Abstract
A methodology to account for the effect of epistemic uncertainty (regarding model parameters) on the strength prediction of carbon fiber reinforced polymer (CFRP) composite laminates is presented. A three-dimensional concurrent multiscale physics modeling framework is considered. A continuum damage mechanics-based constitutive relation is used for multiscale analysis. The parameters for the constitutive model are unknown and need to be calibrated. A least squares-based approach is employed for the calibration of model parameters and a model discrepancy term. The calibrated constitutive model is validated quantitatively using experimental data for both unnotched and open-hole specimens with different composite layups. The quantitative validation results are used to indicate further steps for model improvement.
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References
1.
Liu
,
X.
,
Furrer
,
D.
,
Kosters
,
J.
, and
Holmes
,
J.
, 2018
, “
Vision 2040: A Roadmap for Integrated, Multiscale Modeling and Simulation of Materials and Systems
,” NASA, Hampton, VA, Technical Report No. NASA/CR-2018-219771, E-19477, GRC-E-DAA-TN52454. U2.
Hashin
,
Z.
, 1962
, “
The Elastic Moduli of Heterogeneous Materials
,” ASME J. Appl. Mech.
,
29
(1
), pp. 143
–150
.10.1115/1.36364463.
Hill
,
R.
, 1963
, “
Elastic Properties of Reinforced Solids: Some Theoretical Principles
,” J. Mech. Phys. Solids
,
11
(5
), pp. 357
–372
.10.1016/0022-5096(63)90036-X4.
Mori
,
T.
, and
Tanaka
,
K.
, 1973
, “
Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions
,” Acta Metall.
,
21
(5
), pp. 571
–574
.10.1016/0001-6160(73)90064-35.
Christensen
,
R. M.
, and
Lo
,
K. H.
, 1979
, “
Solution for Effective Shear Properties in Three Phase Sphere and Cylinder Models
,” J. Mech. Phys. Solids
,
27
(4
), pp. 315
–330
.10.1016/0022-5096(79)90032-26.
Hill
,
R.
, 1965
, “
A Self-Consistent Mechanics of Composite Materials
,” J. Mech. Phys. Solids
,
13
(4
), pp. 213
–222
.10.1016/0022-5096(65)90010-47.
Sanchez-Palencia
,
E.
, 1980
, Non-Homogeneous Media and Vibration Theory
(Lecture Notes in Physics, Vol.
127
),
Springer-Verlag
,
Berlin
.8.
Suquet
,
P. M.
, 1987
, “
Elements of Homogenization for Inelastic Solid Mechanics
,” Homogenization Techniques for Composite Media
,
E.
Sanchez-Palencia
and
A.
Zaoui
, eds.,
Springer-Verlag
, Berlin, Germany.9.
Guedes
,
J. M.
, and
Kikuchi
,
N.
, 1990
, “
Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite Element Methods
,” Comput. Methods Appl. Mech. Eng.
,
83
(2
), pp. 143
–198
.10.1016/0045-7825(90)90148-F10.
Terada
,
K.
, and
Kikuchi
,
N.
, 1995
, “
Nonlinear Homogenization Method for Practical Applications
,” Computational Methods in Micromechanics
, Vol.
AMD-212/MD-62
,
S.
Ghosh
and
M.
Ostoja-Starzewski
, eds.,
ASME
, New York, pp. 1
–16
.11.
Matouš
,
K.
,
Geers
,
M. G. D.
,
Kouznetsova
,
V. G.
, and
Gillman
,
A.
, 2017
, “
A Review of Predictive Nonlinear Theories for Multiscale Modeling of Heterogeneous Materials
,” J. Comput. Phys.
,
330
, pp. 192
–220
.10.1016/j.jcp.2016.10.07012.
Hinton
,
M. J.
,
Kaddour
,
A. S.
, and
Soden
,
P. D.
, eds., 2004
, Failure Criteria in Fibre-Reinforced-Polymer Composites: The World-Wide Failure Exercise
,
Elsevier Science
,
Oxford, UK
.13.
Hinton
,
M. J.
, and
Kaddour
,
A. S.
, 2012
, “
The Background to the Second World-Wide Failure Exercise
,” J. Compos. Mater.
,
46
(19–20
), pp. 2283
–2294
.10.1177/002199831244988514.
Kaddour
,
A. S.
,
Hinton
,
M. J.
,
Smith
,
P. A.
, and
Li
,
S.
, 2013
, “
The Background to the Third World-Wide Failure Exercise
,” J. Compos. Mater.
,
47
(20–21
), pp. 2417
–2426
.10.1177/002199831349947515.
Clay
,
S. B.
, and
Engelstad
,
S. P.
, 2017
, “
Benchmarking of Composite Progressive Damage Analysis Methods: The Background
,” J. Compos. Mater.
,
51
(10
), pp. 1325
–1331
.10.1177/002199831667252016.
Clay
,
S. B.
, and
Knoth
,
P. M.
, 2017
, “
Experimental Results of Quasi-Static Testing for Calibration and Validation of Composite Progressive Damage Analysis Methods
,” J. Compos. Mater.
,
51
(10
), pp. 1333
–1353
.10.1177/002199831665853917.
Clay
,
S. B.
, and
Knoth
,
P. M.
, 2017
, “
Experimental Results of Fatigue Testing for Calibration and Validation of Composite Progressive Damage Analysis Methods
,” J. Compos. Mater.
,
51
(15
), pp. 2083
–2100
.10.1177/002199831667013218.
Whiteside
,
M. B.
,
Pinho
,
S. T.
, and
Robinson
,
P.
, 2012
, “
Stochastic Failure Modelling of Unidirectional Composite Ply Failure
,” Reliab. Eng. Syst. Saf.
,
108
, pp. 1
–9
.10.1016/j.ress.2012.05.00619.
Sriramula
,
S.
, and
Chryssanthopoulos
,
M. K.
, 2009
, “
Quantification of Uncertainty Modelling in Stochastic Analysis of FRP Composites
,” Compos. Part A: Appl. Sci. Manuf.
,
40
(11
), pp. 1673
–1684
.10.1016/j.compositesa.2009.08.02020.
Chiachio
,
M.
,
Chiachio
,
J.
, and
Rus
,
G.
, 2012
, “
Reliability in Composites—A Selective Review and Survey of Current Development
,” Compos. Part B: Eng.
,
43
(3
), pp. 902
–913
.10.1016/j.compositesb.2011.10.00721.
Acar
,
P.
, 2020
, “
Recent Progress of Uncertainty Quantification in Small-Scale Materials Science
,” Prog. Mater. Sci.
,
117
, p. 100723
.10.1016/j.pmatsci.2020.10072322.
Bhuiyan
,
F. H.
,
Sanei
,
S. H. R.
, and
Fertig
,
R. S.
, 2020
, “
Predicting Variability in Transverse Effective Elastic Moduli and Failure Initiation Strengths in UD Composite Microstructures Due to Randomness in Fiber Location and Morphology
,” Compos. Struct.
,
237
, p. 111887
.10.1016/j.compstruct.2020.11188723.
Malgioglio
,
F.
,
Pimenta
,
S.
,
Matveeva
,
A.
,
Farkas
,
L.
,
Desmet
,
W.
,
Lomov
,
S. V.
, and
Swolfs
,
Y.
, 2021
, “
Microscale Material Variability and Its Effect on Longitudinal Tensile Failure of Unidirectional Carbon Fibre Composites
,” Compos. Struct.
,
261
, p. 113300
.10.1016/j.compstruct.2020.11330024.
Thapa
,
M.
,
Mulani
,
S. B.
, and
Walters
,
R. W.
, 2019
, “
Stochastic Multi-Scale Modeling of Carbon Fiber Reinforced Composites With Polynomial Chaos
,” Compos. Struct.
,
213
, pp. 82
–97
.10.1016/j.compstruct.2019.01.06825.
Bhattacharyya
,
R.
,
Mahadevan
,
S.
, and
Basu
,
P. K.
, 2022
, “
Computationally Efficient Multiscale Modeling for Probabilistic Analysis of CFRP Composites With Micro-Scale Spatial Randomness
,” Compos. Struct.
,
280
, p. 114884
.10.1016/j.compstruct.2021.11488426.
Bogdanor
,
M. J.
,
Mahadevan
,
S.
, and
Oskay
,
C.
, 2013
, “
Uncertainty Quantification in Damage Modeling of Heterogeneous Materials
,” Int. J. Multiscale Comput. Eng.
,
11
(3
), pp. 289
–307
.10.1615/IntJMultCompEng.201300582127.
Bogdanor
,
M. J.
,
Oskay
,
C.
, and
Clay
,
S. B.
, 2015
, “
Multiscale Modeling of Failure in Composites Under Model Parameter Uncertainty
,” Comput. Mech.
,
56
(3
), pp. 389
–404
.10.1007/s00466-015-1177-728.
Oskay
,
C.
, and
Fish
,
J.
, 2008
, “
On Calibration and Validation of Eigendeformation-Based Multiscale Models for Failure Analysis of Heterogeneous Systems
,” Comput. Mech.
,
42
(2
), pp. 181
–195
.10.1007/s00466-007-0197-329.
Naghipour
,
P.
,
Arnold
,
S. M.
,
Pineda
,
E. J.
,
Stier
,
B.
,
Hansen
,
L. L.
,
Bednarcyk
,
B. A.
, and
Waas
,
A. M.
, 2017
, “
Multiscale Static Analysis of Notched and Unnotched Laminates Using the Generalized Method of Cells
,” J. Compos. Mater.
,
51
(10
), pp. 1433
–1454
.10.1177/002199831665170830.
Yuan
,
Z.
,
Crouch
,
R.
,
Wollschlager
,
J.
,
Shojaei
,
A.
, and
Fish
,
J.
, 2017
, “
Assessment of Altair Multiscale Designer for Damage Tolerant Design Principles (DTDP) of Advanced Composite Aircraft Structures
,” J. Compos. Mater.
,
51
(10
), pp. 1379
–1391
.10.1177/002199831665170731.
Godines
,
C.
,
DorMohammadi
,
S.
,
Abdi
,
F.
,
Villa Montero
,
M.
,
Huang
,
D.
, and
Minnetyan
,
L.
, 2017
, “
Damage Tolerant Composite Design Principles for Aircraft Components Under Static Service Loading Using Multi-Scale Progressive Failure Analysis
,” J. Compos. Mater.
,
51
(10
), pp. 1393
–1419
.10.1177/002199831667157532.
Dalgarno
,
R. W.
,
Action
,
J. E.
,
Robbins
,
D. H.
, and
Engelstad
,
S. P.
, 2017
, “
Failure Simulations of Open-Hole and Unnotched IM7/977-3 Coupons Subjected to Quasi-Static Loading Using Autodesk Helius PFA
,” J. Compos. Mater.
,
51
(10
), pp. 1421
–1432
.10.1177/002199831665317433.
Crouch
,
R. D.
, and
Oskay
,
C.
, 2010
, “
Symmetric Mesomechanical Model for Failure Analysis of Heterogeneous Materials
,” Int. J. Multiscale Comput. Eng.
,
8
(5
), pp. 447
–461
.10.1615/IntJMultCompEng.v8.i5.2034.
Bogdanor
,
M. J.
, and
Oskay
,
C.
, 2017
, “
Prediction of Progressive Damage and Strength of IM7/977-3 Composites Using the Eigendeformation-Based Homogenization Approach: Static Loading
,” J. Compos. Mater.
,
51
(10
), pp. 1455
–1472
.10.1177/002199831665098235.
Bhattacharyya
,
R.
, 2021
, “
Development of Constitutive Relation for Multiscale Modeling of Carbon Fiber Reinforced Polymer Composites Under Uncertainty
,” Ph.D. thesis,
Vanderbilt University
,
Nashville, TN
.36.
Oskay
,
C.
, and
Fish
,
J.
, 2007
, “
Eigendeformation-Based Reduced Order Homogenization for Failure Analysis of Heterogeneous Materials
,” Comput. Methods Appl. Mech. Eng.
,
196
(7
), pp. 1216
–1243
.10.1016/j.cma.2006.08.01537.
Bhattacharyya
,
R.
, and
Basu
,
P. K.
, 2020
, “
Multiscale Progressive Damage Analysis of CFRP Composites Using a Mechanics Based Constitutive Relation
,” Compos. Struct.
,
235
, p. 111759
.10.1016/j.compstruct.2019.11175938.
Bhattacharyya
,
R.
, and
Basu
,
P. K.
, 2020
, “
Multiscale Crack Band Model for Eigenstrain Based Reduced Order Homogenization
,” Comput. Mech.
,
66
(6
), pp. 1237
–1255
.10.1007/s00466-020-01896-039.
Sankararaman
,
S.
,
Ling
,
Y.
, and
Mahadevan
,
S.
, 2011
, “
Uncertainty Quantification and Model Validation of Fatigue Crack Growth Prediction
,” Eng. Fract. Mech.
,
78
(7
), pp. 1487
–1504
.10.1016/j.engfracmech.2011.02.01740.
Sankararaman
,
S.
, and
Mahadevan
,
S.
, 2015
, “
Integration of Model Verification, Validation, and Calibration for Uncertainty Quantification in Engineering Systems
,” Reliab. Eng. Syst. Saf.
,
138
, pp. 194
–209
.10.1016/j.ress.2015.01.02341.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
, 2001
, “
Bayesian Calibration of Computer Models
,” J. R. Stat. Soc. Ser. B
,
63
(3
), pp. 425
–464
.10.1111/1467-9868.0029442.
Ling
,
Y.
,
Mullins
,
J.
, and
Mahadevan
,
S.
, 2014
, “
Selection of Model Discrepancy Priors in Bayesian Calibration
,” J. Comput. Phys.
,
276
, pp. 665
–680
.10.1016/j.jcp.2014.08.00543.
Brynjarsdóttir
,
J.
, and
O'hagan
,
A.
, 2014
, “
Learning About Physical Parameters: The Importance of Model Discrepancy
,” Inverse Probl.
,
30
(11
), p. 114007
.10.1088/0266-5611/30/11/11400744.
Christie
,
M.
,
Glimm
,
J.
,
Grove
,
J. W.
,
Higdon
,
D.
,
Sharp
,
D.
, and
Wood-Schultz
,
M.
, 2005
, “
Error Analysis and Simulations of Complex Phenomena
,” Science-Based Prediction for Complex Systems, Los Alamos National Laboratory, Los Alamos, NM, Vol. 29, pp. 6
–25
.45.
Schervish
,
M. J.
, 1996
, “
P Values: What They Are and What They Are Not
,” Am. Stat.
,
50
(3
), pp. 203
–206
.https://doi.org/10.2307/268465546.
O'Hagan
,
A.
, 1995
, “
Fractional Bayes Factors for Model Comparison
,” J. R. Stat. Soc. Ser. B (Methodol.)
,
57
(1
), pp. 99
–138
.https://www.jstor.org/stable/234608847.
Rebba
,
R.
, and
Mahadevan
,
S.
, 2008
, “
Computational Methods for Model Reliability Assessment
,” Reliab. Eng. Syst. Saf.
,
93
(8
), pp. 1197
–1207
.10.1016/j.ress.2007.08.00148.
Ferson
,
S.
, and
Oberkampf
,
W. L.
, 2009
, “
Validation of Imprecise Probability Models
,” Int. J. Reliab. Saf.
,
3
(1/2/3
), pp. 3
–22
.10.1504/IJRS.2009.02683249.
Marden
,
J. I.
, 2000
, “
Hypothesis Testing: From p Values to Bayes Factors
,” J. Am. Stat. Assoc.
,
95
(452
), pp. 1316
–1320
.https://doi.org/10.2307/266977950.
Ling
,
Y.
, and
Mahadevan
,
S.
, 2013
, “
Quantitative Model Validation Techniques: New Insights
,” Reliab. Eng. Syst. Saf.
,
111
, pp. 217
–231
.10.1016/j.ress.2012.11.01151.
ASTM,
2008
, “Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials
,”
ASTM International
,
West Conshohocken, PA
, Standard No. 3039.52.
ASTM,
2008
, “Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials With Unsupported Gage Section by Shear Loading
,”
ASTM International
,
West Conshohocken, PA
, Standard No. D3410.53.
ASTM,
2005
, “Standard Test Method for Shear Properties of Composite Materials by V-Notched Rail Shear Method
,”
ASTM International
,
West Conshohocken, PA
, Standard No. 7078.54.
Engelstad
,
S. P.
, and
Clay
,
S. B.
, 2017
, “
Comparison of Composite Damage Growth Tools for Static Behavior of Notched Composite Laminates
,” J. Compos. Mater.
,
51
(10
), pp. 1493
–1524
.10.1177/002199831667594555.
Belnoue
,
J. P.-H.
,
Nixon-Pearson
,
O. J.
,
Thompson
,
A. J.
,
Ivanov
,
D. S.
,
Potter
,
K. D.
, and
Hallett
,
S. R.
, 2018
, “
Consolidation-Driven Defect Generation in Thick Composite Parts
,” ASME J. Manuf. Sci. Eng.
,
140
(7
), p. 071006
.10.1115/1.403955556.
Matveev
,
M. Y.
,
Belnoue
,
J. P.-H.
,
Nixon-Pearson
,
O. J.
,
Ivanov
,
D. S.
,
Long
,
A. C.
,
Hallett
,
S. R.
, and
Jones
,
I. A.
, 2019
, “
A Numerical Study of Variability in the Manufacturing Process of Thick Composite Parts
,” Compos. Struct.
,
208
, pp. 23
–32
.10.1016/j.compstruct.2018.09.09257.
Bhattacharyya
,
R.
, and
Adams
,
D.
, 2021
, “
Multiscale Analysis of Multi-Directional Composite Laminates to Predict Stiffness and Strength in the Presence of Micro-Defects
,” Compos. Part C: Open Access
,
6
, p. 100189
.10.1016/j.jcomc.2021.100189Copyright © 2022 by ASME
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