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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. 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