94 chapter 4 the material characterization of afnor 15cdv6...
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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.
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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
4.1.1 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 3.3.2.
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
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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
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Figure 4-2: QLSS test specimen- elastomer
Figure 4-3: Stress Vs Strain for Tensile Test Specimen – Elastomer
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Figure 4-4: Stress Vs Strain for QLSS Specimen - Elastomer
4.1.2 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 3.3.2.
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Table 4-1: Heat Treatment Cycle for AFNOR 15CDV6
Treatment Temperature 0C SoakingTime Cooling Media
Annealing 875 ± 104
minutes/mm
Furnace cooling at therate of 500 C/Hr to room
temperature
Hardening 970 ± 10
1-2minutes/mm
(or 20minutes
min)
Oil Quench
Tempering 610 ± 10
8minutes/mm
(or 120minutes
min)
Oil Quench
The reinforcement material is characterized by carrying out
tensile test on the specimen as per ASTM A 370 – 92. 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
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Figure 4-6: Stress vs. Strain curve for tensile tests - Reinforcement
4.2 MATERIAL MODELING
4.2.1 Elastomer
For modeling elastomer, four material models viz., Mooney-
Rivlin 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
2201110 1133c Jd
IcIW 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
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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
21 1132
Jd
IW 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
k2N
1k k321
N
1i i
i 1Jd13W iii
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, in-
house experiment shows the behavior similar to Neo-Hookean model.
The strain energy potential of Yeoh model is
k2N
1k k
i1
N
1i0i 1J
d13IcW
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,
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deformation of elastomer for the pressure loading is about 30 – 40% in
compression and deformation due to shear is about 100-150 %. 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
4.3.1 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
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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 I3 = 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 = λ3 = 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
10
012
C-1-2 C Eq-(4-9)
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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 = 0.92029 & C10 = 1.8434
Ogden (Eq-4-2) µ = 8.5090 & α = 1.1537
Neo-Hookean (Eq-4-3) µ = 1.81806
Yeoh (Eq-4-4) C10 = 1.81806
The plot of
2
12 vs.
1 gives a straight line (Figure 4-8)
with slope c10 and intercept c01. [123-125].
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
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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
4.3.2 Reinforcement
The design criteria for flex bearings are the margin of safety to
be lower among 0.125 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.
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The need for modeling the reinforcement material as a multi-
linear 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 4-11. The comparison of the results between FEA
and the test data is shown in Figure 4-12. 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.
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Figure 4-10: FE model of the QLSS specimen
Figure 4-11: Displacement plot in a QLSS specimen
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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.