Entropy as an Indication of Damage in Engineering Materials

Transcription

1 Entropy as an Indication of Damage in Engineering Materials Mohammad Modarres Presented at Entropy 2018: From Physics to Information Sciences and Geometry Barcelona, Spain, May 14-16, May 2018 Department of Mechanical Engineering University of Maryland, College Park, MD 20742, USA

2 Acknowledgments The Team: 1. Mr. Huisung Yun (Current PhD candidate) 2. Dr. Anahita Imanian (Former PhD Student) 3. Dr. Victor Ontiveros (Former PhD Student) 4. Ms. Christine Sauerbrunn (Former MS Student) 5. Dr. Mehdi Amiri (Former Postdoc) 6. Dr. Ali Kahirdeh (Former Postdoc) 7. Prof. C. Wang (Corrosion/electrolysis consultant) 8. Prof. Enrique Droguett (Adjunct Associate Professor) 9. Prof. Mohammad Modarres (PI) Grant Funding For This Research is Partly From: Office of Naval Research 2

3 Objectives Describe damage resulted from failure mechanisms within entropic framework Understand sources of irreversible energy dissipation measurements in the fatigue process, i.e. mechanical, thermal, and acoustic Develop entropy for each dissipation measurement representing damage or state of current material based on thermodynamic, information, and statistical mechanics theorems Search for applications to Prognosis and Health Management (PHM) of structures 3

4 Motivation Common definitions of damage are based on observable markers of damage which vary at different geometries and scales Macroscopic Markers of Damage (e.g. crack size, pit densities, weight loss) Macroscopic Fatigues Markers include: crack length, reduction of modulus, reduction of load carrying capacity Issue: When markers of damage observed 80%-90% of life has been expended A 0 D n A0 A A 0 e Continuum damage mechanics [1] Micro-scale Nano-scale [2] Meso-scale [1] J. Lemaitre, A Course on Damage Mechanics, Springer, France, [2] C. Woo & D. Li, A Universal Physically Consistent Definition of Material Damage, Int. J. Solids Structure, V30,

5 0 Motivation (Cont.) Total Strain Energy Expended in 40 P-3 Aircraft with vastly Different Loading Histories when the Miner s Cumulative Damage Reaches 0.5 Energy (MJ/M^3) Aircraft Index Entropy to crack initiation Entropy to Failure (MJ/m 3 K) F=330 MPa F=365 MPa F=405 MPa F=260 MPa F=290 MPa [3] Time (Cycle) 10 4 Fracture Fatigue Failure (MJ m - 3 K -1 ) Number of Cycles to Failure [4] Entropy to Fracture [3] Anahita Imanian and Mohammad Modarres, A Thermodynamic Entropy Approach to Reliability Assessment with Application to Corrosion Fatigue, Entropy (2015): [4] M. Naderi et al., On the Thermodynamic Entropy of Fatigue Fracture, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, (2009):

6 Entropy as the Science of Reliability Hybrid Past Future Recent Physics of Failure Why Entropy? Entropy is independent of the path to failure ending at similar total entropy at failure Entropy accounts for complex synergistic effects of interacting failure mechanisms Entropy is scale independent 6

7 An Entropic Theory of Damage Degradation mechanisms Damage Dissipation energies Entropy generation An entropic theory follows [3] : Damage Entropy Failure occurs when the accumulated total entropy generated exceeds the entropic-endurance of the unit Entropic-endurance describes the capacity of the unit to withstand entropy Entropic-endurance of identical units is equal Entropic-endurance of different units is different Entropic-endurance to failure can be measured (experimentally) and involves stochastic variability In this context we define Damage as: D = γ d γ d 0 γ de γ d 0 Entropy generation, γ d, monotonically increases starting at time zero from a theoretical value of zero or practically some initial entropy, γ 0, to an entropic-endurance value, γ d [3] Anahita Imanian and Mohammad Modarres, A Thermodynamic Entropy Approach to Reliability Assessment with Application to Corrosion Fatigue, Entropy (2015):

8 Sources of Dissipation in Fatigue Process Cyclic loading Acoustic Wave Plastic deformation Thermal dissipation [5] Ali Kahirdeh and M.M. Khonsari, Energy dissipation in the course of the fatigue degradation: Mathematical derivation and experimental quantification, International Journal of Solids and Structures 77 (2015):

9 Entropic Approaches in Fatigue Process Dissipation (Measurement) Source Entropic Approach Related Equation Plastic Deformation Second Law of Thermodynamics σ = 1 T 2 J q. T Σ n k=1 J k μ k + 1 Σ r T j=1 v j A j + 1 Σ h T m=1 T + 1 T τ: c m J m ( ψ) ε p Thermal Information Theory (Shannon) S = p i log p i Acoustic Emission Statistical Mechanics (Crooks) D(P F P R = P F,i ln P F,i P R,i 9

10 Entropy in Thermodynamics Entropy generation σ involves a thermodynamic force, X i, and an entropy flux, J i as: σ = Σ i,j X i J i (X j ) ; (i, j=1,, n) Entropy generation of important dissipation phenomena leading to damage: Thermal Diffusion Plastic deformation σ = 1 J T q. T + Σ n 2 k=1 J k μ k T + 1 T τ: ε p + 1 Σ r T j=1 v j A j + 1 Σ h T m=1c m J m ( ψ) Chemical reaction External fields J n (n = q, k, and m) = thermodynamic fluxes due to heat conduction, diffusion and external fields, T=temperature, μ k = chemical potential, v i =chemical reaction rate, τ =stress tensor, ε p =the plastic strain rate, A j =the chemical affinity or chemical reaction potential difference, ψ =potential of the external field, and c m =coupling constant *, ** S total = Wdiss T = Hysteresis Area T Hysteresis Area: From stress-strain analysis T: From surface temperature measured by infrared camera or thermocouple 10

11 Entropy in Thermodynamics (Cont.) Similarity of the total entropy-to-failure for all tests supports the entropic theory of damage offered proposed More tests needed to reduce the epistemic uncertainties and future confirm the theory Entropy to Failure (MJ/(K*m 3 ) Distribution of entropicendurance F=330 MPa F=365MPa F=405MPa F=260MPa F=290MPa Time(Cycle) x 10 4 [6] Damage P=405 MPa P=365MPa P=330MPa 0.2 P=290MPa P=260MPa P=190MPa P=215MPa Time(Cycle) [7] [6] Mohammad Modarres, A General Entropic Framework of Damage: Theory and Applications to Corrosion-Fatigue, Structural Mechanics TIM 2015, June 2015, Falls Church, VA, USA [7] Anahita Imanian and Mohammad Modarres, Structural Health Monitoring, 2018, Vol. 17(2)

12 Entropy in Information Theory Acoustic emission signals (waveforms) Acoustic emission waveforms Duration of the waveform [µsec] Duration of the experiment Histogram of the AE signals Estimation of the histogram and probability distribution of each individual signal Calculation of P i for each waveform AE signal amplitude [v] Probability distribution (P i ) Entropy S I = n 1 i=1 p i log 2 p i S R = ρ log ρ ρ R dx = = n i=1 p i log 2 (p i ) n i=1 p i log p i p R,i 12

13 Entropy in Information Theory (Cont.) Cumulative AE information entropy correlated measured damage in elastic modulus Information entropy provided better correlation to damage compared with other AE features, e.g. counts [8] [8] Sauerbrunn, Christine M., et al. "Damage Assessment Using Information Entropy of Individual Acoustic Emission Waveforms during Cyclic Fatigue Loading." Applied Sciences 7.6 (2017):

14 Entropy in Statistical Mechanics Relative entropy (Kullback-Leibler divergence) D(P F P R = P F,i ln P F,i P R,i In thermodynamics, relative entropy equals total entropy in forward process or reverse process D(P F P R = β W F β F [8] = β W F β E F + S F System = β Q F + S F System = S F Total For experimental proof, relative entropy is computed by repeating fatigue tests with same conditions and constructing forward / reverse work distributions [8] Gavin E Crooks and David A Sivak, Measures of trajectory ensemble disparity in nonequilibrium statistical dynamics, Journal of Statistical Mechanics: Theory and Experiment, doi: / /2011/06/P

15 Entropy in Statistical Mechanics (Cont.) Analysis Procedure Multiple tests Compute relative entropy Stress 1 β D P F P R = 1 β S F Total B Compose work distributions Curve of reversible process A 1 β D P R P F = 1 β S R Total Strain 15

16 Fatigue Tests Material and Specimen Material: stainless steel 304L Mechanical properties σ U [MPa] σ Y [MPa] Elongation [%] Hardness [RB] Chemical composition [w%] C Cr Cu Mn Mo N Ni P S Si Specimen: notched dogbone K T at notch: 4.04 K T at the hole:

17 Fatigue Tests (Cont.) Test Method Measurements Loading condition Normal Loading 2.4 ~ 24 kn 5 Hz, 1,000 cycle Excitation 1 ~ 10 kn 25 Hz, 500 cycle Extensometer AE sensor Thermocouple Mechanical damper Microscope 17

18 Test Results Test Summary Test Life at initiation Life at.25 mm Life at fracture 1 11,091 11,972 17, ,751 15,025 22, ,398 9,564 14, ,517 8,348 14, ,269 10,099 15, ,319 14,554 21, ,508 11,171 17, ,659 13,459 20, ,434 11,833 17, ,711 9,275 14,381 Average 8,888 9,608 14,607 18

19 Test Results (Cont.) Thermodynamic Entropy 19

20 Test Results (Cont.) AE Information Entropy (Ex: Test 1) 20

21 Test Results (Cont.) AE Relative Entropy (Ex.: Test 1) 21

22 Test Results (Cont.) Relative Entropy in Statistical Mechanics 22

23 Conclusions Three different approaches to derive the entropic damage were investigated: classical thermodynamics, statistical mechanics and information theory A thermodynamic theory of damage proposed and tested Damage model derived from 2 nd law of thermodynamics and used to develop models for reliability of structures The theory was verified through corrosion-fatigue tests The proposed theory offered a more fundamental model of damage and allowed incorporation of all interacting dissipative processes Statistical mechanics-based entropic damage theory was proposed Additional tests and verifications would be needed 23

24 Thank you 24

25 Motivation Common definitions of damage are based on observable markers of damage which vary at different geometries and scales Macroscopic Markers of Damage (e.g. crack size, pit densities, weight loss) Macroscopic Fatigues Markers include: crack length, reduction of modulus, reduction of load carrying capacity Issue: When markers of damage observed 80%-90% of life has been expended 25

26 Entropy in Thermodynamics Electrochemical dissipations σ = 1 T J M,az M FE Mact,a + J M,c z M FE Mact,c + J O,a z O FE Oact,a + J O,c z O FE Oact,c + 1 T J M,cz M FE Mconc,c + z O FJ O,c E Oconc,c Diffusion dissipations + 1 T J M,aα M A M + J M,c 1 α M A M + J O,a α O A O + J M,a 1 α O A O Mechanical dissipations Hydrogen embrittlement dissipation + 1 T ε p: τ + 1 T Y D +σ H Chemical reaction dissipations T = temperature, z M =number of moles of electrons exchanged in the oxidation process, F =Farady number, J M,a and J M,c = irreversible anodic and cathodic activation currents for oxidation reaction, J O,a and J O,c =anodic and cathodic activation currents for reduction reaction, E Mact,a and E Mact,c =anodic and cathodic over-potentials for oxidation reaction, E Oact,a and E Oact,c =anodic and cathodic over-potentials for reduction reaction, E Mconc,c and E Oconc,c =concentration over-potentials for the cathodic oxidation and cathodic reduction reactions, α M and α O =charge transport coefficient for the oxidation and reduction reactions, A M and A O = chemical affinity for the oxidation and reductions, ε p =plastic deformation rate, τ =plastic stress, D =dimensionless damage flux, Y the elastic energy, and σ H =entropy generation due to hydrogen embrittlement. [1] Imanian, A. and Modarres. M, A Thermodynamic Entropy Based Approach for Prognosis andhealth Management with 26

27 Thermodynamics of Damage: A Reliability Perspective Materials, environmental, operational and other types of variabilities in degradation forces impose uncertainties on the total entropic damage Assuming a constant entropic-endurance, D f The reliability function can be expressed as [8] P r T t c R(t c ) = 1 P r T t c = 0 t c g t dt = 1-0 D f =1 f(d)dd = 0 D f =1 f(d)dd T c =Current operating time; g t =distribution of time-to-failure, f(d t)= distribution of damage at t [8] Thermodynamics as a Fundamental Science of Reliability, A. Imanian, M. Modarres, Int. J. of Risk and Reliability, Vol.230(6), pp DOI: / X (2016). 27

28 Entropy Originated from Statistical Mechanics Forward / reverse process representing equations in statistical mechanics [6] π F (+W) π R ( W) = exp W F k B T Crooks fluctuation theorem W F diss k B T = W F F k B T Relative entropy = D π F π R = π F ln π F π R Load Load Load Extensio n = + Extensio n Extensio n Positive Negative [6] C. Jarzynski, Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale, Annu. Rev. Condens. Matter Phys. 2.1 (2011):

29 Statistical Mechanics Entropy Schematics of Entropy Computation Test 1 strain Cycle 1 Cycle 2 Cycle i Cycle f F R F R R F R F R F R cycle Test 2 strain cycle Test n strain F W i,2 R W i,2 cycle ρ ρ R,i ρ F,i W S i = k B log π F,i(+W) π R,i ( W) 29