Transcript Document

Robust Materials Design of
Blast Resistant Panels
Stephanie C. Thompson, Hannah Muchnick, Hae-Jin Choi,
David McDowell
G. W. Woodruff School of Mechanical Engineering, Georgia
Institute of Technology, Atlanta, GA
Janet Allen, Farrokh Mistree
G. W. Woodruff School of Mechanical Engineering, Georgia
Institute of Technology, Savannah, GA
Presented at the 11th AIAA/ISSMO Multidisciplinary Analysis
and Optimization Conference, Portsmouth, Virginia
September 7, 2006
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Outline
• Motivation: Materials Design vs Material
Selection
• Solution Finding Method: cDSP
• Example Problem: BRP design
–
–
–
–
–
Robust Design
BRP Deflection Modeling
Problem Formulation
Design Scenarios
Results and Analysis
• Closing
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Motivation
Material Selection
Robust Design
• Designs are limited by the
finite set of available
materials
•
– Noise factors
– Uncertain control factors
– Model uncertainty
Material Design
•
•
•
Tailoring the properties of
materials to meet product
performance goals
Complex multiscale material
models are needed, but they
increase design complexity
Designers need a method for
choosing between material
design and material selection
Improving product quality by
reducing sensitivity to uncertain
factors
•
Robust design techniques
can be used to find design
solutions that reduce
sensitivity to uncertainty in
material properties, thus
leaving design freedom for
material design or selection
Our goal in this work:
•
Use robust design techniques to design a blast resistant panel
that is robust to uncertainty in material properties and loading
conditions
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The Compromise Decision Support
Problem
• A general framework for
solving multi-objective
non-linear, optimization
problems
–
–
–
–
Given
Find
Satisfy
Minimize
• Hybrid formulation
based on Mathematical
Programming and Goal
Programming
Mistree, F., Smith, and Bras, B. A., "A Decision Based Approach to Concurrent Engineering," Handbook of Concurrent Engineering, edited by
H.R. Paresai and W. Sullivan, Chapman & Hall, New York, 1993, pp. 127-158.
Mistree, F., Hughes, O. F., and Bras, B. A., "The Compromise Decision Support Problem and the Adaptive Linear Programming Algorithm,"
Systems
Structural Optimization: Status and Promise, edited by M.P. Kamat, AIAA, Washington, D.C., 1993, pp. 247-286.
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Example Problem: Blast Resistant
Panels (BRPs)
• Designed to dissipate blast energy through allowable
plastic deformation (crushing) of the core layer
• Three layer metal sandwich panel with a square honeycomb
core
• Design objective: minimize deflection of the back face sheet
while meeting mass and failure constraints
• Robustness : minimize variance of deflection of the back
face sheet due to variation in noise factors
Fleck, N.A. and V.S. Deshpande, 2004, "The Resistance of Clamped Sandwich Beams to Shock Loading". Journal of
Applied Mechanics, Vol. 71, p. 386-401.
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Designing a Robust Solution
Robust Goal:
Noise Factors
In order to design a BRP that is robust to the
blast load, the BRP should be robust to
uncertainty in t0 and p0.
Uncertain Control Factors
In order to maintain design freedom in the
materials of the BRP, the BRP should be robust
to uncertainty in yield strength and density of
each layer.
Minimize variance of
deflection, while
meeting performance
goals
p0, t0
Control Factors
Process
Response
σY,f, ρf, σY,c, ρc, σY,b,
ρb
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Predicting Panel Deflection:
Three-Stage Deformation
The BRP energy absorption and deformation is divided into 3 Stages:
•
Stage I – Fluid-structure interaction
– 0 ~ 0.1ms
– Impulse encounters top face sheet
•
Stage II – Core crushing
– 0.1 ~ 0.4 ms
– Impulse energy is dissipated as the matrix
of the core crushes and deforms
•
Stage III – Plastic bending and stretching
– 0.4 ~ 25 ms
– All remaining impulse energy is dissipated
as the back face sheet stretches and bends
Fleck, N.A. and Deshpande, V.S., 2004, "The Resistance of Clamped Sandwich Beams to Shock Loading".
Journal of Applied Mechanics. 71: p. 386-401.
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Predicting Panel Deflection
Nomenclature:
Stage I: Momentum of blast transferred to
front face sheet
ρf = density of front face sheet
Total kinetic energy/area at end of Stage I:
ρ b = density of back face sheet
p0 = peak pressure of impulse load
2 2
KEI 
2 p0 t0

Y,f
f
t0 = characteristic time of impulse load

δ/L = deflection per length of back face sheet
Stage II: Core crushing dissipates some energy
Crushing strain:
2 2
0 0
2p t

f
 Y , f   c R c Y , c   b Y ,b 
σy,b, = yield strength of back face sheet
hf = front face sheet thickness
hb = back face sheet thickness
H = core thickness
2 p 0 t 0   b Y ,b   c R c Y , c 
hc = core cell wall thickness
 c R c Y , c H  f  Y , f   f  Y , f   b Y ,b   c R c Y , c 
Rc = relative density of core
2 2
c 
σy,f, = yield strength of front face sheet
σy,c, = yield strength of core material
Total kinetic energy /
area at end of Stage II:
K E II 
ρc = density of core
Xue, Z. and Hutchinson, J.W., 2005, "Metal sandwich plates optimized for
pressure impulses". International Journal of Mechanical Sciences. 47: p.
545–569.
λs = core stretching strength factor
λc = core crushing strength factor
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Predicting Panel Deflection
Stage III: Remaining energy is dissipated through bending and
stretching of back face sheet
Plastic work/area in Stage III dissipates the kinetic energy/area
remaining at the end of Stage II:
P
W III
2
H  
 

 Y , f h f   Y , c R c H  S   Y , b h b    4 Y , b h b
 
3
L L
L
2


Equate plastic work/area dissipated in Stage III to the kinetic energy
remaining after Stage II to solve for the deflection of the back face sheet
W III  KE I  KE II
P
Xue, Z. and Hutchinson, J.W., 2005, "Metal sandwich plates optimized for pressure impulses". International Journal of
Mechanical Sciences. 47: p. 545–569.
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Predicting Panel Deflection
Deflection Equation:
 3   Y , b hb H  
 
 3 I 02   Y , f h f   Y , c R c H  S   Y , b hb 
1


2
2
2


9  Y , b hb H  2   
 L  
  f h f   c R c H   b hb 
 
w here H  H 1   c 
Y,f
h f   Y , c R c H  S   Y , b hb 

2

2 I 0   b hb   c R c H 
H 

 c R c  Y , c  f h f   f h f   b hb   c R c H







 
Variance of Deflection:
 

  Y ,b
  Y ,b 

  Y ,c
  Y ,c 

 Y , f
 Y , f 

 b
b 

 c
c 

 f
 f 

p0
 p0 

 t0
 t0
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Compromise DSP (cDSP)
Given
A feasible alternative: 3 layer sandwich
core panel with square honeycomb core
Assumptions:
• air blast impulse pressure of exponential
form
• blast occurs on the order of 10-4 seconds
• no strain hardening
Parameters: area of panel, geometric
constants for analysis, mean and variance
of noise factors
Find
Design variables:
•
•
•
•
•
material density
yield strength
layer thicknesses
cell wall thickness
cell spacing
Deviation variables: underachievement of
goals
Satisfy
Constraints:
• mass/area less than 100 kg/m2
• deflection less than 10% of span, relative
density greater than 0.07 to avoid buckling
• constraints for front face sheet shear-off
Bounds: bounds on design variables and
deviation variables
Goals:
• minimize deflection of back face sheet
• minimize variance of deflection of back
face sheet
Minimize
Weighted sum of the deviation
variables
• Scenario 1: Minimize deflection only
• Scenario 2: Minimize variance of
deflection only
• Scenario 3: Equal priority on both goals
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Design Variables and Bounds
C e rta in D e s ig n V a ria b le s
U pper
Lower
U n its
Bound
Bound
h
h
f
b
H
h
c
B
0 .0 0 1
0 .0 2 5
m
0 .0 0 1
0 .0 2 5
m
0 .0 0 1
0 .0 0 0 1
0 .0 2
0 .0 1
m
m
0 .0 0 5
0 .0 5
m
p(t)
hf
U n c e rta in D e s ig n V a ria b le s
U pper
Lower
Bound
Bound
U n its
ρ
ρ
ρ
f
2000
10000
k g /m
b
2000
10000
k g /m
c
2000
10000
k g /m
3
100
1100
MPa
σ
Y,b
100
1100
MPa
σ
Y,c
100
1100
MPa
0
to
hc
5
Y,f
p
hb
3
σ
N o is e F a c to rs
V a ria n c e
M ean
H
B
U n its
25
3 .7 5
MPa
0 .0 0 0 1
0 .0 0 0 0 1 5
seconds
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Design Constraints
Designer / Customer Requirements:
•
•
Mass per Area – Less than 100 kg/m2
Back Face Sheet Deflection – Less than 10% of Length (10cm)
Constraints to Prohibit Failure:
•
•
Relative Density of Honeycomb Core – Greater than 0.07 to avoid buckling
Front Face Sheet Shearing – Must not fail in shear
–
–
At clamped ends
At the core webs
Constraints on Uncertain Design Variables
•
Uncertain Material Properties – Must remain in bounds
Robust Constraints: All constraints that are a function of uncertain factors
are imposed as “robust” constraints; i.e. the worst case value including the
variation due to uncertain factors is used to evaluate the constraint
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Goals and Scenarios
Deviation Variables: di-, di+
• Measure the deviation from
the goal
• As we minimize the
deviation variables, we
approach the targets
Scenario 1: Optimizing
Goal 1: Minimize Deflection
Scenario 2: Stabilizing

•
•
W1 = 1
W2 = 0

T /   d 1  d 1  1
T  0.05 m
•
•
W1 = 0
W2 = 1
Goal 2: Minimize Variance of
Deflection


T  /    d 2  d 2  1
Deviation Function:
All priority on
robust goal; none
on deflection goal
Scenario 3: Robust
•
•
T   0.005 m

All priority on
deflection goal; none
on robust goal
W1 = 0.5
W2 = 0.5
Equal priority on
both goals

Z  W1  d 1  W 2  d 2
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Verifying the cDSP Results
B R P D im e n s io n s
C e ll S p a c in g , B
C o re H e ig h t, H
C e ll W a ll T h ic k n e s s , h
c
U n its
C o n s tra in t
T yp e
Bounds
Lower
U pper
S ta rtin g P o in t
Lower
M id
U pper
mm
mm
b e tw e e n
b e tw e e n
1
5
20
50
20
1 5 .7 6 7
20
1 4 .4 6 3 3
20
1 3 .0 3 5
mm
b e tw e e n
0 .1
10
2 .2 9 2 5
3 .9 3 6 4
6 .2 3 8 6
F ro n t F a c e S h e e t T h ic k n e s s , h
f
mm
b e tw e e n
1
25
1 3 .4 4 3
8 .3 2 8 5
6 .5 9 7 7
B a c k F a c e S h e e t T h ic k n e s s , h
b
mm
b e tw e e n
1
25
2 1 .6 1 1
25
25
MPa
MPa
MPa
b e tw e e n
b e tw e e n
b e tw e e n
100
100
100
1100
1100
1100
1100
1100
1100
1100
1100
1100
1100
1100
1100
b e tw e e n
2000
10000
2000
2000
2000
b e tw e e n
2000
10000
2000
2000
2000
b e tw e e n
2000
10000
2000
2000
2000
100
0 .1 0 0
100
0 .0 9 0 5
0 .2 1 6 1
0 .2 5 3 5
0 .1 9 6 9
100
0 .0 9 1 2
0 .3 5 4 9
0 .6 1 6 3
0 .3 1 7 8
100
0 .0 9 2 9
0 .5 2 6 6
1 .0 4 0 3
0 .4 0 1 2
W o rs t C a s e M a te ria l P ro p e rtie s
Y ie ld S tre n g th , B a c k
Y ie ld S tre n g th , C o re
Y ie ld S tre n g th , F ro n t
D e n s ity, B a c k
D e n s ity, C o re
D e n s ity, F ro n t
W o rs t C a s e C o n s tra in t A n a lys is
M ass
D e fle c tio n
R e la tive D e n s ity o f th e C o re
Mu
G am m a
k g /m
k g /m
k g /m
k g /m
3
3
3
2
m
n o u n it
n o u n it
n o u n it
at m ost
at m ost
a t le a s t
at m ost
at m ost
0 .0 7 0
2 .3 0 9
0 .6 0 0
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Verifying the cDSP Results
0 .9
Lower
0 .8
M id
Upper
D e via tio n F u n ctio n V a lu e
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0
0
4
8
12
16
20
24
28
32
36
Ite ra tio n
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Comparing the Scenarios:
What is a Robust Solution?
0 .1 0 0 0
D e fle c tio n (m )
0 .0 8 0 0
0 .0 6 0 0
0 .0 4 0 0
V a ria n c e o f D e fle c tio n
0 .0 2 0 0
N o m in a l D e fle c tio n
0 .0 0 0 0
S c e n a rio 1 :
S c e n a rio 2 :
S c e n a rio 3 :
O p tim iz in g
S ta b iliz in g
Robust
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Investigating the Need for New
Materials
All the solutions tend towards higher yield strengths and lower
3
densities
Y ie ld S tre n g th (M P a )
D e n s ity (k g /m )
S c e n a rio 1 : O p tim izin g
S c e n a rio 2 : S ta b ilizin g
S c e n a rio 3 : R o b u s t
Back
1 0 7 5 .0 0
1 0 7 5 .0 0
1 0 7 5 .0 0
C o re
1 0 7 5 .0 0
1 0 6 2 .0 2
1 0 7 5 .0 0
F ro n t
1 0 7 5 .0 0
8 2 3 .5 3
1 0 7 5 .0 0
Back
2 2 0 0 .0 0
2 4 3 4 .2 3
2 2 0 0 .0 0
C o re
2 2 0 0 .0 0
3 1 4 2 .3 9
2 2 0 0 .0 0
F ro n t
2 2 0 0 .0 0
2 2 0 0 .0 0
2 2 0 0 .0 0
Since there are no materials that have properties that satisfy these
ranges of material properties, a material must be designed or
problem formulation must be reevaluated
Back
C o re
F ro n t
C om m on
Y ie ld S tre n g th (M P a )
Lower
U pper
1050
1100
1050
1100
1050
1100
1050
1100
3
D e n s ity (k g /m )
Lower
U pper
2000
2400
2000
2400
2000
2400
2000
2400
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Investigating the Impact of the Mass
Constraint
Robust solutions were found for three values of the mass constraint
M a s s p e r A re a
C o n s tra in t
85
100
115
Y ie ld S tre n g th (M P a )
Back
C o re
F ro n t
1 0 7 3 .2 9
6 7 6 .5 6
1 0 7 4 .8 0
1 0 7 5 .0 0
1 0 7 5 .0 0
1 0 7 5 .0 0
1 0 7 5 .0 0
1 0 7 5 .0 0
1 0 7 5 .0 0
M a ss p e r A re a
D e n s ity (k g /m
Back
C o re
2 2 2 6 .9 4
2 5 8 6 .7 8
2 2 0 0 .0 0
2 2 0 0 .0 0
2 2 0 0 .0 0
2 2 0 0 .0 0
L a ye r H e ig h t (m m )
C o n stra in t
B a ck
C o re
F ro n t
85
100
115
2 4 .4 2 7
2 5 .0 0 0
2 5 .0 0 0
4 4 .5 9 4
1 4 .4 6 3
1 3 .4 0 6
4 .4 0 6
8 .3 2 9
1 4 .3 1 8
M ass per A rea C onstraint
85
100
115
D eflection (m )
0.0856
0.0661
0.0570
3
)
F ro n t
2 2 1 1 .0 2
2 2 0 0 .0 0
2 2 0 0 .0 0
C o re T o p o lo g y (m m )
B
hc
1 4 .8 9 8
2 0 .0 0 0
2 0 .0 0 0
V ariance of D eflection (m )
0.0139
0.0251
0.0221
0 .5 3 1
3 .9 3 6
4 .0 8 1
T otal (m )
0.0995
0.0912
0.0791
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Closure
• We have presented a methodology for the design of a
blast resistant panel that is robust to uncertainty in
loading conditions and material properties.
• A cDSP was formulated to balance the goals of
minimizing deflection and minimizing the variance of
deflection.
• Three design scenarios were presented to compare
optimizing, stabilizing, and robust solutions.
• The impact of the mass constraint was examined for
three values of the constraint.
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Acknowledgments
Stephanie Thompson and Hannah Muchnick both gratefully acknowledge
Graduate Research Fellowships from the National Science Foundation.
Financial support from AFOSR MURI (1606U81) is also gratefully
acknowledged.
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