International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 1521-1527, Article ID: IJMET_10_01_154
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=01
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
NUMERICAL ANALYSIS OF VISCOELASTIC
PROPELLANT GRAINS SUBJECTED TO
PRESSURE LOAD
Gowrishankar M C, Nitesh kumar* and Jayashree P K
Department of Mechanical and Manufacturing Engineering, Manipal Institute of Technology
Manipal Academy of Higher Education, Manipal, India.
*corresponding author
ABSTRACT
Idealized solid propellant is models is analysed for thick sphere subjected to internal
pressure, whose casing is made of composite materials,. Results obtained from present
elements are compared with analytical results and 8 noded counterpart of MARC. Solid
Rocket Motor (SRM) is developed based on casting method where solid propellant grains
are cast into a composite or metallic casing. Generally, SRMs are exposed to extreme
loading scenarios during storage, transportation, and firing, leading to cracks in the solid
propellants. In this paper, Computational Finite Element Analysis is performed with
developed 8 node quadrilateral, 9 node quadrilateral and 6 node triangular elements
using Herrmann formulation to analyze stress and strain variations in the mid segments
of the typical SRM subjected to pressure loading. The obtained results are compared with
commercially available Finite element software.
Keywords – Finite element, Rocket motor, Internal pressure, Pressure loads, Propellant
grain.
Cite this Article: Gowrishankar M C, Nitesh kumar and Jayashree P K, Numerical
Analysis of Viscoelastic Propellant Grains Subjected to Pressure Load, International
Journal of Mechanical Engineering and Technology, 10(01), 2019, pp.1521–1527
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&Type=01
1. INTRODUCTION
Solid rocket motor (SRM) structural design is currently is based on concept of a mechanically
weak solid propellant grain cast into a stronger metallic or composite case. The outer case
provides the essential structural resistance against service and operational loads, and the inner
propellant grain’s low strength is used for transmission of loads from grain surface to outer case.
In general, solid rocket motors are subjected to diverse loading during transportation, storage and
firing. It is well known that under these loading conditions, cracks can develop in solid
propellants because of excessive loads. Therefore, in order to determine the integrity and the
http://www.iaeme.com/IJMET/index.asp
1521
editor@iaeme.com
Gowrishankar M C, Nitesh kumar and Jayashree P K
ultimate service life of solid rocket motors, studies are conducted to evaluate the significance of
the value and distribution of stress and strain [1].
The finite element method has the capability to deal with complex loading conditions,
material behavior and practical geometries. Commercial codes (viz., MARC, NASTRAN, NISA,
ANSYS, etc.) are available to solve structural problems. Based on the nature of the final matrix
equations the finite element methods are classified as Displacement Method [2], Force method
[3] and mixed method [4]. The major structural elements in rocket motor are motor case, in
compressible liner and propellant. The propellant is viscoelastic in nature. The analysis of a
viscoelastic structure may be reduced to elastic analysis by following the concept of Schapary [6]
described in detail [8].
The finite element developed by displacement method was studied by many researchers [2].
The disadvantages of this method are that it cannot handle the nearly incompressible materials
like that of the solid propellants. For such materials (Poisson Ratio approximately 0.5), the strain
components are not independent anymore. Therefore, the principle of minimum potential energy
and the corresponding displacement method experience locking and the resulting solutions and
erroneous [7]. Locking occurs when the element formulation is not sufficient for capturing the
appropriate displacements. In these materials, the volumetric strain is nearly zero, hence using
displacement method based finite elements results in zero displacement and the calculated
stresses are under predicted and unreliable when low order displacement interpolations are used.
Although higher order interpolations, such as biquadratic interpolation or mesh refinement may
be used, the displacement solution is generally not accurate. The solution to element locking in
case of incompressibility is to break the strain field down to its fundamental components. In the
case of any deformation, there are deviatoric and volumetric strain components. Deviatoric
strains determine the shape change of the body and volumetric strains determine the volume
change (dilatation) of the body. The volume change occurs due to a hydrostatic pressure. The
trouble caused by the displacement based finite element formulation for the incompressible
material can be understood by examining the familiar elasticity relationship,
(1)
For nearly incompressible materials, the Bulk modulus becomes large relative to the Shear
modulus. In the limit, when the material is completely incompressible (ν=0.5), all hydrostatic
deformations are precluded. In this limiting case, it is therefore not possible to determine the
complete state of stress from strain alone. Therefore, special formulations are required to account
for the hydrostatic deformations as well as to predict the actual state of stress for such materials.
Thus, in this study, a computational study based on Herrmann Formulation is presented to
overcome the limitation of the direct method. Here an 8 node quadrilateral, 9 node quadrilateral,
and a 6 node triangular element are developed and studied for stress and strain variations for midsection of SRM subjected to the pressure load.
2. FORMULATION
In this formulation volumetric strain is imposed as a constraint to the total potential given by
(2)
http://www.iaeme.com/IJMET/index.asp
1522
editor@iaeme.com
Numerical Analysis of Viscoelastic Propellant Grains Subjected to Pressure Load
Where {ε} is the strain vector, {σ} is the stress vector, P is the hydrostatic pressure, εv is the
volumetric strain, K is the bulk modulus, WP is the work potential.
Replacing
The total potential takes the form
in (1)
(3)
Taking the variation of equation (2) with respect to nodal displacement {q} and nodal
pressure {p}, one can write
(4)
Where {q} is the elemental displacement vector, {p} is the elemental hydrostatic
pressure,{FM} is the load vector due to mechanical load, {FT} is the load vector due to thermal
load, where α is the linear coefficient of thermal expansion, ΔT is the difference in
temperature.[Kuu] is the element stiffness matrix corresponding to element displacement, [Kpp]
is the element stiffness matrix corresponding to element pressure, [Kup] is the element stiffness
matrix corresponding to the cross coefficient {q} and {p}. Element stiffness matrix, consistent
nodal forces for each element corresponding to point loads, gravity loads and distributed element
edge loading are evaluated and assembled as per the standard procedures.
http://www.iaeme.com/IJMET/index.asp
1523
editor@iaeme.com
Gowrishankar M C, Nitesh kumar and Jayashree P K
3. VALIDATION
A thick walled sphere subjected to internal pressure is analysed for different values of the
Poisson’s ratio. The results are compared with the existing analytical results [9].
Figure 1: Thick Sphere subjected to internal pressure
Fig. 1 shows the finite element model of a thick walled sphere subjected to unit internal
pressure. It includes the details of geometry, material and boundary conditions. The displacement
at the inner radius obtained with the present elements (8 Node quad, 9 node quad, 6 node
triangular) are found to be in good agreement with the following analytical solution [9] given in
equation (5) for various values of ν ε [0, 0.5].
(5)
4. INFINITELY LONG 3 LAYERED CYLINDRICAL SHELL
SUBJECTED TO INTERNAL PRESSURE
An infinitely long 3 layered cylindrical shell is taken whose upper and lower boundaries are
suppressed to allow only radial movement as shown on Figure 2. A pressure of 4.905 N/mm2 is
applied at the inner port of the propellant. It is made of composite casing. The other materials in
the rocket motor are solid propellant grain and insulation. The results obtained at various
locations of the configuration shown in path plot (see Fig. 11 to Fig. 13) are in good agreement
with those of MARC.
http://www.iaeme.com/IJMET/index.asp
1524
editor@iaeme.com
Numerical Analysis of Viscoelastic Propellant Grains Subjected to Pressure Load
Figure 2: An infinitely long 3 layered cylindrical shell subjected to internal pressure with boundary
conditions
The geometric details and material properties are as given below
Geometric details
Inner radius of propellant = 500 mm.
Outer radius of propellant = 1389 mm.
Outer radius of liner = 1394 mm.
Outer radius of casing = 1401.8 mm.
Table 1: Material properties of the 3 layered cylindrical shell subjected to internal pressure
Material
Propellant
Liner
casing
Young’s modulus
E (N/mm2)
4.905
1.962
186390
Poisson ratio
ν
0.499
0.5
0.3
Coeff. Of Thermal
Exp α (/°C)
0.0001
0.0003
1.1E-5
Weight Density
ρ (N/mm3)
1.7461E-5
1.7461E-5
7.6518E-5
The node associated to one particular material is tied to the material adjacent to it so as to
avoid averaging of the stresses and displacements at the interface of these elements. When these
nodes are merged the matrices gets averaged and hence gives erroneous results. When we need
to force two or more degrees of freedom (DOFs) to take on the same (but unknown) value, you
can tie these DOFs together. A set of coupled DOFs contains a prime DOF, and one or more other
DOFs. Coupling will cause only the prime DOF to be retained in your analysis' matrix equations,
http://www.iaeme.com/IJMET/index.asp
1525
editor@iaeme.com
Gowrishankar M C, Nitesh kumar and Jayashree P K
and will cause all the other DOFs in a coupled set to be eliminated. The value calculated for the
prime DOF will then be assigned to all the other DOFs in a coupled set.
Table 2 shows the displacements and strains at inner port and stresses at the outer port
of the propellant which matches accurately with the analytical results. Figure 3 shows
the displacement contour.
Table 2: Radial displacements at the inner port of the propellant after tying the interfacial nodes
Radius
500
1389
1394
1401.8
Analytical
25.891
5.82759
5.80669
5.79275
MARC
25.8911
5.82745
5.80654
5.7926
8 node quad
25.893
5.8282
5.80677
5.7927
9 node quad
25.893
5.82781
5.80662
5.79268
6 node tria
25.8599
5.82785
5.80662
5.79268
Figure 3: radial displacement contour
The results of the analysis are shown in the displacement contour (Figure 3). The variation of
resultant displacement, at the inner port of propellant show higher displacements at the inner
section of the motor and at the outer end of the mid-section due pressure and resistance of the
case, which results in higher stresses. It is observed from the results obtained based on Herrmann
formulation finite elements (present study) are in close agreement with those results obtained
from MARC software.
5. CONCLUSIONS
Herrmann formulation based Axisymmetric FE on is validated by comparing the results with the
eight-node quadrilateral axisymmetric Hermann element of MARC. It can be observed that while
using MARC software package, the casing material (which is compressible in nature) is idealized
using the standard eight-node iso parametric quadrilateral axisymmetric element having two
degrees of freedom (w, u), while the propellant grain (which is nearly incompressible in nature)
is idealized using the Hermann element having three degrees of freedom (w, u, σ mean). Here σ
mean is the hydrostatic pressure known as the mean pressure. For the present case bonded
cylindrical solid propellant grain, the interface nodes should be connected using tying option to
http://www.iaeme.com/IJMET/index.asp
1526
editor@iaeme.com
Numerical Analysis of Viscoelastic Propellant Grains Subjected to Pressure Load
take care of the mismatch between two degrees of freedom of casing element and three degrees
of freedom of nearly incompressible propellant element [9]. The present study does not require
tying of the nodes, whether the structure is made of compressible or nearly incompressible
materials. It can be concluded from the above-considered numerical problems that the present
axisymmetric element can be used for examining the structural behavior of rocket motors having
nearly incompressible and incompressible materials.
REFERENECS
[1]
J.E Fitzerad, and W.L.Hufferd “Handbook for the Engineering Structural Analysis of Solid
Propellant”, CPIA publication 214, 1971
[2]
O. C. Zienkiewicz., “The Finite Element Method”, 3d ed. New York: McGraw-Hill, 1977.
[3]
Walter. C. Hurty and Moshe. F. Rubinstein, “Dynamics of Structures”, Prentice-Hall of India
Private Limited, New Delhi, 1967.
[4]
T. J. R. Hughes, “The Finite Element Method – Linear Static and Dynamic Finite Element
Analysis”, Englewood Cliffs, NJ: Prentice-Hall, 1987.
[5]
E.H. Lee, “Stress analysis in Viscoelastic Bodies”, Quarterly of Applied Mathematics,
Vol.13, pp 183-190, 1955-1956.
[6]
R.A. Schapary, “Two Simple Approximate Methods of Laplace Transform Inversion for
Viscoelastic Stress Analysis”, California Institute Technical Report, SM 61-23. Graduate
Aeronautics Laboratories, 1961
[7]
L.R. Herrmann, “Elasticity Equations for Incompressible and Nearly Incompressible
Materials by a Variational Theorem”, AIAA Journal, Vol.3, pp.1896-1900, 1965.
[8]
K.Kanakaraju, B.Nageswara Rao and R.Marimuthu, “Hybrid Stress-Displacement Finite
Elements for Viscoelastic Analysis”, In: FINITE ELEMENTS (Nikos Mastorakis, olga
Martin Editors), Published by World Scientific and Engineering Academy and Society
(WSEAS), pp. 43-53, 2007.
[9]
E. P. Kasper and R. L. Taylor, “Mixed-enhanced formulation for geometrically linear
axisymmetric problems”, International Journal for Numerical Methods in Engineering 2002:
53:2061-2086
[10]
H. Ouabel, A. Zadjaoui, Contribution of Pressure Meter and Static Penetrometer Tests for
Bearing Capacity Estimation-Numerical Analysis, International Journal of Civil Engineering
and Technology, 9(10), 2018, pp. 2087– 2097.
[11]
Nazik Abdulwahid Jebur, Fadhel Abbas Abdulla and Ammar Fadhil Hussein, Experimental
and Numerical Analysis of Below Knee Prosthetic Socket, International Journal of
Mechanical Engineering and Technology, 9(8), 2018, pp. 1–8.
[12]
Vipin Kumar, Sugandh Gupta and Ankur Sachdeva Experimental and Numerical Analysis of
Ground Effect of A Typical RIV (Reusable Launch Vehicle). International Journal of
Mechanical Engineering and Technology, 8(7), 2017, pp. 1755–1762.
[13]
Madhwesh N and K. Vasudeva Karanth, Numerical Analysis of a Solar Air Heater for
Improved Performance using Cylindrical Turbulators with Ribs, International Journal of
Mechanical Engineering and Technology 9(8), 2018, pp. 296–309.
http://www.iaeme.com/IJMET/index.asp
1527
editor@iaeme.com