Structural Integrity Assessment on Solid Propellant Rocket Motors

B. Nageswara Rao K L University

Presentation in Pravartana 2016: Symposium on Mechanics at IIT Kanpur during February 12-14, 2016

In a solid rocket motor, the combustion reaction generates a large amount of thermal/potential energy that

is converted to kinetic energy by expansion through a nozzle, whereby the required lift-off thrust is created.

For the solid rocket motor to perform successfully in its mission, it is necessary for it to retain its structural integrity under a wide variety of mechanical loads, that are imposed on it during storage and operational phases.

This lecture deals with solid rocket motor propellant grain structural integrity analysis, including materials characterization, structural analysis, and failure criteria for margin/factor of safety estimation.

Figure-1.1: Free- standing grains and case-bonded grains

Figure-1.2: Cross-section of a typical case-bonded solid rocket motor. (A) Chamber; (B) Head end dome; (C) Nozzle; (D) Igniter; (E) Nozzle convergent portion; (F) Nozzle divergent portion; (G) Port; (H) Inhibitor; (I) Nozzle throat insert; (J) Lining; (K) Insulation; (L) Propellant; (M) Nozzle exit plane; (N) SITVC system; (O) Segment joint.

Figure-1.3: Evaluation of Structural Integrity.

For selection of grain configuration, the main factors taken into account are: �• Volume available for the propellant grain �• Grain length to diameter ratio �• Grain diameter to web thickness ratio �• Thrust versus time curve, which gives a good idea of what should be the Burning area versus web burned curve (see Figure-2.1) �• Volumetric loading fraction which can be estimated from required total impulse and actual specific impulse of available propellants �• Critical loads such as thermal cycles, pressure rise at ignition, acceleration, internal flow

Figure-2.1: Progressive, regressive and neutral burning

Figure-2.3 Typical solid propellant grain geometries.

Axisymmetric configurations

Cylindrical configurations

Three-dimensional geometries

Propellants

• CTPB (carboxy-terminated polybutadiene); • HTPB (hydroxy-terminated polybutadiene); • PBAN (polybutadiene acrylonitrile), • PS (polysulfide); • PVC (polyvinyl chloride).

Loads • pressure load

• Thermal load

• Gravity load

Figure-2.4: Basic geometric parameters of a right-circular cylinder geometry.

Materials Characterization

Structural Analysis

Failure Criteria

Structural Integrity Analysis

Modeling of Structural Response with the Development of Computational Methods

Observation of Response Phenomena

Development of Computational Models

Development / Assembly of Software / Hardware to implement the Computational Models

Post-processing and Interpretation of Results

Use of Computational Models in the Analysis and Design of Structures

Based on the nature of the final matrix equations, finite element methods are often referred to as:

Need experience selection of suitable elements for modeling

specification of boundary conditions for the intended

structural analysis under the specified loading conditions

interpretation of finite element analysis results

displacement method

force method

mixed method Commercial codes (viz., MARC, NASTRAN, NISA, ANSYS, etc.) and user’s guides are currently available to solve structural problems.

Materials Characterization

Figure-6.1: Tensile Specimens.

Figure-6.2: Uniaxial stress-strain behavior at constant strain-rate.

property change due to ageing 00

Ptt logKP +=

Figure-6.5: Master stress relaxation modulus curve with reduced time

Figure-6.3: Failure boundary envelope for HTPB propellant from

fracture data of uniaxial tensile specimens.

Figure-6.4: Variation of strain with temperature reduced

strain rate

Structural Response under various loads

Effect of Thermal Shrinkage

The Effects of gravity

Pressure rise at firing

STAR GRAIN CONFIGURATION FOR IGNITER MOTOR

GRAIN CONFIGURATION OF A TYPICAL MOTOR

HEAD END GRAIN CONFIGURATION

MID SEGMENT GRAIN CONFIGURATION

NOZZLE END SEGMENT GRAIN CONFIGURATION

* To idealize the grain configuration, the following elements are required (i) Axi-symmetric element(2-D model) (ii) Plane- strain element(2-D model) (iii) 3-D Brick element

* TYPES OF LOADS ( i ) Pressure load

( ii ) Thermal load

( iii ) Gravity load

MATERIALS IN SRM. ( i ) Casing (isotropic/orthotropic) ( ii ) Liner (Insulation material) ( iii ) Solid Propellant material

Young’s Modulus and Poisson’s ratio for the solid propellant material will be specified from Master stress relaxation modulus(MSRM) curve and the Bulk-modulus(K).

Master stress relaxation modulus (MSRM)

curve of a HTPB-based propellant grain

(1)

τ

−∑+=∞

=∞

kT1kkrel a2

texpAE)t(E

Where E∞ is the equilibrium modulus,

t is the time

τk ’S are relaxation times

)TT(c)TT(c)a(log

R2R1

Te -+−

= (2)

* The relaxation modulus curve is represented by means of Prony series in the form

c1,c2 are material constants TR – Reference temperature. T- Temperature * For the specified time, the modulus of the propellant material is

t2/1srel ))]t(E(sL[E == (3)

−=

K3E1

21ν (4)

•Hybrid - stress - displacement formulation: (1) Strain – displacement relation

-

{ } [ ]{ }qB=ε (5)

(2) Stress function

{ } [ ]{ }βP=σ (6)

#The elements in [ P ] matrix are functions of co-ordinates.

#These functions are obtained from equilibrium quations(without body forces) and Compatibility equations.

#{β}’s are unknown constants of element which will be expressed in terms of element displacement{q}

(3)Element stiffness matrix:

:

[ ] [ ] [ ] [ ]GHGk 1T −= (7)

Where

And [C] is a compliance matrix [ ] [ ] [ ][ ]dvPCPH T

v∫= (8)

-

Matrix

[ ] [ ] [ ]dvBPG T

v∫=

(9)

{ } [ ] [ ]{ }qGHβ 1−= (10)

(4) Load vector computations are as per standard procedures available in FEAST-C

(5) Assembly of element stiffness matrices and load vector are as per FEAST-C Architecture

* Solution of displacements by solving the following system of linear equations

-

[ ] { } { }GG FδK = (11)

Through,

(i) Frontal solver

(or)

Cholesky solver(Band solver)

available in FEAST-C.

(10) From the displacements for each element,{β} are computed as

{ } [ ] [ ]{ }qGHβ 1−= (12)

Using {β}:

Stresses in the element :

{ } [ ]{ }βP=σ

Strains in the element :

{ } [ ]{ }σCε =

TYPES OF ELEMENTS 4 Node iso-parametric axi-symmetric element

4 Node iso-parametric plane strain element

8 Node 3-D brick element

calculation of relaxation modulus at particular time from prony series

PROBLEM DESCRIPTION • Three layered infinitely long thick cylindrical shell of propellant grain, insulation and casing under

Case (a) pressure load

Case (b) Thermal load

Case (c) Gravity load

• Inside layer is propellant grain, middle layer is insulation and outer layer is casing

Validation of axi-symmetric,Plane strain and 3 D brick element with closed form solution and MARC software

GEOMETRICAL DETAILS •Grain inner radius = 50 cm

•Grain outer radius = 138.9 cm

• Insulation thickness = 0.5 cm

• Casing thickness = 0.78 cm

LOAD DETAILS Case (a) Internal pressure = 50ksc

Case (b) Thermal load of –38.0 oC

Case ( c) Gravity load of 1 g acting downward

Material Young’s modulus (KSC)

Poisson’s ratio

Density Kg/cm3

Coefficient of thermal expansion (/oC)

Casing 190000 0.3 0.0078 0.000011

Insulation 20 0.499 0.00178 0.0003

Propellant Case (a) 50 Case (b) 20 Case (c) 20

0.499 0.00178 0.0001

MATERIAL PROPERTIES

Location ANALYTICAL [ref 1]

Axi-symmetric

Plane Strain

3 D Brick Element

Radial disp. at inner port (cm)

2.6086 2.609 2.6086 (MARC)

2.606 2.558 (MARC)

2.631

Hoop strain at inner port (%)

5.21 5.149 5.1810 (MARC)

5.095 5.506 (MARC)

5.18

Comparison of results with closed form solution for Pressure load

Location ANALYTICAL [ref 1]

Axi-symmetric

Plane Strain

3 D Brick Element

Maximum radial Stress at the interface of propellant and insulation (ksc)

0.4545 0.4617 0.4545 (MARC)

0.462 0.4662 (MARC)

0.4564

Hoop strain at inner port (%)

3.35 3.365 3.327 (MARC)

3.32 3.26 (MARC)

3.326

Comparison of results with closed form solution for Thermal load

Location ANALYTICAL [ref 1]

Axi-symmetric

3 D Brick Element

Maximum slump displacement for vertical storage,w (cm)

0.7875

0.7895 0.7892 (MARC)

0.7859

Comparison of results with closed form solution for gravity load

Time march analysis for slump estimation of S200 Mid-Segement

-8.00 -4.00 0.00 4.00 8.00t in seconds (log base 10 scale)

0.00

0.40

0.80

1.20

1.60

Slump

Wma

x in c

m

Slump Vs Time

Axi-symmetric model3-D model

•STAR SEGEMENT GRAIN

PROBLEM DESCRIPTION: Thermal shrinkage analysis due to cooling from the stress free temperature of 68 oC to the room temperature of 30 oC for star segment grain

GEOMETRICAL DETAILS:

Grain length = 291.3 cm; Grain outer diameter (OD) = 314.4 cm; Star OD fore - end = 266 cm; Star OD aft - end = 270 cm;Insulation thickness=0.5 cm; and Casing thickness=0.78 cm.

•LOAD DETAILS: The nodal temperature difference of –47.5 oC (38*1.25 = 47.5 oC) is applied on each nodes of the propellant, insulation and casing. Here 1.25 is the factor of safety

BOUNDARY CONDITIONS

Symmetry boundary conditions is applied.

DETAILS OF MESH IDEALISATION

One half of a star is idealized using 4 node iso-parametric plane strain element;Type of element: 4 node iso-parametric plane strain element of type 103 *Total No of elements = 60; and Total No of Nodes = 78

•Finite element idealisation

•Deformed configuration

•Resultant displacement under thermal load

Slump estimation analysis of cylindrical segment grain for horizontal storage under gravity load

GEOMETRICAL DETAILS:Grain length = 800.0 cm; Length of the segment = 800 cm; Port diameter = 105 - 125 cm; Grain outer diameter (OD) = 317.4 cm; Insulation thickness=0.5 cm; and Casing thickness=0.78 cm.

LOAD DETAILS: Gravity load of 1 g acting downward direction

BOUNDARY CONDITIONS: Under horizontal storage the cylindrical - segment casing is supported at the bottom 90o arc at both ends. Symmetry boundary conditions are applied at both symmetry plane

Finite element idealisation of Cylindrical segment grain for horizontal storage under gravity load

•4 node axi-symmetric iso-parametric of same type has been used for casing, insulation and propellant using FEAST software * 8 node iso-parametric Hermann element (Type 33 ) has been used for propellant & insulation and 8 node iso-parametric general element of type 28 for casing using MARC software * Tying option is required at the interface of insulation and casing in MARC software for Hermann element whereas no tying option is required for feast-visco element * Results obtained using FEAST-VISCO element with 4 node iso-parametric element is as accurate as with results obtained with 8 node iso-parametric Hermann element using MARC

Additional advantages of FEAST-VISCO Elements

Failure Modes

Two types of failure criteria recognized by rocket industry, are yielding and fracture.

Failure due to yielding is applied to a criterion in which some functional of the stress or strain is exceeded

Fracture is applied to a criterion in which an already existing crack extends according to energy balance hypothesis.

Failure Investigations Experimentation with a variety of materials would show that the theory works well for certain materials but not very well for others.

Designer has to use / establish a suitable failure theory for the intended materials.

APPLICATION OF FRACTURE MECHANICS TECHNOLOGY TO PRESSURE VESSEL DESIGN AND MATERIAL SELECTION DATES BACK TO THE MISSILE MOTOR CASES OF THE EARLY 1960’S IN THE AEROSPACE INDUSTRY AND BRITTLE FRACTURES IN PETROCHEMICAL PLANTS. AFTER EXPERIENCING THE EXPLOSION OF AMMONIA PRESSURE VESSEL AND THE FAILURE OF A SECOND STAGE MISSILE MOTOR CASE DURING HYDROSTATIC PROOF PRESSURE TESTING

Figure 1. Failed Motor Casing Figure 2. Failed ammonia pressure vessel

HISTORY

Hydro-burst pressure tested AFNOR 15CDV6 steel chamber

Hydro-burst pressure tested ESR 15CDV6 steel chamber

A 300mm diameter maraging steel motor case after burst

BURST PRESSURE = 15.2 MPa MAXIMUM PRESSURE ESTIMATED = 18.9 MPa WITHOUT CONSIDERING MISMATCH (24.4% HIGH) WITH ELASTIC STRESS CONCENTRATION = 11.7 MPa FACTOR (Ke) (22.5% LOW) WITH PLASTIC STRESS CONCENTRATION = 14.6 MPa FACTOR (4% LOWER)

A 2000mm diameter maraging steel motor case after burst.

BURST PRESSURE = 10.3 MPa MAXIMUM PRESSURE ESTIMATED = 10.7 MPa WITHOUT CONSIDERING MISMATCH (5% HIGH) WITH ELASTIC STRESS CONCENTRATION = 9.4 MPa FACTOR (Ke) (8.7% LOW) WITH PLASTIC STRESS CONCENTRATION = 10.3 MPa FACTOR (COINCIDING)

p

Radial

Axial

0

200

400

600

800

1000

0 2 4 6 8 10

Strain x 103

Stre

ss, M

Pa

Equation (12)Test [4]

A typical axi-symmetric finite element model of a cylindrical pressure vessel

Stress-strain curve of Afnor15CDV6 Steel

0

5

10

15

20

25

30

0 4 8 12 16 20

Strain x103

Inte

rnal

pre

ssur

e, M

Pa

FEAFEA with spherical endTest [4]

Hoop strain at the outer surface in the cylindrical shell with internal pressure

0

200

400

600

800

1000

0 200 400 600 800 1000 1200

Internal Pressure, MPa

Effe

ctive

Stre

ss, M

Pa

Inner surface

Midddle layer

Outer surface

Variation of the effective stress from inner surface to outer surface of the thick- walled cylindrical vessel with the applied internal pressure upto the global plastic deformation.

0

200

400

600

800

1000

1200

0 2 4 6 8 10

Strain x 103

Pres

sure

, MPa

FEATest [2]

Hoop strain at the outer surface of the thick- walled cylindrical vessel with the applied internal pressure.

STRESS CONCENTRATION AND STRESS INTENSITY FACTOR

The Kmax - σf relationship

( )

−−

−=

p

u

f

u

fF mmKK

σσ

σσ

11max

ASTM STANDARDS FOR FRACTURE TOUGHNESS EVALUATION AND FATIGUE CRACK GROWTH

ASTM E399 - PLANE STRAIN FRACTURE TOUGHNESS OF METALLIC MATERIALS

ASTM E561 - STANDARD PRACTICE FOR DETERMINATION R-CURVE

ASTM E813 - JIC, A MEASUREMENT OF FRACTURE TOUGHNESS

ASTM E740 - FRACTURE TESTING WITH SURFACE CRACK SPECIMENS

ASTM E645& E646 - FRACTURE TOUGHNESS TESTING OF ALUMINIUM ALLOYS

ASTM E647 - CONSTANT LOAD AMPLITUDE FATIGUE CRACK GROWTH RATES ABOVE 10-8 M/CYCLE

ASTM E812 - CRACK STRENGTH OF SLOW BEND PRE- CRACKED

CHARPY SPECIMENS OF HIGH STRENGTH METALLIC MATERIALS

RUPTURED STEEL CYLINDERS

Failure assessment diagram for M300 grade maraging steel

CRACK GROWTH PHENOMENON

FRACTURE MECHANICS RELATED TO FATIGUE DEALS WITH THE CRACK GROWTH.

IN TERMS OF FRACTURE MECHANICS, THE FATIGUE BEHAVIOR CAN BE EXPRESSED IN THE FOLLOWING CYCLIC LOAD CRACK GROWTH .

1)(arg −=FactorLoadDesignXInduced

AllowableMSSafetyofinM

CONCLUDING REMARKS

Design of the propellant grains involves

vast knowledge and numerous techniques due to the nature of propellants, the geometry and architecture of propellant grains and to their operation modes in rocket motors.

A solid rocket motor is essentially a one – shot proposition.

Despite the advent of reusable motor cases,

a complete rocket motor is used only once, and cannot be pre –tested in full operation.

As a result, individual rocket motor reliability

must be assured by assuming the structural integrity of entire populations of motors on en – masse basis.

Heavy reliance on engineering design

verification processes is unavoidable.

THANK YOU