CHAPTER 4 MATERIAL CHARACTERISATION. The material characterization of AFNOR 15CDV6 steel and the

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1 94 CHAPTER 4 MATERIAL CHARACTERISATION The material characterization of AFNOR 15CDV6 steel and the natural rubber based elastomer developed for the use in flex nozzle are discussed in this chapter. This includes material testing, modeling and selection of material model suitable for analysis and validation of material model [118]. The elastomer selected should have certain specific properties. The joint spring torque is directly proportional to the elastomer shear modulus which implies that elastomer should have as low a shear modulus as possible. The shear stress in the material is due to the motor pressure and vectoring. For high pressure and high thrust vectoring motors, the shear strength required should be as high as possible. In case of storable solid rocket motors, ageing plays a major role. The elastomer should have less effect on the mechanical properties due to ageing. Other requisite properties are good bonding with the reinforcement material and reproducibility of properties. The reinforcement should have a high material yield and ultimate strength, ease of machining, availability of fabrication expertise, infrastructure and simple heat treatment cycle. In the finite element analysis (FEA) of rubber engineering components, proper selection of a rubber elastic material model and the material parameters plays an important role. The deformation state of these components is often a complex three-dimensional one.

2 95 However, measurements of the mechanical behavior are often performed for simple deformation states. Mooney-Rivlin, Ogden, Yeoh and Neo-Hookean models [119, 120] are the most widely used among the classical rubber elasticity models. In this chapter, the characterization aspects of the natural rubber based elastomer and AFNOR 15CDV6 reinforcement, developed for the use in flex bearing are discussed. 4.1 MATERIAL CHARACTERIZATION Elastomer The major ingredients of the elastomer are natural rubber, carbon black filler, plasticizer, sulphur, accelerator and anti-oxidant. The elastomer is loaded in bulk compression and in shear direction during motor operation and vectoring of the nozzle. Material modeling of the elastomer requires data in all loading modes in which the elastomer is loaded during the rocket motor operation. For this, the elastomer is tested in tensile and shear directions [121]. The mechanical properties of the elastomer are given in section Tensile test on the elastomer is carried out using a dumbbell specimen as per ASTM D412. The tensile test specimens for are shown in Figure 4-1. The shear strength and shear modulus are evaluated using Quadruple Lap Shear Specimen (QLSS) as per BS 903 Part A14. The QLSS test specimens are shown in Figure 4-2. Stress vs. strain curves obtained in tensile test and QLSS test for three specimens are given in Figure 4-3 & Figure 4-4 respectively. The

3 96 failure modes in the elastomer & reinforcement combinations are adhesive bond failure and cohesive elastomer failure. The bond strength for the chosen adhesive system of Chemlok 205 and 220 is 40 kgf/cm2 (3.92 MPa) minimum and shear strength is 27 kgf/cm2 (2.65 MPa). By design, the failure occurs within the elastomer (cohesive). To ensure this, all the failures in the QLSS specimen must be cohesive during testing at specimen level. Figure 4-1: Tensile test specimen - elastomer

4 97 Figure 4-2: QLSS test specimen- elastomer Figure 4-3: Stress Vs Strain for Tensile Test Specimen Elastomer

5 98 Figure 4-4: Stress Vs Strain for QLSS Specimen - Elastomer Reinforcement The reinforcement material used in the flex seal is AFNOR 15CDV6 steel. The basic constituents of the material are carbon (0.15%), chromium (1.5%), molybdenum and vanadium ( together 1.5%). The reinforcement is also loaded in bulk compression and in shear direction during motor operation and vectoring of the nozzle. The reinforcement experiences compressive hoop stresses on the ID which are dominating than the other components of stresses. The heat treatment cycle is given in Table 4-1. The mechanical properties of the steel reinforcement are given in section

6 99 Table 4-1: Heat Treatment Cycle for AFNOR 15CDV6 Treatment Temperature 0C Soaking Time Cooling Media Annealing 875 ± 10 4 minutes/mm Furnace cooling at the rate of 500 C/Hr to room temperature Hardening 970 ± 10 Tempering 610 ± minutes/mm (or 20 minutes min) 8 minutes/mm (or 120 minutes min) Oil Quench Oil Quench The reinforcement material is characterized by carrying out tensile test on the specimen as per ASTM A The specimens are shown in Figure 4-5. Stress vs strain curve for three tested specimens are shown in Figure 4-6. Figure 4-5: Tensile Test Specimen - Reinforcement

7 100 Figure 4-6: Stress vs. Strain curve for tensile tests - Reinforcement 4.2 MATERIAL MODELING Elastomer For modeling elastomer, four material models viz., MooneyRivlin Model, Neo-Hookean, Yeoh and Ogden material models are considered. A Mooney-Rivlin model [122] exhibits a constant shear modulus and gives good correlation up to 150 % strain in uniaxial tension. In house experimental data shows good match upto 200% strain. The form of strain energy potential for 2 parameter model is W c10 I 1 3 c 01 I J 1 2 d Eq-(4-1) A Neo-Hookean material model [122] exhibits a constant shear modulus and gives good correlation with experimental data up to 40 % strain in uni-axial tension and up to 90 % in simple shear. However

8 101 the elastomer developed and tested for this purpose shows good match between 80% to 120% strain and deviating in other regions. The strain energy potential for Neo-Hookean material model is W I J 1 2 d Eq-(4-2) Literature shows that the Ogden material model [122] gives good correlation with test data in simple tension up to 700 %. It also accommodates non-constant shear modulus and slightly compressible material behavior. In-house experiments show a deviation from 80% strain onwards. The form of strain energy potential for Ogden material model is N i i 1 2k 1 2 i 3 i 3 J 1 i 1 i k 1 dk N W Eq-(4-3) The Yeoh model [122] has been demonstrated to fit various modes of deformation using data from a uniaxial tension test only. This model should be used with caution at low strains. However, inhouse experiment shows the behavior similar to Neo-Hookean model. The strain energy potential of Yeoh model is N 1 J 1 2k k 1 dk N W ci0 I1 3 i i 1 Eq-(4-4) The comparison of the experimental data with all the four material models [118,122] is given in Figure 4-7. It is clearly evident that Mooney-Rivlin and Ogden are having a good correlation with experimental data. The percentage variation of Mooney-Rivlin and Ogden material model with respect to experimental data is 4.94% and 9.75% respectively. From the practical usage point of view,

9 102 deformation of elastomer for the pressure loading is about 30 40% in compression and deformation due to shear is about %. Based on these conditions, the Mooney-Rivlin model has been selected to simulate the elastomer. Figure 4-7: Comparison of Material Models 4.3 MATERIAL CONSTANTS EVALUATION Elastomer The constants of the materials are derived using experimental stress-strain data. It is recommended that this test data be taken from several modes of deformation over a wide range of strain values. To achieve material stability, the constants should be fit using test data in at least as many deformation states as will be experienced in the model. For characterizing hyper elastic materials, six different deformation modes can be used. They are uniaxial tension, uniaxial

10 103 compression, equibiaxial tension, equibiaxial compression, planar tension and planar compression. Combinations of data from multiple tests will enhance the characterization of the hyper elastic behavior of the material. Once the strain energy function is defined, the stress is obtained by differentiating the strain energy with respect to the strain σ = W / ε Eq-(4-5) For an incompressible material in uniaxial tensile state, the volume change due to deformation is zero and the ratio of original to deformed configuration volume is one. This can be represented as the third invariant of strain I 3 = 1, i.e, λ12 λ22 λ32 = 1 Eq-(4-6) These results are incorporated into the strain energy function considering the case of a rubber rod subject to uniaxial tension along its longitudinal axis. Let λ1 be the stretch ratio along the longitudinal axis and λ2 and λ3 are the stretch ratios along the lateral axes. For uniaxial tension, the deformation state is represented by, λ1 = λ Eq-(4-7) and, λ2 = λ 3 = 1 / λ Eq-(4-8) By differentiating the strain energy potential and substituting the above expressions, the stress for a 2 parameter Mooney-Rivlin model is C C01-10 Eq-(4-9)

11 104 A similar procedure is adopted to derive the relations for other material models. The material constants for all the models are given in Table 4-2. Table 4-2: Material constants of Elastomer models Material Model Constants Mooney-Rivlin (Eq-4-1) C01 = & C10 = Ogden (Eq-4-2) µ = & α = Neo-Hookean (Eq-4-3) µ = Yeoh (Eq-4-4) C10 = The plot of 1 1 gives a straight line (Figure 4-8) 2 vs. 2 with slope c10 and intercept c01. [ ]. Figure 4-8: Mooney Rivlin Constants for Elastomer For analysis of the elastomer in axial compression, data from uniaxial tensile test data is used. For analysis of the rubber in both axial compression and vectoring mode, uniaxial tensile data is not

12 105 sufficient to predict the behavior of the elastomer. Mooney-Rivlin model has to be fitted by having uniaxial tensile data and shear data. The plot of uniaxial test data and shear test data are fitted with experimental data for two parameter Mooney-Rivlin model and is shown in Figure 4-9. Figure 4-9: Two Parameter Mooney Rivlin Model Reinforcement The design criteria for flex bearings are the margin of safety to be lower among on yield strength and 0.25 on ultimate tensile strength. The design is carried out to have a working stress close to yield strength of the material. The material is non linear all through the stress strain curve.

13 106 The need for modeling the reinforcement material as a multilinear elastic material model arises because of the working stresses close to yield strength of the material. The model considers constant young s modulus between two points on the stress strain curve. To have a better approximation, stress at every 100 µ strains data has been taken into account. 4.4 VALIDATION OF MATERIAL MODELS The material models have to be validated before using it in analysis. The validation of the material models with the test data is done by modeling the test specimens in ANSYS and loading is applied in steps to get the strain induced. The stress strain curve for every 100 µ strains of the reinforcement material is given as input for metal. The FE model is shown in the Figure 4-10 and the displacement plot for a load of 25 N is shown in Figure The comparison of the results between FEA and the test data is shown in Figure The curve clearly shows a good match upto 400 % elongation for 2-parameter Mooney-Rivlin material model. The shear stress in the elastomer for failure load is more than the shear strength of the material which is observed in the corners of the elastomer metal combination. The corner stresses on the elastomer are neglected because of stress singularity.

14 107 Figure 4-10: FE model of the QLSS specimen Figure 4-11: Displacement plot in a QLSS specimen

15 108 Figure 4-12: Comparison of FEA predictions and Experimental Results for a QLSS Specimen 4.5 CONCLUDING REMARKS The material characterization of elastomer and reinforcement shims used in flex bearings of solid rocket motors are carried out. The hyper-elastic modeling of elastomer was studied with four material models and 2-parameter Mooney-Rivlin model was found to be most suitable for modeling the flex bearing. The material models are validated at the specimen level. The predictions from the models are having a very close match with the test data. The model fitted data varies within 10-11% with respect to the experimental data. The Finite Element Modelling and Analysis of the flex seal using the validated material models will be detailed in the next chapter.

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