Full- field measurements & IdenHficaHon in solid mechanics TEDDINGTON Thermomechanical analysis of material behaviour J. Barton & A. Chrysochoos Part 2: Dissipa,on to characterise irreversible deforma,on mechanisms A. Chrysochoos LMGC Montpellier University - France GDR CNRS 2519 1
Goals Observing kinematic and energetic effects accompanying the material transformations Constructing energy balances consistency of behavioral models Quantitative imaging techniques full-field measurements IRT : since 1983 DIC : since 1991 DIC & IRT : since 2 DIC - Mechanics - displacement fields - strain and strain rate - stress - deformation energy rate IRT - Thermodynamics - absolute temperature - heat source (via the heat equation ) - intrinsic dissipation - coupling sources 2
Outline Theoretical background - dissipation, energy storage - thermomechanical (thm) coupling effects Experimental tools - temperature fields - displacement fields Focusing on cyclic loading Case study #1 : HCF of steel - material vs. structure effects - properties of intrinsic dissipation fields Case study #2 : LCF of rubber - intrinsic dissipation vs. strong coupling + thermal dissipation Concluding comments 3
Theoretical background Thm constitutive equations Generalised standard material formalism [Halphen & Nguyen, 75] state variables internal/free energy poten9al state equa9ons dissipa9on poten9al evolu9on equa9ons { T, ε, α} e( s, εα, ) ψ( T, εα, ) s = ψ,t T T r σ =ρψ, ε ϕ( q, ε, α;t,... ) A α =ρψ,α = ϕ,q σ ir =ϕ X, ε α =ϕ, α Irreversibility Material degradation Heat diffusion d 1 = σ ir : ε + X α. α d 2 = ϕ, q.q = T T.q 4
Energy balance (I) wi def = σ : ε = σ r : ε + σ ir : ε = σ r : ε + A α. α + d 1 wi e +ws i i w e : rate of elastic energy d 1 : intrinsic dissipation wi s : rate of strored energy incomplete balance!! 5
Energy balance (II) rate of internal energy ρ e = ρc T + (σ r : ε + A. α) (Tσ,T r = ρct + w i e + w i i s w thc : ε +TA,T. α) «thc» = thermomechanical couplings heat equation ρct + divq = σir : ε A. α + T σ r,t : ε + T A,T. α + r e comments d 1 C.1: C specific heat C.2 : q = k.gradt kinematics required C.3: T = T t + v. T 6
Experimental tools Quantitative imaging full field measurement system IRFPA camera [3,5] mm max frame rate 25 Hz 64 32 pixels 18 23 µm2 14 bits δx.1 mm (min 25 µm) δt.2 C CCD camera max frame rate 2 Hz 128 124 pixels 13 13 µm2 14 bits δx.1 mm (min 15 µm) Refs [Wa<risse et al., Exp. Mech, 21] [Chryso et al., IJES, 2] Hydraulic testing machine load cell : ± 25 kn frame : ± 1 kn max(fl)= 5 Hz en Rs= -1 7
Combining DIC & IRT Speckle image IR image x DIC (pixel) Grey scale 4 2 Ut ( x, y ) DIC, DIC DIC Map-to-map correspondence x IRT (pixel) T ( C) 3 26 y DIC (pixel) Tt ( x, y ) IRT, IRT IRT y IRT (pixel) 22 [Chryso et al., JoMMS, 21] tdic synchrocam t IRT (5 µs) ( xdic, ydic) target ( xirt, yirt) (.5 pixel) 8
Stress derivation stress triaxiality neglected plane stress Thin, flat sample σxx ( xyt,, ) σxy ( xyt,, ) + = x y σxy ( xyt,, ) σyy ( xyt,, ) + = [Chryso et al., x y JoMMS, 21] no volume variation e: thickness, w: width, L: length uniform distribution F( t) of tensile stress σ xx ( xt, ) = exp ( εxx ( xt, )) S over a cross-section εxx ( xt, ) σ xy ( xyt,, ) = σxx ( xt, ) y no overall shear loading x (, ) (, ) (, ) 2 xx xt xx xt w xt 2 (,, ) σ ε σ. no lateral stress yy xyt = y x 2 x 4 9
Heat rate Heat equation averaged over the sample thickness!!!!!!!!!!!!!!" +!! +!! +! + =!!!!!!"!"!"!!!!!!! Direct estimate of heat sources using noisy and discrete thermal data Thermal noise uniform power spectrum Gaussian probability distribution Estimate of the partial derivative operators Refs: IJES : 2 EXP- MECH : 27 JoMMS : 21 EXP- MECH : 214 thermal data projection onto spectral solutions (1995) periodic expansion and convolutive filtering by DFT (2) local approximation of θ using l.sq. fitting (24) POD: pre-filtering of thermal fields (213) 1
Focusing on cyclic tests Monochroma9c uniaxial loading 11
Focus on a load-unload cycle σ σ σ ε ε ε A (Τ, α) (i) A B B A (Τ, α) B (ii) ε A = ε B A = B (Τ, α) (iii) A = B (i) (ii) (iii) w def = t B σ : ε dt = d 1 dt + (ρ e t A Hysteresis loop : w def = A h ( for uniaxial loading) Load-unload cycle = thermodynamic cycle t B t A t B t A ρct + wi thc )dt t B w def = d t 1 dt + A wi t thc dt A 12 t B
IR image processing ρ + + = + 2 2 θ θ k θ θ C d 2 2 1 sthe t τth ρ C x y x θ ( xyt,, ) y Slow evolution of mean dissipation t d 1 = Local approximation function Linear PDE + Linear BC In phase with loading f L d 1 dτ w the = f L s the dτ cycle θ = θ d + θ the cycle θ app (x,y,t) = p 1 (x,y) cos(2π f L t) + p 2 (x,y) sin(2π f L t) + p 3 (x,y) t + p 4 (x,y) periodic response p i (x,y), i=1,..,4, are 2nd order polynomials of x and y drift 13
Fatigue loading m i p i m i+1 [Boulanger, PhD 24] [Berthel, PhD 28] [Blanche, PhD 212] 6 4 2 56 MPa Δσ(MPa) 175 MPa Blocks m i : series of mini cycle blocks (3 cycles) at different stress ranges: energy balance at constant fatigue state p i : large blocks (1 cycles) at constant stress range: energy balance evolution induced by fatigue mechanisms highest stress range fatigue limit 14
Thermal and calorific effects HCF test on DP 6 steel θ = T - T ( C).4 C 3! Thermoelastic effect!! Δσ = 5 MPa, R =σ σ = 1 et f =3 Hz σ min max L Dissipative effect time (s) 16! θ ( C) 3.2 f L -1 3 2.8 1/6 s time.4 C every 1/6 s Δs the ρc 75 C.s 1 3 C within 16 s d 1 ρc.1 C.s 1 15
Dissipation properties (I) 12,8 x (mm).2 C/s.1.4 C/s.25.1 6,9 6,9 6,9 y (mm) Δσ = 26MPa Δσ = 34MPa Δσ = 512MPa Ø heterogeneous distribution Ø similar distribution, whatever Δσ Ø non linear, 12,8 d 1ρC d1ρc d 1 (Δσ) d 1 f L Ø slow time evolution of dissipation Ø plastic shakedown? (TQ ratio 5 %!) 12,8 12,8 f L = 3Hz and R σ = -1 d 1ρC d 1ρC 6,9.15 C/s.1.5.15 C/s.1.5 5. cycles 5. cycles 16
Dissipation properties (II) Interpretation of curves m i = dissipation induced by activated micro-defects at constant fatigue state for different stress ranges p i = dissipation drift at constant stress range, reflecting a slow evolution of the fatigue state 4! 3! 2! 1!!! d 1 ρc ( C.s 1) 375k cycles! 26k cycles! 145k cycles! 3k cycles! m 3! m 1! m 2! 2! 3! 4! 5! 6! Δσ!(MPa)!! p 3! p 2! m! p 1! Ø energy safeguard: kinetics of fatigue progress 17
Case study #2 : back to thm couplings The most simplistic non-adiabatic thermoelastic (the) model ε = σ E + λ th (T T ) T + T T = Eλ th T ε τ th ρc E = 1 MPa ρ = 1 kg.m -3 C = 1 J.kg -1.K -1 λ th = 5 1-5 K -1 1 τ th = 3 s T = 294 K -5.5 «D» approach linear heat losses 5 σ (MPa) ε (%) θ ( C) -.5 1 William Thomson Lord Kelvin (1824-197) -1 Toward a stabilised cycle Thm couplings + thermal dissipation w def = A h = w the 18
Thm couplings vs. viscosity Thermoelastic coupling (i.e. d 1 =) anisothermal framework Viscous dissipation Isothermal framework σ E θ λ th σ σ E h µ σ ε state variables (θ, ε) ε = σ E + λ th θ θ + θ = Eλ th (T + θ) ε τ th ρ C rheological equation σ + τ th σ Eε + Eτ th 1+ χ ( ) ε σ + µ ε state variables(ε, ε v ) σ = E(ε ε v ) σ = hε v + µ ε v rheological equation σ E + h = Eh E + h ε + Eµ ε E + h Visco-analysis of polymers - Dynamic Mechanical Analysis (DMA) 19
Entropic elasticity of natural rubber F (N) 4.2 Energy rate (K.s-1 ) w def 2 w h 4 θ ( C) -4 λ 5 -.2 t (s) 5 1 15 Gough (185) Joule (1857) Treloar (196) e(s,ε) = e nr (T) perfect gas analogy ψ nr (T,ε) = TK 1 (ε) + K 2 (T) w def = w h [Saurel, PhD 99] [Honorat, PhD 6] [Caborgan, PhD 11] 2
Concluding comments q Full-field measurements Material vs. Structure effects q Temperature, the 1 st state variable thermal effects vs. calorimetric effects not totally intrinsic (heat diffusion) q Heat sources of different nature Thm coupling sources : thermo-sensitivity dissipation sources : irreversibility of material deformation q Energy balance and constitutive equations stored energy / state laws dissipated energy / evolution law q Rate-dependent behaviour d 1 (viscosity) vs. [thm coupling + d 2 (heat diffusion)] 21