Uncertainties in material models and their influence on structural behavior and reliability are important considerations in analysis and design of structures. In this article, a methodology based on the evidence theory is presented for uncertainty quantification of constitutive models. The proposed methodology is applied to Johnson–Cook plasticity model while considering various sources of uncertainty emanating from experimental stress–strain data as well as method of fitting the model constants and representation of the nondimensional temperature. All uncertain parameters are represented in interval form. Rules for agreement, conflict, and ignorance relationships in the data are discussed and subsequently used to construct a belief structure for each uncertain material parameter. The material model uncertainties are propagated through nonlinear crush simulation of an aluminium alloy 6061-T6 circular tube under axial impact load. Surrogate modeling and global optimization techniques are used for efficient calculation of the propagated belief structure of the tube response, whereas Yager’s aggregation rule of evidence is used for multi-model consideration. Evidence-based uncertainty in the structural response is measured and presented in terms of belief, plausibility, and plausibility-decision values.
JohnsonGRCookWH. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, 1983. Proceedings of 7th international symposium on Ballistics, The Hague, The Netherlands541–547.
2.
HolmquistTJJohnsonGR. Determination of constants and comparison of results for various constitutive models. J Phys IV France1991; 1(C3): 853–860.
3.
ShirakashiTMaekawaKUsuiE. Flow stress of low carbon steel at high temperature and strain rate, part 1: property of incremental strain method in impact compression test with rapid heating and cooling systems. Bull Jpn Soc Precis Eng1983; 3: 161–166.
4.
ZerilliFJArmstrongRW. Dislocation-mechanics-based constitutive relations for material dynamics calculations. J Appl Phys1987; 61: 1816–1825.
5.
BammannDJChiesaMLJohnsonGCModeling large deformation and failure in manufacturing processes. In: TatsumiTWannabeEKambeT (eds). Theoretical and applied mechanics, New York: Elsevier Science, 1996, pp. 359–376.
6.
GrayGTChenSRWrightW. Constitutive equations for annealed metals under compression at high strain rates and high temperatures. Los Alamos National Laboratory, LA-12669-MS1994.
7.
MeyersMA. Dynamic behavior of materials, New York: Wiley, 1994, pp. 328–328.
8.
RuleWK. A numerical scheme for extracting strength model coefficients from Taylor test data. Int J Impact Eng1987; 19(9-10): 797–810.
9.
SolankiKNHorstemeyerMFSteeleWG. Calibration, validation, and verification including uncertainty of a physically motivated internal state variable plasticity and damage model. Int J Solid Struct2010; 47: 186–203.
10.
SolankiKNAcarERais-RohaniM. Product design optimization with microstructure-property modeling and associated uncertainties. Int J Design Eng2009; 1(2): 47–79.
11.
HeltonJCJohnsonJDOberkampfWL. An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab Eng Syst Saf2004; 85: 39–71.
12.
OberkampfWLHeltonJCJoslynCA. Challenge problems: uncertainty in systems response given uncertain parameters. Reliab Engng Syst Safety2004; 85(3): 11–19.
13.
TannerABMcGintyRDMcDowellDL. Modeling temperature and strain rate history effects in OFHC Cu. Int J Plast1999; 15: 575–603.
14.
GottsteinGKocksaUF. Dynamic recrystallization and dynamic recovery in <111> single crystals of nickel and copper. Acta Metall1983; 31: 175–188.
15.
JohnsonGRCookWH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech1985; 21: 31–48.
NicholasTMaterial behavior at high strain rates. In: ZukasJA (ed.) Impact dynamics, New York: John Wiley, 1982, pp. 27–40.
18.
HoltDLBabcockSGGreenSJ. The strain-rate dependence of the flow stress in some aluminum alloys. ASM Trans Quart1967; 60: 152–159.
19.
JohnsonGRHolmquistTJ. Test data and computational strength and fracture model constants for 23 materials subjected to large strains, high strain rates, and high temperatures. LA-11463-MS, Los Alamos National Laboratory1989.
20.
HogeKG. Influence of strain rate on mechanical properties of 6061-T6 aluminum under uniaxial and biaxial states of stress. Exp Mech1966; 6: 204–211.
21.
LeeWShyuJChiouS. Effect of strain rate on impact response and dislocation substructure of 6061-T6 aluminum alloy. Scripta Mater2000; 42: 51–56.
22.
SalehghaffariSRais-RohaniM. Epistemic uncertainty modeling of Johnson-Cook plasticity model using evidence theory. 13th AIAA/ISSMO multidisciplinary analysis and optimization conference, 9–13 August 2010, Fort Worth, Texas.
23.
ChoquetG. Theory of capacities. Annales de L’Institut Fourier1953–1954; 5: 131–295.
ShaferG. A mathematical theory of evidence, Princeton, NJ: Princeton University Press, 1976.
26.
DempsterAP. A generalization of Bayesian inference. J Roy Stat Soc Ser B1968; 30: 205–247.
27.
MooreRE. Methods and applications of interval analysis, Philadelphia, PA: SIAM, 1979.
28.
DuboisDPradeH. Possibility theory: an approach to computerized processing of uncertainty, New York: Plenum Press, 1988.
29.
WalleyP. Statistical reasoning with imprecise probabilities, London: Chapman and Hall, 1991.
30.
KlirGLSmithRM. On measuring uncertainty and uncertainty-based information: recent developments. Ann Math Artif Intell2001; 32: 5–33.
31.
BaeHGrandhiRVCanfieldRA. An approximation approach for uncertainty quantification using evidence theory. Reliab Eng Syst Saf2004; 86: 215–225.
32.
BaeHGrandhiRVCanfieldRA. Sensitivity analysis of structural response uncertainty propagation using evidence theory. Struct Multidisc Optim2006; 31: 270–279.
33.
AlyanakEGrandhiRVBaeH. Gradient projection for reliability-based design optimization using evidence theory. Eng Optim2008; 13: 923–935.
34.
AgarwalHRenaudJEPrestonEL. Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab Eng Syst Saf2004; 85: 281–294.
35.
YagerRKacprzykJFedrizziM. Advances in the Dempster-Shafer theory of evidence, New York: Wiley, 1994.
36.
YagerR. Quasi-associative operations in the combination of evidence. Kybernetes1987; 16: 37–41.
37.
YagerR. On the Dempster-Shafer framework and new combination rules. Inform Sci1987; 41: 93–137.
38.
DongWMShahHC. Vertex method for computing functions of fuzzy variable. Fuzzy Sets Syst1987; 24: 65–78.
39.
SalehghaffariSRais-RohaniM. Analysis and optimization of externally stiffened crush tubes. Thin-Wall Struct2011; 49(3): 397–408.
40.
FangHRais-RohaniMLiuZ. A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput Struct2005; 83: 2121–2136.
41.
Rais-RohaniMBammannDHauptT. Epistemic uncertainty modeling of plasticity material models. NSF CMMI research and innovation conference, 4–7 January 2011, Atlanta, GA.
42.
FlorenceALGoodierJN. Dynamic plastic buckling of cylindrical shells in sustained axial compressive flow. J Appl Mech1968; 23: 80–86.
