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Lecture 7: Viscoelasticity and Relaxation MIT 3.071 Amorphous Materials 7: Viscoelasticity and Relaxation Juejun (JJ) Hu 1 After-class reading list Fundamentals of Inorganic Glasses Ch. 8, Ch. 13, Appendix A Introduction to Glass Science and Technology Ch. 9 (does not cover relaxation) Mathematics is the language with which God wrote the Universe. Galileo Galilei 2 Where and why does liquid end and glass begin? “What don’t we know?” Science 309, 83 (2005). 3 Is glass a solid or a viscous liquid? V Supercooled Solid Liquid liquid Elasticity Viscosity instantaneous, time-dependent, transient permanent E Glass xx xy xy xy G xy t E : Young’s : viscosity modulus (unit: Pa·s or Poise) G : shear modulus Tm T 4 Viscoelasticity: complex shear modulus Consider a sinusoidally varying shear strain xy 0 expit Elastic response: xyG xy G 0 exp i t Viscous response: xy i exp i t i xy t 0 xy Loss modulus In a general viscoelastic solid: * * xy G i xy G xy G G i G ' iG" G* : complex shear modulus Shear/storage modulus 5 Phenomenological models of viscoelastic materials Elasticity: Hookean spring xy G xy Viscosity: Newtonian dashpot xy xy t Models assume linear material response or infinitesimal stress Each dashpot element corresponds to a relaxation mechanism 6 The Maxwell element G V EE V t Serial connection of a Hookean spring and a Newtonian dashpot Total stress: E V Total strain: E V 7 The Maxwell element G V EE V t Constant stress (creep): E V constant t 0 E V t G Constant strain (stress relaxation): E V constant t 0 t t Relaxation E expV 1 exp G time 8 The Maxwell element G V EE V t Oscillatory strain: E V 0 expit 22 G ' G 122 G" G 122 9 The Voigt-Kelvin element Parallel connection of a Hookean spring and a Newtonian dashpot Total stress: V E V EE G V Total strain:t E V Constant strain: G Constant stress: t 1 exp G Oscillatory strain: 0 expit G ' G G" G 10 Generalized Maxwell model For each Maxwell component: G i 11 i t Gtii 1 i i 1 G1 i tG ii Gi 1 ... i ... 2 G2 Total stress: … G1 ... i ... 11 Generalized Maxwell model Stress relaxation: constant t 0 G Prony series: G ... ... 1 G1 1 i t GGexp i i0 i 2 G2 Gexp tG R … In real solids, a multitude of microscopic relaxation processes give rise to dispersion of relaxation time (stretched exponential) 12 Elastic, viscoelastic, and viscous responses Stress Stress Stress t t t Elastic strain Viscoelastic strain Viscous strain t t t 13 Viscoelastic materials Mozzarella cheese Human skin Turbine blades Volcanic lava Memory foams Naval ship propellers Image of Naval ship propellers is in the public domain. Various images © unknown. Lava image © Lavapix on YouTube. All rights reserved. This content is excludedfrom our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 14 Boltzmann superposition principle In the linear viscoelastic regime, the stress (strain) responses to successive strain (stress) stimuli are additive Strain Strain Strain 2 1 1 t 2 t t t1 t2 Stress Stress Stress t t t t1 t2 1 G1 t t1 2 G 2 t t2 12 15 Boltzmann superposition principle In the linear viscoelastic regime, the stress (strain) responses to successive strain (stress) stimuli are additive Strain Viscoelastic response i i is history-dependent Relaxation function t d i dt ' dictates time-domain t 0 dt ' response Stress i G i t ti i i ttd d t i dt G t t ' dt ' 00dt dt ' 16 V Structural relaxation in glass Supercooled liquid Glass transition Glassy state T Elastic regime Viscoelastic regime Viscous regime re tobs re ~ tobs re tobs All mechanical and Glass structure and Structural changes thermal effects only properties are history- are instantaneous: affect atomic vibrations dependent equilibrium state can be quickly reached Ergodicity breakdown 17 V Structural relaxation in glass Supercooled liquid Glass transition Glassy state T Elastic regime Viscoelastic regime Viscous regime DN 1 DN ~1 DN 1 Debroah number (DN): DN re t obs “… the mountains flowed before the Lord…” Prophetess Deborah (Judges 5:5) 18 Comparing stress relaxation and structural relaxation V T “Equilibrium” state “Equilibrium” state Zero stress state Supercooled liquid state Driving force Driving force Residual stress Free volume Vfe VV Relaxation kinetics Relaxation kinetics Exponential decay with Exponential decay with a single relaxation time a single relaxation time Relaxation rate scales Relaxation rate scales with driving force with driving force 19 Free volume model of relaxation (first order kinetic model) Relaxation kinetics at V constant temperature Vfet Vt V Vtf 0 expt re VV T ff t re Temperature dependence of relaxation EEaaV VVff re 0 exp exp kTBt t 0 kTB 20 Model predicted relaxation kinetics Cooling rate: 10 °C/s Slope: volume CTE Varying reheating rate of supercooled liquid 10 °C/s 1 °C/s 0.1 °C/s 21 Model predicted relaxation kinetics Varying cooling rate Reheating rate: 1 °C/s 100 °C/s 10 °C/s Fictive temperature and glass structure are functions of cooling rate 1 °C/s 22 Tool’s Fictive temperature theory Fictive temperature T : the V, H Supercooled f temperature on the supercooled liquid liquid curve at which the glass Increasing would find itself in equilibrium cooling rate with the supercooled liquid state if brought suddenly to it 3 With increasing cooling rate: 2 Tf1 < Tf2 < Tf3 1 A glass state is fully described by thermodynamic parameters Tf1 Tf2 Tf3 T (T, P) and Tf Glass properties are functions of VVTT , f HHTT , f temperature and Tf (structure) 23 Tool’s Fictive temperature theory Glass property change in the V, H Supercooled glass transition range consists liquid of two components Increasing Temperature-dependent cooling rate property evolution without modifying glass structure 3 VT Volume: T Vg, 2 f Enthalpy: HTC T Pg, 1 f Property change due to Tf1 Tf2 Tf3 T relaxation (Tf change) Volume: VT f V,, e V g VVTT , f HHTT , f T Enthalpy: HTCC f T P,, e P g 24 Tool’s Fictive temperature theory Glass property change in the V, H Supercooled glass transition range consists liquid of two components Increasing cooling rate dV T, Tf VVdTf dT T T dT Tf f T 3 dTf dT V,,, g V e V g 2 dt dt 1 dH T, Tf HHdTf Tf1 Tf2 Tf3 T dT T T dT Tf f T dTf dT VVTT , f HHTT , f CCCP,,, g P e P g dt dt 25 Predicting glass structure evolution due to relaxation: Tool’s equation Consider a glass sample Pe Pg with Fictive temperature T PPge TTf f TT P (V, H, S, etc.) Take time derivative: Supercooled dP P dT liquid g Pe g f dt T T dt Assume first-order relaxation: dPg P g P e Glass dt re Pg - Pe T - T dTff T T f Tool’s equation dt re T Tf 26 Tool’s Fictive temperature theory Line D'M: Tf T Tool’s equation dH T, Tf HHdTf dT T T dT Tf f T dT dTff T T f dT CCCP,,, g P e P g dt dt dt J. Am. Ceram.re Soc. 29, 240 (1946) 27 Difficulties with Tool’s Tf theory Ritland experiment: two groups of glass samples of identical composition were heat treated to obtain the same refractive index via two different routes Group A: kept at 530°C for 24 h Group B: cooled at 16°C/h through the glass transition range Both groups were then placed in a furnace standing at 530°C and their refractive indices were measured as a function of heat treatment time Glass structure cannot be fully characterized by the single parameter Tf J. Am. Ceram. Soc. 39, 403 (1956). 28 Explaining the Ritland experiment n expt re,1 t expt re,2 Structural relaxation in glass involves multiple structural entities and is characterized by a multitude of relaxation time scales Tool-Narayanaswamy-Moynihan (TNM) model 29 What is relaxation? 30 Relaxation: return of a perturbed system into equilibrium Examples Stress and strain relaxation in viscoelastic solids Free volume relaxation in glasses near Tg Glass structural relaxation (Tf change) Time-dependent, occurs even after stimulus is removed Debroah Number: DN re t obs DN >> 1: negligible relaxation due to sluggish kinetics DN << 1: system always in equilibrium DN ~ 1: system behavior dominated by relaxation 31 P PP Driving Modeling relaxation Relaxation e force rate t re Maxwellian relaxation models t P t P0 Pee exp P re t P t P0 Pee exp P re Boltzmann superposition principle in linear systems d G tt ' t R IRF t t' S t dt ttd d t dt G tt ' dt ' R IRFt t ' S t ' dt ' 00dt dt ' 0 32 “The Nature of Glass Remains Anything but Clear” Free volume relaxation theory V Vf E exp a t 0 kB T Tool’s Fictive temperature theory Uses a single parameter Tf to label glass structure Tool’s equation (of Tf relaxation) Structural relaxation in glass involves multiple structural entities and is characterized by a multitude of relaxation time scales The Ritland experiment dT T T ff dt re 33 MIT OpenCourseWare http://ocw.mit.edu 3.071 Amorphous Materials Fall 2015 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms..
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