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29.7.2004 Fracture Mechanics of solid rocket propellants

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Fracture Mechanics – Presentation of the course report

The application of fracture mechanics to the structural assessment of solid propellant rocket motors

Giuseppe Tussiwand

SP Lab, Politecnico di Milano (aerospace, energetics)


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Presentation outline

Introduction on Solid Rocket Technology, Effect of a crack in a motor

Material properties

Manufacturing problems

Literature Survey

Crack Deterioration Experiments

Subcritical Propagation, Fatigue and Service Life

Toughness Testing and unstable propagation limits


3

Solid Propellant Technology

Applications:

Mining

Solid Rocket Motors

Airbags

Gun propellant

Rescue Systems


4

Solid Rocket Motor Operation

motornozzlecombustion t

mmm∂∂=−

nb

solid aprdt

dx ==:

dtbb cAprA ⋅⋅=⋅⋅ρρ⋅⋅= bbcombustion Arm

dtnozzle cApm ⋅⋅=

pAcT tF ⋅⋅=

n

t

b

d AA

cap −

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅= 11

ρ

casegrain

nozzleportigniter


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The effect of a crack…

)1/(1 n

designb

crackedb

design

crackedA

Ap

p−

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

• Increase of the motor’s equilibrium pressure:

• Too much thrust, loss of the mission

• Volume combustion and burn through

• Instantaneous or delayed burst of the case

• Deflagration to Detonation Transition with mass detonation

The pressure increase depends on the propellant combustion:

AP-HTPB – n=0.4; A = 110% p =117% ;

DB / HNF / Nitramines: n=0.8; A = 110% p =160% ;


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Solid Rocket Propellant – Material Properties

Secondary explosive, ignitable by friction and shock

Heterogeneous: 80-90 % solid particles, 10%-15% binder

Other constituents: catalyzers, stabilizers, curing agent, bonding agent, plasticizer


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A filled elastomer with:

Time Dependence: Temperature Dependent Viscoelasticity

Plastic effects: softening; damage, healing, aging

Glass Transition

Pressure dependence; Pre-Strain Dependence

Poisson’s Ratio variation with strain

Stiffness dependence on strain (Payne-Effect,Mullins-Effect…)


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( ) ( ) ττετσ dtEt

t

rel ∂∂−= ∫

0

Relaxation

εησ = stress = elastic +viscous component


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Tref

T1 < Tref

Log time

E(t)

log(aT) for T1 vs.Tref


Tatt =*

Definition of a material time


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Modelling of the shift factor with the WLF equation

( )ref

refT TTC

TTCTa

−+−

−=2

1 )()(log


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Failure by:

• Disentanglement

• Chain scission

• Shear yielding

• Crazing

• Dewetting


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( ) ∑=

−∞ +=

n

k

t

krel keEEtE1

τ

Linear, temperature dependent viscoelasticity:

( ) ( ) ττετσ dtEt

t

rel ∂∂−= ∫

0

Shapery’s principle of correspondence:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−= ∑

=

−n

k

t

k keEEtE1

0 )1(~1)( τ

( )∫ −=t

effective dtEt1 E

0*** ττ virtual strain history – the same stresses

)( Treleffective aE E ⋅= ε

The method of reduced variables:

Valid for: E,σmax, ε


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Load cases for a motor

Thermal Cycling due to different CTE for case and propellant (high strains in the bore – low-medium number of cycles)

Vibrations (Transport,Handling, Captive Carriage…-moderate stresses, but a lot of cycles)

Ignition pressurization (very high stresses, low strains, just once)

If there is a crack, even microscopic…

how fast does it propagate with cycling? (subcritical propagation)

How big can it be ? (ballistics,critical propagation,structural-ballistic coupling)


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Manufacturing Flaws

• Mixing of the ingredients

• Casting (different methods)

• Curing at 60+ °C

• Cooling

• Extraction of the mandrel …troubles!


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Manufacturing Flaws

The extraction might cause microcracks at the bore: significant service life reduction;


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Cracks in the bore of the motor!


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Cracks in the bore of the motor


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Manufacturing Damage


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Solid Propellant Fracture, Literature Survey

Essentially: course material, Anderson, viscoelastic fracture, exact solutions.

Application and development of the work of Shapery & students allowed many predictions:

- qualitative and quantitative predictions on the P-E parameters, the toughness, the general behaviour

- scaling rules

- revision of the ASTM standard for toughness

These predictions were checked experimentally


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Solid Propellant Fracture, Literature Survey

∫Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂−= ds

xundyJ

ei

jijev σω

αGka =

0.20167n

IC

IC

K

K =

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2

1

2

εε

ε

ε

mCa *γ=

Shapery: LEFM valid for a linear viscoelastic material

Shapery: continuous FPZ advancement if the crack is loaded (even a little) – creep

Scaling law for toughness at different strain rates


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Solid Propellant Fracture Deterioration Analysis


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Microcracks and partialdewetting spotsFPZ: Damage

zone /microcrack zone

FPZ: bridging zone

True crack

Similarities to non-reinforced concrete!


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End of the FPZ

σfailure, chain scission

σdewetting

σ

w1 w2 COD

Cohesive Crack; Analogy to Hilleborg.


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Solid Propellant Testing – specimen reqs

- ease, accuracy and safety of manufacturing

- safety issues during test (the material is a secondary explosive)

- plane strain conditions

2a

B

2W

Standard MT geometry:

2 scales, 2 crack lengths

3 geometries


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Solid Propellant Testing – tabbed MT specimen

resin

Aluminium alloy

Cracked propellant sample


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Shape imperfection:

Misalignement of the crack: since

Non-constant crack length across the thickness:

Cracking at a large oxidiser grain: inserting the blade, a different crack edge radius

The crack is not perfectly straight because the notching blades (being very thin) bend with use

The crack advanced at the specimen surface because the sample was manipulated and it bent a bit.

Sources of dispersion


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Solid Propellant Testing, Fatigue test procedure

• Digital measurement of the crack length before testing under an optical microscope

• Insertion of the sample in the machine and application of a very small pre-load (0.5 N) to make sure the specimen was rigidly fixed in the grips.

• Cycling between a maximum and a minimum load previously chosen, at a frequency of about 0.4-0.5 Hz.

• Extraction of the sample after the application of the cycles, and digital crack length measurement with the same optical microscope.


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Solid Propellant Testing, Fatigue

Load input


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Solid Propellant Testing, Fatigue

Load input


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Fatigue, crack length measurements


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After the test…


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subcritical crack growth

solid propellantAP-HTPB-Al

high burning rate

R 0.1

da/dN = 0.0809(∆Κ)7.5011

R2 = 0.9533

0

0,05

0,1

0,15

0,2

0,25

0,3

0 0,2 0,4 0,6 0,8 1 1,2 1,4

∆Κ, MPa*mm0.5

da/d

N, m

m/c

ycle

Experimental results


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Fatigue, service life computations

Thermal cycling

Initial crack length Number of sustained loads

0.1 mm 863

0.2 mm 132

0.4 mm 22

0.6 mm 9

Computation of the induced stress

Application of exact solutions (2-4)

Propagation until a = a critical, pressurisation


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Fatigue, service life computations


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Solid Propellant Testing, Toughness

Initial crack length

Number of sustained loads

Hours of life

0.1 mm 1.096 e9 6087 hrs

0.2 mm 162’810’000 904.5 hrs

0.4 mm 24’131’000 134.1 hrs

0.6 mm 7'869’000 43.7 hrs

0.8 mm 3'530’400 19.6 hrs

Vibrations, (monochromatic, 50 Hz)


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ASTM E 399 – D 5045 91a

Pmax

5% secant

P5%=PQ

displacement

load


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( ) ( ) ( ) ( ) ττεττεσ dtEttdtEt

t

rel

t

rel ∫∫ −=−=00

( )

( ) 0100

0 10

1lim

11lim

EdteEdtEt

tE

EeEt

dtEt

tE

n

k

t

k

t

t

t n

k

t

k

t

t

k

k

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

=+=

∑∫

∫∑∫

=

−∞

→

∞=

−∞

∞→

τ

τ

Viscoelastic stiffness drop


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ε

σ

E0Max. stress,Pmax

ε @ max. stress

PQ’

Average stiffness computed with the viscoelastic model.

PQ

At low temperatures and/or high strain rates


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Temperature Strain Rate Initial Stiffness Maximum stress

71 °C 0.01 mm/min. 2.9 MPa 0.54 MPa

-30 °C 50 mm/min. 15.77 MPa 2.09 MPa

Simulated Toughness test at –30°C and 50 mm/min.


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Toughness, new data analysis methodology


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Tested strain rates: 50 mm/min. 2000 mm/min.

Tested temperatures: 71°C, 20°C, -30°C

2 different geometries; values at Tg close to PVC/PC KIC vs r e d u ce d s tr ain r ate

y = 0,2488x 2 - 0,3894x + 1,3433R2 = 0,9944

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

lo g (e 'aT)

K IC


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KIC, temperature dependence

0

1

2

3

4

5

6

-40 -20 0 20 40 60 80

T, °C

K IC, M

Pa m

m0.

5

Temperature dependence


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Kic, strain rate dependence

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4

log(strain rate)

K IC, M

Pa m

m0.

5

Strain rate dependence


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Cooling and Ignition - ComputationCombustion gases under pressure

Flame front; propellant ballistics (burning rate vs.pressure) is known

LEFM crack tip, orNLFM FPZ (using acohesive crack model like Hilleborg’s)

3 solutions:

ac = 2.4 – 3.4 mm


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Conclusions

• Fracture mechanics explains many structural failures

• It can be applied to constitutive and failure modelling of thematerial

• LEFM gives sensible results, FE Analysis with NLFM cohesive elements required

• In Europe we have a similar situation to the one with metals before FM and with Woehler curves / Goodman diagrams

• Application is badly required to reduce testing, improve safety and increase the performance of motors!

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The application of fracture mechanics to the structural ...civil.colorado.edu/~saouma/FM/fracture-in-solid-rocket-propellants.pdf · The application of fracture mechanics to the