An Overview of Multicontinuum Theory with Application to Progressive Failure of Large Scale Composite Structures Don Robbins Chief Engineer Firehole Technologies,

An Overview of Multicontinuum Theory with Application to Progressive Failure of

Large Scale Composite Structures

Don RobbinsChief EngineerFirehole Technologies, Inc.Laramie, Wyoming

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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


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


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.


Consequences of Matrix Damage/Failure Isotropic Stiffness Degradation

9


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


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


11

Independent Variables for Predicting Damage/Failure in Composites

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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?

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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.).


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


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Filtering Characteristics of Volume Average Stress StatesFiltering Characteristics of Volume Average Stress States

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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

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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!

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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

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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.


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

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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

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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.

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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

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• 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


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


37



22

2323


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


38

2

3

2323


zero



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


39

2323




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

Documents

An Overview of Multicontinuum Theory with Application to Progressive Failure of Large Scale Composite Structures Don Robbins Chief Engineer Firehole Technologies,