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Epistemic Uncertainty Quantification of Product-Material Systems Grant No. 826547 CMMI, Engineering Design and Innovation Shahabedin Salehghaffari PhD Student, Computational Engineering Masoud Rais-Rohani (PI, Research Advisor) Douglas J. Bammann (Co-PI) Prof. of Aerospace Engineering Prof. of Mechanical Engineering Masoud@ae.msstate.edu Bammann@me.msstate.edu Esteban B. Marin (Co-PI) Tomasz A. Haupt (Co-PI) Research Associate Prof. Ebmarin@cavs.msstate.edu Haupt@cavs.msstate.edu Center for Advanced Vehicular Systems Bagley College of Engineering

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2 Abstract Principles of evidence theory are used to develop a methodology for quantifying epistemic uncertainty in constitutive models that are often used in nonlinear finite element analysis involving large plastic deformation. The developed methodology is used for modeling epistemic uncertainty in Johnson-Cook plasticity model. All sources of uncertainty emanating from experimental stress-strain data at different temperatures and strain rates, as well as expert opinions for method of fitting the model constants and the representation of homologous temperature are considered. The five Johnson-Cook model constants are determined in interval form and the presented methodology is used to find the basic belief assignment (BBA) for them. The represented uncertainty in intervals with assigned BBA are propagated through the non-linear crushing simulation of an Aluminum 6061-T6 circular tube. Comparing the propagated uncertainty with belief structure of the crushing response—constructed by collection of all available experimental, numerical and analytical sources—the amount of epistemic uncertainty in Johnson-Cook model is estimated.

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3 Sources of Uncertainties in Plasticity Models Uncertainties in Simulation of Large Deformation Process Model Selection Uncertainty caused by different choices of Plasticity Models (Johnson-Cook, EMMI, BCJ, …) Uncertain Material Parameters reflecting incomplete knowledge of the defamation mechanism of metals Different Expert Opinions for fitting method of material constants Different Choices of Experimental Data (stress-strain curves): Types, Strain Rates, Temperatures Uncertainties in Experimental Data method of Experimentation, Measuring stress Model Form Uncertainty caused by making simplifications in mathematical representation of deformation process

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4 Uncertainty Modeling 1.Uncertainty Representation: –Establishment of an informative methodology for construction of Basic Belief Assignment (BBA) using available sources of experimental data as well as different expert opinions. –Using a proper aggregation rule to combine evidence from different sources with conflicting BBA. –Uncertainty representation of Johnson-Cook models in intervals with assigned BBA using the established methodology by collection of evidence from different experimental sources and fitting approaches of material constants. 2.Uncertainty Propagation: –Propagation of the represented uncertainty through the non-linear crushing simulation of an Aluminum 6061-T6 circular tube. –Obtaining bounds of simulation responses due to the variation of material constants in intervals using Design and Analysis of Computer Experiments to determine propagated belief structure. 3.Modeling Model Selection Uncertainty: –Using Yager’s aggregation rule to combine the propagated belief structure obtained from different formulations of Johnson-Cook models. 4.Uncertainty Quantification: –Constructing belief structure of the simulation response through consideration of available experimental, numerical and analytical sources of evidence.

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5 Data, Opinion Values of Constan tC2 Values of Constan tCn Values of Constan t C1 0.1 0.7 0.2 0.4 0. 5 0. 1 0.05 0.25 0.1 0.3 I1I1 I2I2 I3I3 I3I3 I2I2 I1I1 I1I1 I2I2 I3I3 I4I4 I5I5 Joint Belief (BBA) [I 1 (C 1 ), I 5 (C 2 ),…, I 3 (C n )] 0. 0036 m{[I 1 (C 1 ), I 5 (C 2 ),…, I 3 (C n )]} = m{[I 1 (C 1 )}×m{[I 5 (C 5 )}× … ×m {[I 3 (C n )} Propagated BBA I1I1 I2I2 I3I3 I4I4 InIn I5I5 From Evidence Collection to Evidence Propagation 0. 0009

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6 Mathematical Tools of Evidence Theory Consider Θ = {θ 1, θ 2,..., θ n } as exhaustive set of mutually exclusive events. Frame of Discernment is defined as –2 Θ = { , {θ 1 }, …, {θ n }, {θ 1, θ 2 }, …, {θ 1, θ 2,... θ n } } The basic belief assignment (BBA), represented as m, assigns a belief number [0,1] to every member of 2 Θ such that the numbers sum to 1. The probability of event A lies within the following interval –Bel(A) ≤ p(A) ≤ Pl(A) Belief (Bel) represents the total belief committed to event A Plausibility (Pl) represents the total belief that Intersects event A Bel(A) Bel(Ā) Pl(A) 01 Epistemic Uncertainty

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Relationship Types Between Uncertainty Intervals 7 Ignorance Relationship BBA: m({I 1 })=A / (A+B), m({I 2 })= 0, m({I 1,I 2 })=B / (A+B) Bel: Bel({I 1 })=A / (A+B), Bel({I 2 })= 0, Bel({I 1,I 2 })=1 Pl: Pl({I 1 })=1, Pl({I 2 })= B / (A+B), Pl({I 1,I 2 })=1 Agreement Relationship Since two disjoint intervals are combined into a single interval, BBA structure construction is meaningless Conflict Relationship BBA: m({I 1 })=A / (A+B), m({I 2 })= B / (A+B), m({I 1,I 2 })= 0 Bel: Bel({I 1 })=A / (A+B), Bel({I 2 })= B / (A+B), Bel({I 1,I 2 })=1 Pl: pl({I 1 })= A / (A+B), Pl({I 2 })= B / (A+B), Pl({I 1,I 2 })=1 Data Points in interval 1 (I 1 ) = A Data Points in interval 2 (I 2 ) = B Total Data points = A+B (B/A < 0.5) (B/A > 0.8) (0.5 ≤ B/A ≤ 0.8) Ignorance Agreement Conflict BBA Structure AA A B B B I1I1 I1I1 I1I1 I2I2 I2I2 I2I2

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Different Types of BBA 8 Bayesian: all intervals of uncertainty are disjointed and treated as having conflict. Consonant: Similar to the case of ignorance, all intervals of uncertainty in consonant BBA structure are in ignorance. General: Intervals of uncertainty can be in both forms of ignorance and conflict. It is more prevalent in uncertainty quantification of physical systems.

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Methodology for BBA Construction in Intervals 9 Step 1: Collect all possible values of uncertain data and determine the interval of uncertainty that represents the universal set. Step 2: Plot a histogram (bar chart) of the collected data. Step 3: Identify adjacent intervals of uncertainty that are in agreement and combine them. Step 4: Identify the interval with highest number of data points (I m ) and recognize its relationship with each of the adjacent intervals to its immediate left and right (I a ),and construct the associating BBA Step 5: Consider the adjacent interval (I c ) to interval (I a ) –I a and I m are in ignorance relationship: recognize relationship type between intervals I c and I m and construct the associating BBA. –I a and I m are in conflict relationship: recognize relationship type between intervals I a and I c and construct the associating BBA.

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Aggregation of Evidence 10 Yager’s rule BBA of conflict between Information from Multiple Sources is assigned to the Universal Set (X) and interpreted as degree of Ignorance

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Uncertainty Representation of Johnson-Cook Models Expert Opinion 1: Johnson-Cook Model form − A -> yield stress − B and n -> strain hardening − C -> strain rate − m -> temperature Strain Rate Term Opinions – Log-Linear Jonson-Cook, 1983 – Log-Quadratic Huh-Kang, 2002 – Exponential Allen-Rule-Jones, 1997 – Exponential Cowper-Symonds, 1985 Temperature Term Opinions Expert Opinion 2: Fitting Methods Method 1: Fit constants simultaneously Method 2: Fit in three separate stages Expert Opinion 3: Choice of experimental test system Expert Opinion 4: Choice of stress-strain curve sets to fit constants Unknown Constants to be determined by fitting methods 11

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Uncertainty Representation of Johnson-Cook Models 12 Curve # Experimental Source 1 Curve # Experimental Source 1 Type Strain Rate (s -1 ) Temperature (K) Type Strain Rate (s -1 ) Temperature (K) 1Tension63460511Torsion11293 2Tension62750512Torsion1293 3Tension62447213Torsion0.001293 4Tension62229314Torsion0.1293 5Torsion9929315Compression800293 6Torsion4829316Compression0.008293 7Torsion3929317Compression40293 8Torsion23929318Compression2293 9Torsion13029319Compression0.1293 10Torsion126293---- Experimental Source 2Experimental Source 3 1Tension4.8e-52971Compression1000298 2Tension282972Compression2000298 3Tension652973Compression3000298 4Tension1e-055334Compression4000298 5Tension185335Tension5.7E-04373 6Tension1305336Tension1500373 7Tension1e-056447Tension5.7E-04473 8Tension236448Tension1500473 9Tension54644---- Testing Requirements − Produce the required dynamic loads − Determine the stress state at a desired point of a specimen − Measure the stress and strain rates at the above point Resulting test data by different approaches always subject to epistemic uncertainty Test Data for Aluminum Alloy 6061-T6

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Uncertainty Representation Procedure 13 Histograms for Model 1, Source 1, Fitting Method 1 Experimental Source 1 Combinations Experimental Source 3 Experimental Source 2 Experimental Source 1 BBA for M2 BBA for M1 BBA for M2 BBA for M1 BBA Source 1 BBA Source 2 BBA Source 3 Intervals of Uncertainty With Assigned BBA for Each Type Johnson-Cook Model Combinations Histograms m ([200.74, 274.29])= (1330+1395)/4220=0.646 m ([274.29, 311.07])= 920/4220=0.218 m ([163.96, 274.29])= 245/4220=0.058 m ([90.4, 274.29])= (210+120)/4220=0.078 Agreement Conflict Ignorance BBA Construction for Constant A Model 1 Method 1 A1 A2 A4 A3 A B n C m

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Uncertainty Propagation m({A1,B1,C1,n1,m1}) m({A1,B1,C1,n2,m1}) m({A3,B2,C3,n3,m1}) m({A(i),B(j),C(k),n(l),m(o)})= m ({A(i)})×m ({B(j)}) ×m ({C(k)})× m ({n(l)})× m ({m(o)}) Consider All Sets of Uncertain Variables BBA Structure for Johnson-Cook Model 1 m({A1}) m({A2}) m({A3}) m({B2})m({B1}) m({C2})m({C1})m({C3}) m({n1})m({n2}) m({n3}) m({m1}) Generate Random Samples for each Set of Uncertain Variables Perform Crush Simulations to Obtain Output of Interest (Mean and Maximum Crush Force) Establish metamodels Between Uncertain Variables and output of interest for each set Perform global optimization analysis using the established metamodel To obtain intervals for output of interests Assign a BBA to each obtained interval for output of interests Aggregate Propagated BBA from different sources 14

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15 Uncertainty Propagation Finite Element Model Random Samples Variables: Material Constants Outputs: Time Duration & Crush Length Simulation Description Tube Length: 76.2 mm Tube Thickness: 2.4mm Tube Mean Radius: 11.5 mm Attached Mass: 127 g Mass Velocity: 101.3 m/s Element Number: 1500 Final representation of uncertainty for outputs of interest (final BBA structure for Mean or Maximum Crush Load) Modeling Model Selection Uncertainty of Johnson-Cook (JC) based Material Models BBA Structure of output of interest using JC Type#1 BBA Structure of output of interest using JC Type#2 BBA Structure of output of interest using JC Type#3 BBA Aggregation

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Uncertainty Propagation 16 Metamodeling Technique – Radial Basis Functions (RBF) with Multi-quadric Formulation r = normalized X r = normalized X Material Constants Design Variables: Material Constants Crush Length Simulation Response: Crush Length Collapsed shapes of some samples

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17 Construction of Belief Structure for Crush Length BBA Aggregation Available Sources of Evidence for Crush Length: Experimental (E): 13.9 Analytical: 13.1 Numerical: 12.03 1212.51313.514 0.0359 0.2278 0.2178 0.2985 0.22

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Belief: Epistemic Uncertainty: Belief Complement: Universal set: Element of Belief Structure for Crush Length: 18 Uncertainty Quantification 1212.51313.514 0.0359 0.2278 0.2178 0.2985 0.22 Belief Structure for Crush Length Propagated Belief Structure for Crush Length

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19 Developed Approach for Uncertainty Modeling Experimental Stress-Strain Curves Intervals of Uncertainty with Assigned BBA FE Simulation of Crush Tubes Using Material Models Propagated Intervals of Uncertainty with Assigned BBA Available Evidences for Crush Length Intervals of Uncertainty with Assigned BBA Uncertainty Representation Uncertainty Propagation Fully Covered: Increase Belief Not Covered: Decrease Belief Partially Covered: Increase Plausibility and Ignorance Uncertainty Representation of Output of Interests Comparison Propagated Belief Structure Belief Structure for Crush Length Comparison

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20 References Salehghaffari, S., Rais-Rohani, M., “Epistemic Uncertainty Modeling of Johnson-Cook Plasticity Model, Part 1: Evidence Collection and Basic Belief Assignment Construction ”, International Journal of Reliability Engineering & System Safety (under review), 2010. Salehghaffari, S., Rais-Rohani, M.,“Epistemic Uncertainty Modeling of Johnson-Cook Plasticity Model, Part 2: Propagation and quantification of uncertainty”, International Journal of Reliability Engineering & System Safety (under review), 2010. Johnson, G.R., Cook W.H., “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures”,In: Proceedings of 7th international symposium on Ballistics, The Hague, The Netherlands 1983;. 4, 1999, pp. 557–564. Hoge, K.G., “ Influence of strain rate on mechanical properties of 6061-T6 aluminum under uniaxial and biaxial states of stress”, Experimental Mechanics, 1966; 6: 204-211. Nicholas, T., “ Material behavior at high strain rates”, In: Zukas, J.A. et al., 1982. Impact Dynamics, John Wiley, New York, 27–40. Helton, J.C., Johnson, J.D., Oberkampf, W.L., “An exploration of alternative approaches to the representation of uncertainty in model predictions”, International Journal of Reliability Engineering & System Safety,2004; 85: 39–71. Shafer, G., “A mathematical theory of evidence”, Princeton, NJ: Princeton University Press; 1976. Yager, R., “On the Dempster-Shafer Framework and New Combination Rules”, Information Sciences, 1987; 41: 93-137. Bae, H., Grandhi, R.V., Canfield, R.A., “Epistemic uncertainty quantification techniques including evidence theory for large- scale structures”, Computers & Structures, 2004; 82: 1101–1112. Bae, H., Grandhi, R.V., Canfield, R.A., “An approximation approach for uncertainty quantification using evidence theory”, International Journal of Reliability Engineering & System Safety, 2004; 86: 215–225.

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