An Overview of Multicontinuum Theory with Application to Progressive Failure of
Large Scale Composite Structures
Don RobbinsChief EngineerFirehole Technologies, Inc.Laramie, Wyoming
1
2
Helius:MCT is a software module that integrates seamlessly with commercial F.E. codes, providing accurate multiscale material response for progress failure analysis of composite structures.
TM
ProgressiveFailure
Simulation
Based on MultiContinuum TheorySimple to UseProven Accuracy for Progressive Failure SimulationExtremely Robust Convergence
3
Requirements for Effective Finite Element Analysis of Progressive Failure of Composite Structures:
3
a) Deformation must be represented at the appropriate scale (dictated entirely by mesh density and element type)b) Material Response must be predicted accuratelyc) Loading and Constraints must be realistic d) Must use an Effective Nonlinear Solution Strategy (e.g., incrementation scheme, regularization, etc.)
• Mesh discretization typically can not reach the material ply level must use ply grouping (sublaminates)• Lack of practical constitutive relations that accurately represent material degradation (damage/failure)• Convergence is very difficult to achieve
Common Difficulties:
4
1. Idealization of Failure in Composite Materials
2. Independent Variables for Predicting Failure
3. Failure Criteria, and the Consequences of Failure
4. MCT Characterization of Composite Materials
5. Selected Demonstration Problems
OUTLINE
5
Idealization of Failure in Composite Materials
Matrix damage/failure occurs due to stressesin the Matrix
Matrix damage/failure degraded Matrix properties
Fiber
Matrix
Fiber damage/failureoccurs due to stressesin the Fibers
In a heterogeneous composite material, failure is assumed In a heterogeneous composite material, failure is assumed to occur at the to occur at the constituent material levelconstituent material level. .
Fiber damage/failure degraded Fiber properties
The homogenizedcomposite stress
state does notprovide this info.
6
Idealization of Failure in Composite Materials
Degraded Matrix Properties+
Degraded Fiber Properties
Degraded Composite Properties
Micromechanical Finite Element
Model
A A micromechanical finite element modelmicromechanical finite element model is used to is used to establish consistency between the homogenized composite establish consistency between the homogenized composite properties and the damaged or failed constituent properties. properties and the damaged or failed constituent properties.
7
Idealization of Failure in Composite Materials
7
Matrix Failure Matrix Failure
Should matrix stiffness be degraded Isotropically or Orthotropically?
Fiber FailureFiber Failure
Should fiber stiffness be degradedIsotropically or Orthotropically?
How should constituent material stiffness be reduced in the event of a constituent damage/failure event?
8
1. Isotropic degradation requires less experimental data, and there is usually a lack of available experimental data
2. Isotropic degradation does degrade the stiffness of the primary load path. causes redistribution of the primary load path.
3. Isotropic degradation also degrades the stiffness of non-primary load paths. largely inconsequential for monotonic loading.
Idealization of Failure in Composite Materials
Consequences of Matrix Damage/Failure Isotropic Stiffness Degradation
9
Idealization of Failure in Composite Materials
Consequences of Fiber Damage/Failure Orthotropic Stiffness Degradation
1. Fiber breakage produces a very strong reduction in the axial stiffness of the fiber constituent. large degradation of E11.
2. Fiber breakage produces a significant reduction in the longitudinal shear stiffness of the fiber constituent. intermediate degradation of G12, G13.
3. Fiber breakage does not produce a significant reduction in the transverse normal or transverse shear stiffness of the fiber constituent. insignificant degradation of E22, E33, G23.
f f f
f f
f
10
Idealization of Failure in Composite Materials
10
In Situ Matrix Properties
In Situ Fiber Properties
MeasuredCompositeProperties
Micromechanical F.E. Model
Degraded MatrixProperties
Degraded FiberProperties
DegradedCompositeProperties
1
3
2
4
5
Hypothesize the modesand consequences of constituent damage
or failure
1. Measured Composite Properties2. In Situ Constituent Properties3. Constituent Damage or Failure Model4. Degraded In Situ Constituent Properties5. Degraded Composite Properties
Micromechanical F.E. Model
11
Independent Variables for Predicting Damage/Failure in Composites
11
Logical Candidates: Stress, Strain, or Both
… But exactly which measures of stress or strain?
Constituent Failure = f ( ? )
Homogenized Laminate-Level Stress (Laminate Average Stress)Homogenized Composite Stress (Composite Average Stress)
.
.Constituent Average Stress
.Actual stress field within the constituents of the microstructure
i.e., at what scale should strain & stress be represented?
12
Desired attributes for the independent variables used to predict composite material response
The variables chosen for predicting material responseThe variables chosen for predicting material responsemust be must be physically relevantphysically relevant to the material considered. to the material considered.
Calculation of the variables should be Calculation of the variables should be efficientefficient,, adding minimal computational burden to the structuraladding minimal computational burden to the structurallevel finite element analysis.level finite element analysis.
Calculation of the variables should be Calculation of the variables should be consistentconsistent;; the calculated variables should not be overly sensitive to the calculated variables should not be overly sensitive to a) idealization of the microstructural architectural, ora) idealization of the microstructural architectural, or b) micromechanical mesh-related issues.b) micromechanical mesh-related issues.
Method used to calculate the variables should be Method used to calculate the variables should be scalablescalable as the microstructural architecture becomes more complex as the microstructural architecture becomes more complex (e.g. unidirectional (e.g. unidirectional woven woven braided, etc.).braided, etc.).
Independent Variables for Predicting Damage/Failure in Composites
1313
Constituent Average Strain & Stress States
(j j T)i
=1,2…,# of constituents
Retains a significant level of physical relevance+
Scalable (same basic calculation method for unidirectional, woven, braided composites, etc.)
Efficient Calculation via MCT decomposition
Consistent (stability w/r to idealization and meshing of microstructure)
+
+
+
Cij
=
MCT decomposition requires linearized constitutive relations
i,j = 1,2,…,6
Independent Variables for Predicting Damage/Failure in Composites
14
Filtering Characteristics of Volume Average Stress StatesFiltering Characteristics of Volume Average Stress States
14
Constituent Average Constituent Average Stress StatesStress States
Filters out all stress components that are self-equilibrating over each individual constituent material.
Retains Poisson interactions between constituents.
Retains thermal interactions between constituents caused by differences in thermal expansion coefficients.
Composite Average Composite Average Stress StatesStress States
Filters out all stress components that areself-equilibrating over the entire RVE
Filters out self-equilibrating shear stresses that arise solely to satisfy local equilibrium.
Filters out Poisson interactions between constituents.
Filters out thermal interactions betweenconstituents caused by differences in thermal expansion coefficients.
+
+
Filters out self-equilibrating shear stresses that arise solely to satisfy local equilibrium.
Computation of Constituent Average Stress States
15
DDmm
mm
= = 1 1 dvdvijij VVmm
matrix average stress state
ijij
DDff
f f
= = 1 1 dvdvijij VVff
fiber average stress state
ijij
Composite RVEComposite RVE
DDcc
c c
= = 1 1 dvdvijij VVcc
composite composite averageaverage stress statestress state
ijij
DDcc = D = Dmm D Dff
It is NOT necessary to integrate stresses and strains over the micromechanical F.E. model.
MCT Decomposition
Instead, we use transfer functions [Hill Hill (1963), Garnich & Hansen (1990s)] (1963), Garnich & Hansen (1990s)] to accurately & efficiently decompose the composite average strain state into the constituent average strain states.
MCT Decomposition
16
fiber averagestrain state
compositecompositeaverageaverage
strain statestrain state
cc
transfer functiontransfer function[Hill (1963), Garnich & Hansen (1990s)][Hill (1963), Garnich & Hansen (1990s)]
cc
= = mmm m
+ + ffff
matrixaverage
strain state
mmcc
TTmm((CC, , CC, , CC, , , , , , , , ) )c m f c m f
c m f mc m f m
mm
= = CCmm ((m m
mm
)) ff
= = CCff ((f f
ff
))
linearizedlinearized about as many different about as many different discrete damaged statesdiscrete damaged states as desired as desired
matrix average stress state fiber average stress state
This process adds less than 3% to the overall cost of an equilibrium iteration In a typical F.E. analysis of a composite structure!
17
Idealization of Failure in Composite Materials
Damaged State 1Damaged State 1Undamaged matrix,Undamaged matrix,Undamaged fibersUndamaged fibers
Damaged State 2 Failed matrix,Undamaged fibers
Damaged State 3Damaged State 3 Failed matrix,Failed matrix,Failed fibersFailed fibers
matrixfailureevent
fiberfailureevent
A Simple Case: Three Discrete Damaged States
c
33
c
11
22
matrix failure event
fiber failure eventResponse of the composite
to imposed deformation
Constituent Failure Criteria
18
IfIf m m
((mm
) ) 1, 1,Matrix Failure CriterionMatrix Failure Criterion
IfIf f f
((ff
) ) 1, 1,ThenThen Matrix properties areMatrix properties areisotropically degradedisotropically degradedby a user-specifiedby a user-specifiedamount.amount.
Degraded Composite Properties
ThenThen Fiber properties areFiber properties areorthotropically degradedorthotropically degradedby a user-specifiedby a user-specifiedamount.amount.
Micromechanical F.E. Model
Fiber Failure CriterionFiber Failure Criterion
Constituent Failure criteria
• Fiber failure
• Matrix failure– Comprehensive
– Simplified
12154433222
211
mmmmmmmmmmm IIAIAIAIAIA
1
2
4
66
max2
2233322
233223322
3
223
233222
213
212
23311
222111
m
mm
A
AA
1442
11 mmmm IAIA
213
2124
223
233
2223
33222
111
2
I
I
I
I
Transversely isotropicstress invariants
213
2124
223
233
2223
33222
111
2
I
I
I
I
Transversely isotropicstress invariants
20
MCT Material Characterization
CCmm
,, mm
CCf f
,, ff
CCcc
,, cc
MicromechanicalMicromechanicalFinite ElementFinite ElementModel of RVEModel of RVE
in situin situconstituentconstituentpropertiesproperties
homogenizedhomogenizedcompositecompositepropertiesproperties
Step 1.Step 1. Optimize the Optimize the in situin situ constituent properties constituent properties so that the so that themicromechanical finite element model matches the micromechanical finite element model matches the measuredmeasured properties of the composite material properties of the composite material
EE1111, E, E2222, E, E3333, G, G1212, G, G1313, G, G2323 c c c c c c c c c c ccmeasuredmeasured
compositecompositepropertiesproperties 1212, , 1313, , 2323, , 1111, , 2222, , 3333
c c c c c c c c c c cc
21
MCT Material Characterization
21
mm 11ff 11
DecompositionDecompositionmeasuredmeasuredcompositecompositestrengthsstrengths
Step 2.Step 2.
Determine the coefficients of the constituent failure Determine the coefficients of the constituent failure criteria so that the micromechanical finite element criteria so that the micromechanical finite element model matches the model matches the measured strengths of the measured strengths of the composite materialcomposite material
SS1111, S, S1111, S, S2222, S, S2222, S, S1212, S, S2323 c+ c+ cc c+ c+ cc c c cc
measuredmeasuredcompositecompositestrengthsstrengths
constituentconstituentfailure criteriafailure criteria
Summary
• Micromechanical F.E. models are only used during the material characterization process, not during the actual structural-level finite element analysis.
• During the material characterization process, the micromechanical F.E. model is used to establish in situ constituent properties and homogenized composite properties for a finite set of discrete damage states (3).
• These properties are stored in a database and can be quickly accessed by the structural-level finite element model as dictated by the outcome of the constituent failure criteria.
• The coefficients of the constituent failure criteria are determined using only industry standard strength tests.
• The entire process of computing the constituent average stress states, evaluating the constituent failure criteria, and identifying the damaged properties of the composite material adds less than 3% to the total cost of an equilibrium Iteration in a structural-level finite element analysis.
22
Example: Atlas V CCB Conical ISA(Used on all Atlas V 400 series launches)
Loading: Combined vertical compression & horizontal shear,designed to drive failure in the top corner of the access door.
Diameter: 12.5’ to 10’ Height: 65 inchesGraphite/epoxy and honeycomb core
0
20
40
60
80
100
120
140
160
180
200
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11vertical displacement of load head (in)
% o
f fl
igh
t lo
adStructural Response Predicted with Helius:MCTTM
Failure State
Fiber FailureMatrix FailureNo Failure
190% Flight Load
185% Flight Load170% Flight Load
impending global failurestructural response softeningbecomes detectable
The modified ISA was tested to failure at AFRL Kirtland (Oct. 2008). Ultimate failure measured at 183% of Flight Load The ISA exhibited a nearly linear response up to ultimate failure Final failure process was very rapid (almost instantaneous) Failure initiated at door corners and progressed circumferentially
Failure initiated at door corners Rapidly propagated around circumference
0
20
40
60
80
100
120
140
160
180
200
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11vertical displacement of load head (in)
% o
f fl
igh
t lo
admeasured global
failure load
structural responsepredicted with MCT
190% Flight Load
Excellent agreement was achieved for:1) Location of Failure Initiation2) Failure Evolution Behavior3) Ultimate Load
Example: Unlined Cryogenic Composite Pressure Vessel
Loading: 1. Submerge tank in liquid nitrogen (T = -216C)2. Pressurize tank until a constant leakage rate was detected
Six tanks were tested with an average leak pressure of 1233 psi.
Measured: 1233 psiHelius:MCT: 1215 psi (-1.5%) LARC 02: 900 psi (-27.0%)
LeakPressure
c
c
No Damage MatrixDamage
FiberDamage
Crack Saturation / Leakage
MCT Model
Observedpermeation
Post-test procedure for detecting leak
locations
Good correlation between predicted region of permeation and observed region of permeation
crack saturationobserved
permeation
c
c
No Damage
MatrixDamage
FiberDamage
Sponsorship
The work presented herein has been sponsored by numerous DoD agencies as well as internal R&D
at Firehole Technologies.
AFRL & AFOSR via contract number FA9550-09-C-0074. Directors: Dr. David Stargel & Dr. Victor Giurgiutiu.
AFRL(Space Vehicles Directorate) via contract number FA9453-07-C-0191.Directors: Dr. Tom Murphey & Dr. Jeff Welsh.
NASA’s Exploration Systems Mission Directorate Director: Wyoming NASA Space Grant Consortium
Current government sponsorship includes:
The Firehole River (Yellowstone National Park, Wyoming)
The End
The remaining slides are extras to be used as needed
34
• Failure of the Homogenized Composite Material vs. Failure of the Heterogeneous Composite Material • Modes of Damage/Failure Addressed
• Consequences of Damage/Failure (i.e. stiffness degradation) Isotropic Degradation vs. Orthotropic Degradation Continuous Degradation vs. Discrete Degradation
• Local vs. non-local damage/failure
Idealization of Failure in Composite Materials
Issues to Consider
The Micromechanical F.E. Model represents an idealized microstructurethat is unlikely to accurately representa) the actual fiber distribution statisticsb) the actual distribution of micro-defects caused by manufacturing & curing
The Micromechanical F.E. Model does not accurately represent thefiber/matrix interphasea) the model often does not explicitly include the interphaseb) knowledge of interphase properties is typically absent or incomplete
The properties of the Matrix constituent material are sensitive to curing conditions (e.g., temperature, pressure, deformation, chemical environment). It is unlikely that a sample of bulk matrix material has been subjected to the same curing conditions as the matrix material in a fiber reinforced composite.
Use of Micromechanical Finite Element Models
Why do we need In Situ Constituent Properties?Aren’t Bulk Constituent Properties good enough?
36
Imposed Uniform Temperature Reduction in an Unconstrained Composite
2323
CompositeAverage Stress
Micromechanical Stress Field
Constituent Average Stress
2
3
11
zero
tension
compression
Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent.Constituent averaging process retains thermal interactions.Composite averaging process filters out thermal interactions.
37
Imposed Uniform Temperature Reduction in an Unconstrained Composite
37
Micromechanical Stress Field
Constituent Average Stress
22
2323
CompositeAverage Stress
2
3
zero
tension
compression
Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent.
Constituent averaging process retains thermal interactions.Composite averaging process filters out thermal interactions.
38
Imposed Uniform Temperature Reduction in an Unconstrained Composite
38
2
3
2323
CompositeAverage Stress
zero
Micromechanical Stress Field
Constituent Average Stress
33
tension
compression
Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent.
Constituent averaging process retains thermal interactions.Composite averaging process filters out thermal interactions.
39
Imposed Uniform Temperature Reduction in an Unconstrained Composite
39
2323
CompositeAverage Stress
Micromechanical Stress Field
Constituent Average Stress
2323
2
3
23Both the constituent averaging process and the composite averaging process filter out the transverse shear stress since it is self-equilibrating within each individual constituentas well as self-equilibrating over the entire RVE.
zerozero
Example: Cryogenic cooling of a composite
40
all all c c
= = 00 ijijf f
= = f f
= = 25.87 MPa 25.87 MPa 2222
f f
= = 66.75 MPa 66.75 MPa 1111
3333
fiber average stress statefiber average stress state
m m
= = m m
= = 17.25 MPa 17.25 MPa 2222
m m
= = 44.5 MPa 44.5 MPa 1111
3333
matrix average stress statematrix average stress stateσ33 = -10 MPa
σ22 = -10 MPa
TT
= = 217217CC
The constituent average stress states are inherently triaxial due to the thermal interactions between
constituents!
Example: Composite under biaxial compression
41
σ33 = -10 MPa
σ22 = -10 MPa
c c
= = 10 MPa10 MPa 2222
c c
= = 10 MPa10 MPa 3333
all other all other c c
= = 00 ijij
f f
= = f f
= = 10.9 MPa 10.9 MPa 2222
f f
= += +3.8 MPa 3.8 MPa 1111
3333
fiber average stress statefiber average stress state
m m
= = m m
= = 8.7 MPa 8.7 MPa 2222
m m
= = 5.73 MPa 5.73 MPa 1111
3333
matrix average stress statematrix average stress state
The constituent average stress states are inherently triaxial due to the Poisson interactions between
constituents!
Example: Composite Adapter for Shared PAyload Rides
Two identical laminated composite monocoque shellsIM7/8552 unidirectional tape (up to 64 plies thick)60 inches tall, 74 inches in diameter
CASPAR
Predicted Observed
Helius:MCT Progressive Failure Simulation
Initial Matrix CrackingOccasional
Matrix Cracking
Noise
Continuous Matrix Cracking Noise
Door Debonding
Lapband Gapping
Initial Matrix Failure
Initial Fiber Failure
Lower Radius Failure
Helius:MCT Ultimate Failure
Helius:MCT Predictionvs.
Experiment Observation
First significant slope increase
Test Stopped
Flight Load Limit (%)
Co
mp
ress
ive
Dis
pla
cem
ent
(in)
CASPAR was successfully tested to ultimate failure on April 14, 2008
Lower radius failure at 792% of FLL
Failure StateFiber FailureMatrix FailureNo Failure
Fiber Failure predicted in lower radius at 800% of FLL
0
0.5
1
1.5
2
2.5
3
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Com
pres
sive
Disp
lace
men
t (in
)
Flight Load (%)
Init
iati
on
of
Fib
er F
ailu
re :
74
0
Has
hin
Mat
rix:
110
0
Has
hin
Fib
er:
130
0
1st S
ign
if.
Slo
pe
Ch
ang
e:
980
Has
hin
Ult
imat
e: 1
950
??
?
Tes
t S
top
ped
: 8
47
Init
iati
on
of
Mat
rix
Fai
lure
:
260
Helius:MCT PredictionExperimental ObservationHashin Damage Evolution