the results

generated by these equations. To substantiate the authenticity of the mathematical relations devised and hence the results generated by the MATLAB code, various star geometries were taken into consideration and a large-scale comparison of the burning perimeter and port area profiles predicted using above equations (via programming in MATLAB) and measured directly on drawings generated using Autodesk Mechanical Desktop. The impeccable accordance of calculated and measured results makes the developed equations credible.

III. RESULTS AND DISCUSSION

A. Model Grain Configuration Geometry: Two geometries were considered for analysis; first one

is the classical truncated 7-star points with circular tip (with zero cusp radius) and the other one is the equivalent truncated 7-star point geometry with elliptical tip. Following figures (Fig. 6 and 7) and tables (Table 1 and

2) describe the main features of the cross section geometry as function of the grain outer radius ‘R’.

TABLE I.

MAIN DIMENSIONS OF CROSS SECTION (CIRCULAR TIP STAR) Designation Value

R Outside radius 1R

r1 Fillet radius 0.05R

Rp Port radius 0.214R

ε Secant fillet angle 24.45°

N Number of star points 7

Figure 6. π/N cross section of truncated star with circular tip.

TABLE II.

MAIN DIMENSIONS OF CROSS SECTION (ELLIPTICAL TIP STAR) Designation Value

R Outside radius 1R

ea Major ellipse axis 0.058R

eb Minor ellipse axis 0.035R

Rp Port radius 0.214R

N Number of star points 7

Figure 7. π/N cross section of truncated star with elliptical tip.

For the selection of an equivalent truncated star geometry with elliptical tip, previous research work [6, 9] has been taken into consideration. The value of elliptical ratio eb/ea = 0.6 was chosen because of its proven minimum stress configuration [6, 9].

B. Ballistic Performance Comparison: Web burning profiles for both the cases (elliptical and

circular tip star configuration) were calculated employing the already developed MATLAB code. For circular tip star configuration, already defined burn perimeter


equations [1, 7] were employed and rewritten in the MATLAB code. In Fig 8, comparison of burning profiles of truncated star with circular fillet tip and equivalent elliptical tip is presented. The result shows a congruence of the two burn perimeter profiles. This consolidates the claim that the two geometries will produce similar ballistic performance results.

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w/l

S/l

with circular filletwith elliptical tip

Figure 8. Comparison of burn perimeter evolution of truncated star

with circular (black line) and ellliptical tip (red line).

C. Ignition Pressurization Stresses and Strains in Circular Star Tip:

Our main aim is to assess the stresses and strains generating in the two equivalent star configurations when the propellant grain is subjected to ignition pressurization. For this, first we only consider the circular tip truncated star geometry. We will simulate the linear viscoelastic behavior of the 2D geometry of the propellant grain employing ABAQUS-CAE software. Solid propellant grain model properties:

Solid Propellant selected was the HTPB encased in a high strength steel chamber case. Following are the material properties used:

TABLE III.

MATERIAL PROPERTIES OF SOLID POROPELLANT MOTOR Steel Case Propellant

Material HSS HTPB

Young’s modulus, E (MPa) 210,000 19

Poisson ratio, ν 0.31 0.499

Relaxation modulus curve × Fig. 9

Time shift factor curve × Fig. 10

-3 -2 -1 0 1 2 3 4 5 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Log Reduced Time Scale Log[t/at]

Log

Str

ess

Rel

axat

ion

Mod

ulus

Log

[E(t)

]

Figure 9. Relaxation modulus verses the temperature-reduced time. Solid line shows the prony fit to the data.

-40 -20 0 20 40 60 80-2

-1

0

1

2

3

4

Temperature (T)

Log

Shi

ft Fa

ctor

Log

(aT

)

Figure 10. Log shift factor “Log (aT)” verses the temperature. Solid red

line shows the WLF fit.

Propellant Grain material mechanical characterization

was done via the tensile load relaxation tests conducted on INSTRON universal testing machine. Rectangular propellant samples were subjected to a minimal strain (1% – 5%) at various temperatures ranging from –40°C to 70°C and the relaxing values of stresses were noted at various time intervals ranging from 0sec to 1000 sec. Relaxation modulus of the propellant was then calculated from these test results. A master curve was then plotted at 20°C. Fig. 9 shows the relaxation modulus verses the temperature-reduced time scale. Here, the temperature-reduced time is the time required to reach equilibrium time value t*, which is defined by the shift factor aT, as shown in Fig. 10. Shift factor aT, is defined by the WLF (Williams-Landel-Ferry) equation [2] as:

( )

ref

refT TTC

TTCaLog

−+

−−=

2

1 ………. (38)

with C1 = 5.47, C2 = 160.5, and Tref = 20 °C. The relaxation modulus can be expressed as:

)/( *TatEE = .................... (39)

Time-Temperature shift data of the relaxation modulus master curve was fitted with a prony series with eight terms of the following form:

)(exp)(1

τtn

iiEEtE −

=∞ ∑+= .................... (40)

From the above equation, the instantaneous propellant elastic modulus (E0) was determined as the sum of E terms at zero time instant (see Fig. 9). The calculated value is E0 = 19 MPa as shown in Table III. Viscoelastic model to be used in ABAQUS-CAE:

For the usage of the relaxation modulus data in ABAQUS for FEA modeling and simulation, the elastic stress relaxation modulus was converted into shear relaxation modulus by the following relation:

)1(2 ν+=

EG .......................... (41)

The linear viscoelastic behavior of polymers is described in the FEA program ABAQUS by a shear relaxation modulus given by [10]:


( )⎟⎠

⎞⎜⎝

⎛−−= ∑

=

n

iii tgGtG

10 /exp1)( τ .............. (42)

Where, G0 is the instantaneous shear modulus, and the gi are dimensionless shear relaxation moduli associated with relaxation times τi. It is important to note that the sum of the gi cannot exceed 1.0 since the long term modulus G∞ is defined by [10]:

⎟⎠

⎞⎜⎝

⎛−= ∑

=∞

n

iigGG

10 1 ................. (43)

Similarly the dimensionless parameter gR(t) used in ABAQUS is defined by:

0

)()(G

tGtg R = ......................... (44)

We made use of the parameters gR(t) and τi extracted from the conversion of the tensile relaxation modulus master curve into shear modulus curve. The dimensionless parameters of the prony series employed in ABAQUS are listed down in Table-IV:

TABLE IV. PRONY SERIES PARAMETERS USED IN ABAQUS

gi – (prony) gR(t) – (prony) τi (sec)

0.400 0.725 0.001

0.218 0.450 0.01

0.126 0.296 0.1

0.064 0.212 1

0.040 0.165 10

0.021 0.137 100

0.018 0.118 1000

0.015 0.101 10000

0.020 0.083 100000

FEA in ABAQUS-CAE standard:

Simple 2-D planar propellant grain geometry encased in steel shell was considered for analysis. Model was constructed in ABAQUS using its modeling tools. Properties of the propellant grain and steel case were used as mentioned in Table-III and IV. A zero value displacement boundary condition was applied on the outer steel case wall. A steady state Ignition pressure was applied on the internal bore of the 2-D grain geometry. Analysis was done in two steps:

1. Step-I: Static loading, step time = 0.001 sec. 2. Step II: Visco propagation, step time = 1 sec. Automatic step increment technique was adopted for

the convergence of the solution. The meshing element type scheme adopted was of standard 2-D plane strain quad-dominated with advancing fronts (automatic mapping where required) elements of linear geometric order. For this, the ABAQUS standard software generated a 4-node bilinear plane strain quadrilateral, reduced integration; hourglass control (CPE4R) meshing elements. Fig. 11 shows the meshing of the 2-D geometry, and Fig. 12 shows the pressure application and the boundary conditions.

Figure 11. 4-node bilinear plane strain quadrilateral, reduced

integration; hourglass control (CPE4R) meshing.

Figure 12. Internal pressure loading and zero-displacement boundary

condition application. The presented grain geometry was subjected to an

ignition pressure of 6MPa for one sec duration. The step chosen for studying the pressurization affects is ‘visco’ to include the linear viscoelastic response of the grain structure. The numerical simulation results employing ABAQUS-CAE are shown in Fig. 13 and 14. Fig. 13 shows the computational stress distribution i.e., Von-Mises stresses, and Fig. 14 shows the deformation (strain) distribution i.e., the maximum in-plane principle strain of the solid propellant geometry under investigation. It is evident from the figures that the star circular tip region contains the highest magnitude of the stresses and strains. For the geometry under consideration, the maximum stress is 36.15 KPa and strain percentage is 0.7 %.


Figure 13. Stress distribution in circular tip 7-point truncated star

geometry

Figure 14. Strain distribution in circular tip 7-point truncated star

geometry

D. Comparison with Various Elliptical Ratios Star Tip: As has already been described, the equivalent

elliptical-tip star geometry with elliptical ratio (ellipse minor to major axis ratio) of 0.6 was chosen because: (i)- ballistic performance similarity with circular tip geometry and (ii)- an expected reduction in the stresses and strains as compared to the circular tip geometry. It has been already shown in an important previous research work [6, 9] that the parameter ‘elliptical ratio’ plays a significant role in reducing the stress and strains in the critical regions of the star geometry. Milos [6] showed that the configuration of minimum stress occurs for a value of elliptical ratio in the range of 0.35 < elliptical ratio < 0.7. Employing Pro/Mechanica simulation model, he [6] showed that the minimum stress concentration factor value will occur for elliptical ratio of 0.58. This was the reason that equivalent elliptical star tip geometry with elliptical ratio of 0.6 was selected for ballistic performance comparison. However, this needs to be validated in terms of the strain values also. For this, we

investigated the percentage change or improvement in the maximum strain values of the various elliptical ratio truncated star tip geometry to find out the compatible geometry with minimal strain values.

Same model FEA simulation considerations were

adopted as has already been described for circular tip star geometry with internal pressure as the specified load on 2D plain strain model geometry. The parameter ‘elliptical ratio’ was varied from 0.55 to 1 with 0.05 steps. For an easy comparison purpose, we normalized the strain values with the maximum strain value of the circular tip truncated star geometry. The results of change in the percentage strain (normalized by the strain value of circular tip star geometry) as the elliptical ratio changes are depicted in Fig. 15.

0.8

0.85

0.9

0.95

1

1.05

0.5 0.6 0.7 0.8 0.9 1 1.1

Nor

mal

ized

max

imal

stra

inva

lue

at st

ar ti

p

Ellipse minor to major axis ratio Figure 15. Elliptical star tip simulation results of the normalized strain

verses the elliptical ratio (ellipse minor to major axis ratio).

From Fig. 15 it can be seen that minimal strain as compared to that of circular tip truncated star geometry will occur for elliptical ratio value of ~0.6. At this value of elliptical ratio the percentage maximal strain at elliptical star tip is ~0.85 of the value of maximal strain at circular star tip. These results consolidate the previous work done by Milos [6] in which he exploited the stress concentration factors of various elliptical ratio star tips to that of the circular tip star geometry. In both the cases (this case and Milos) the most optimal configuration considering the stresses and strain values is with elliptical ratio of 0.6.

IV. CONCLUSION Mathematical relations for burn perimeter evolution

with time for truncated star with elliptical tip were developed and were written in a MATLAB code. The results from MATLAB program output were compared to that of the AutoCAD generated results of the burn perimeter evolution. The congruence of the two results shows the generalized verity of the developed equations. Two truncated star geometries, one with circular tip and other with elliptical tip were considered for ballistic performance comparison. Both the geometries showed a superimposed burn perimeter evolution curve which makes sure their ballistic performance equivalence. This


means that a circular tip star geometry can well be replaced with elliptical tip geometry. This is of great significance in propellant grain mechanics as the elliptical tip generates smaller magnitudes of stresses and strains. For the selection of optimum elliptical ratio truncated star-tip geometry, FEA simulations were conducted on ABAQUS-CAE standard software considering a number of star geometries with various ellipse ratios. Maximal stress and strains resulting from the ignition pressurization loading of a 2D truncated star geometry with circular tip and elliptical-tip (with various elliptical ratios) were compared. Our results consolidate the previous claim of the optimal elliptical ratio (ellipse minor to major axis ratio), in terms of minimal stresses and strains, to be existing at an elliptical ratio of 0.6.

ACKNOWLEDGMENTS We thank Mr. Muhammad Azam, Dr. Hassan Ijaz, Dr.

Wajid Ali Khan, Dr. Aurengzeb and Dr. Zain-ul-Abedeen of NESCOM, Islamabad-Pakistan for sparing their valuable time in helping us with FEA and its application in ABAQUS-CAE software pertaining to polymers and composites. We are grateful to Mr. Noorullah (PoP) of NESCOM for his immense support in finalizing this research work.

REFERENCES

[1] Hartfield, R. J., Jenkins, R.M., Burkhalter, J.E., and Foster, W., " A Review of Analytical Methods for Solid Rocket Motor Grain

Analysis” AIAA Paper 2003-4506, 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 20-23 July 2003, Huntsville, Alabama.

[2] Sutton, G., and Biblarz, O., Rocket Propulsion Elements, 7th ed.,Wiley, New York, 200.

[3] Davenas, A., Solid Rocket Propulsion Technology, Pergamon, Oxford, England, UK, 1993.

[4] Solid Propellant Grain Design and Internal Ballistics,” NASA SP 8076, URL: library.msfc.nasa.gov/cgi-bin/lsp8000.

[5] Solid Rocket Motor Performance Analysis and Prediction, NASA SP 8039, URL: library.msfc.nasa.gov/cgi-bin/lsp8000.

[6] Predrag Milos, “Geometry Optimization of Star Shaped Propellant Grain with Special Attention to Minimization of Stress and Starin”, FME Transactions (2007) 35, 35-40.

[7] Rajesh K. K, Anantharam S, and Subhash Chandran B. S., “Analytical Expressions for Burning Perimeter and Port Area of a Solid Propellant Finocyl Grain”, AIAA Paper 2007-5784, 43rd AIAA/ASME SAE/ASEE Joint Propulsion Conference and Exhibit, 8-11 July 2007, Cincinnati, OH.

[8] George B. Thomas, Ross L. Finney, “Calculus and Analytic Geometry”, 6th edition, Addison-Wesley Publishing Company Inc. (City), 1983, Page 338.

[9] Fitzgerald, J. E., Handbook for the Engineering Structural Analysis of Solid Propellants, University of Utah Lake City, Utah, 1971.

[10] Mills, N. J., Finite Element Models for the Viscoelasticity of Open-Cell Polyurethane Foam, Cellular Polymers, Vol. 25, No. 5, 2006.

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17 Waqas Ahmed