Enhancing Durability and Mobility
Through Optimized Plasticity
Natasha Vermaak
Assistant Professor, Lehigh University
Mechanical Engineering & Mechanics Department
2015, 2016 AFOSR Summer Faculty Fellowship Program
WPAFB/RQVC Advisor: Dr. Phil Beran
July 18-22, 2016
Multi-Scale Structural Mechanics & Prognosis
Program Review: Mr. James Fillerup
The Need
Many Air Force platforms involve structural components
that are subjected to cyclic loading conditions where
thermal management is required
Current designs for metallic structures based on elastic
(yield-limited) analysis lead to bulky structures that are suboptimal for weight-critical applications (hypersonic thermal
protection systems, stiffeners for exhaust wash structures).
The goal of this work is to overcome these limitations by
employing experimentally validated elastoplastic analysis in
parametric, shape and topology optimization.
2/11
Project Overview
Overall Objective: develop validated structural design methodologies that exploit
shakedown theorems for thermostructural performance
Main Tasks:
(1) Create parametric and shape/topology
optimization protocols for elastoplastic
shakedown response
(2) Use full-field experimental measurements
to assess the shakedown behavior of
structures in combined environments.
(3) Investigate the role of material processing,
geometry, loading and variability on hightemp. shakedown behavior (600—1000 C).
USAF Relevance: use advanced plastic design/optimization tools to reduce
weight, moderate maintenance/repair, enhance mobility/durability
3/11
Parameters of Interest (Task 1):
Optimization: several objectives and constraints will be explored while
enforcing shakedown behavior
minimizing mass,
maximizing thermal conductance,
achieving target structural stiffness, and
limiting plastic deformations.
Candidate materials may include nickel-based superalloys, cobalt-based
alloys, refractory alloys with different processing conditions as well as
multi-material systems.
Initial analyses restricted to elastic-perfectly-plastic assumptions; will
progressively add temperature-dependent properties, hardening, buckling.
Test Cases of interest include design of brackets, beams and plates in
curved and flat configurations as well as cylindrical and rectangular
panels with internal architecture (active cooling).
Background on elastoplastic design: Bree Diagram
s
e
DT
P
Time
s
e
J. Bree. Journal of Strain Analysis, 1967.
5
Lower Bound Shakedown Theorem (Melan, 1938)
Koiter, W. T. (1960). General theorems for elastic-plastic solids (pp. 165-220). Amsterdam: North-Holland.
Applied Mechanics of Solids by Allan F. Bower, Ch. 6.2.7
The EPP solid is guaranteed to shake down if any time independent
residual stress field ( ij ) can be found which satisfies:
1. Equilibrium,
ij x j  0
2. Boundary conditions, ij n j  0 on  2 R (where part of the boundary is
subject to a prescribed cycle of traction).
3. The residual stress combined with the elastic solution*  yield during
the entire load cycle: f s e + r £ 0
(
ij
ij
)
*Note there are many theorem extensions to hardening, temperature-dependent properties, creep, etc.
*Is a direct methods because information about the exact load response history is not needed
H. Abdalla, M. Y. A. Younan, and M. M. Megahed. Shakedown limit load determination for a kinematically hardening 90 deg pipe bend
subjected to steady internal pressures and cyclic bending moments. Journal of Pressure Vessel Technology, 133:051212–1–10, 2011.
A. Oueslati and G. de Saxcé. Static shakedown theorem for solids with temperaturedependent elastic modulus. In Limit States of
Materials and Structures, pages 157–178. Springer, 2009
D. K. Vu and M. Staat. Shakedown analysis of structures made of materials with temperature-dependent yield stress. International
Journal of Solids and Structures, 44(13):4524–4540, June 2007.
[52] M. Boulbibane and A. Ponter. A method for the evaluation of design limits for structural materials in a cyclic state of creep.
European Journal of Mechanics-A/Solids, 21:899–914, 2002.
6
Acreage example: Bree-analysis
Simplified: Fixed-fixed beam, uniformly heated/cooled
Material: Inconel 625
ρ = 8440 kg/m3
σo = 425 Mpa
E = 165 GPa
CTE = 14*10-6 (1/K)
S. Cinoglu, M. R. Begley, R. M. McMeeking, , N. Vermaak, Elastoplastic design of aerothermally heated aircraft surfaces, In preparation.
7
Pressure
Demonstration of Shakedown
d
t
t
a
b
c
d
c
c
d
8
Mechanical Stresses:
plane stress bending,
ignore shear
d 4u P

4
EI
dx
Moment
d 2u P
M ( x)  EI 2  12 x 2  L2 
24
dx
y
Py
s xx ( x, y)  M   3 12 x 2  L2 
I
2t
9
Thermal Stresses:
plane stress,
ignore shear
σzz = 0
εxx = 0
Using Stress-Strain Relations
s zz   (s xx  s yy )  EDT
1
s xx  (s yy  s zz )   DT
E
E
E
EDT

e xx  e yy  e zz  
e yy 

(1  )(1  2 )
(1  )
(1  2 )
e xx 
s yy
...
Solve for stress and strain components:
e xx  0
e yy  e zz  (1  )DT
s zz  s yy  0
s xx   EDT
10
Elastic Limit:
Py
s xx ( x, y)   3 12 x 2  L2   EDT
2t
Critical location is at the wall
L t
2 2
s xx ( , , DT )  s o
PL2
 2  EDT  s o
2t
PL2
 2 EDT  2s o
2
t
Elastic Limit
11
Lower Bound Shakedown Theorem (Melan, 1938)
Koiter, W. T. (1960). General theorems for elastic-plastic solids (pp. 165-220). Amsterdam: North-Holland.
Applied Mechanics of Solids by Allan F. Bower, Ch. 6.2.7
The EPP solid is guaranteed to shake down if any time independent
residual stress field ( ij ) can be found which satisfies:
1. Equilibrium,
ij x j  0
2. Boundary conditions, ij n j  0 on  2 R (where part of the boundary is
subject to a prescribed cycle of traction).
3. The residual stress combined with the elastic solution*  yield during
the entire load cycle: f s e + r £ 0
(
ij
ij
)
*Note there are many theorem extensions to hardening, temperature-dependent properties, creep, etc.
*Is a direct methods because information about the exact load response history is not needed
H. Abdalla, M. Y. A. Younan, and M. M. Megahed. Shakedown limit load determination for a kinematically hardening 90 deg pipe bend
subjected to steady internal pressures and cyclic bending moments. Journal of Pressure Vessel Technology, 133:051212–1–10, 2011.
A. Oueslati and G. de Saxcé. Static shakedown theorem for solids with temperaturedependent elastic modulus. In Limit States of
Materials and Structures, pages 157–178. Springer, 2009
D. K. Vu and M. Staat. Shakedown analysis of structures made of materials with temperature-dependent yield stress. International
Journal of Solids and Structures, 44(13):4524–4540, June 2007.
[52] M. Boulbibane and A. Ponter. A method for the evaluation of design limits for structural materials in a cyclic state of creep.
European Journal of Mechanics-A/Solids, 21:899–914, 2002.
12
Shakedown Limit:
Critical conditions:
Don’t go beyond yield in compression on loading
and beyond yield in tension on unloading
c
PL2
+ EDT = 2s o
2
t
Shakedown Limit
13
Interaction Diagram (Bree):
DTo 
so
E
Elastic Limit
PL2
 2EDT  2s o
2
t
Shakedown Limit
PL2
+ EDT = 2s o
2
t
t
Po  2s o  
L
2
14
Designing beyond first-yield can save weight
ΔT = 160 K; P = 1 MPa
Yield-limited
Design
Design beyond
first-yield
27% weight reduction
Level sets
of mass
m =  Lt
Shaded regions are infeasible
15
Shape/Topology Optimization Components
16/25
The level set method
Method for tracking evolving interfaces
S. Osher, UCLA, http://www.math.ucla.edu/~sjo/
J.A. Sethian, Berkeley,http://math.berkeley.edu/~sethian/level_set.html
17/25
Shape/topology optimization algorithm
1. Initialize the shape, W0
1. Iterate until convergence for
• Evaluate the advection velocity (via shape gradient)
• Transport the shape by
to obtain a new shape Wk+1
Gregoire Alliaire, Shape and Topology Optimization, Ecole Polytechnique, http://www.cmap.polytechnique.fr/~optopo/level_en.html
18/25
Incorporating lower-bound shakedown limits in
level-set based topology optimization schemes:
Initial Case:
minimize weight
under shakedown
constraints
Shakedown determination based on elastic analyses (Melan’s theorem).
F. Feppon , G. Michailidis, M. A. Sidebottom, G. Allaire, B. A. Krick, N. Vermaak, Introducing a level-set based shape and topology optimization
method for the wear of composite materials with geometric constraints, Structural and Multidisciplinary Optimization, doi:10.1007/s00158-0161512-4, 2016
R. Estevez, A. Faure, G. Michailidis, G. Parry, N. Vermaak, Design of thermoelastic multi-material structures with graded interfaces using topology
optimization, Structural and Multidisciplinary Optimization, Submitted.
N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Bréchet, Material interface effects on the topology optimization of multi-phase
structures using a level set method. Structural and Multidisciplinary Optimization, 50(4), pp.623-644, 2014.
N. Vermaak, G. Michailidis, A. Faure, G. Parry, R. Estevez, F. Jouve, G. Allaire, Y. Brechet, Topological Optimization with Interfaces, In: Archimats:
Architectured Materials in Nature and Engineering, Editors: Y. Estrin, J. Dunlop, P. Fratzl and Y. Brechet, Springer 2016, In press.
N. Vermaak, L. Valdevit, A. Evans, F. Zok, and R. McMeeking. Implications of shakedown for design of actively cooled thermostructural panels.
Journal of Mechanics of Materials and Structures, 6(9):1313–1327, 2012.
N. Vermaak, M. Boissier, L. Valdevit, and R. M. McMeeking, Some Graphical Interpretations of Melan’s Theorem for Shakedown Design, In: Direct
Methods of Structural Analysis, Editors: A.R.S. Ponter, Olga Berrera, A. Cocks, Springer 2016, In press.
Towards experimental validation of shakedown:
Heitzer, Michael, M. Staat, H. Reiners, and F. Schubert.
"Shakedown and ratchetting under tension–torsion
loadings: analysis and experiments."Nuclear engineering
and design 225, no. 1 (2003): 11-26.
Vermaak Tension/Torsion MTS
testing with furnaces ~ 1100 C,
High-temperature
extensometry and DIC
Enhancing Durability and Mobility
Through Optimized Plasticity
Natasha Vermaak ( [email protected] )
Assistant Professor, Lehigh University
Mechanical Engineering & Mechanics Department
2015, 2016 AFOSR Summer Faculty Fellowship Program
WPAFB/RQVC Advisor: Dr. Phil Beran
July 18-22, 2016
Multi-Scale Structural Mechanics & Prognosis
Program Review: Mr. James Fillerup