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EMA6165-lect-14 - Agenda sway-3 Introduction Thermo-Elastic...

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Unformatted text preview: Agenda sway-3% - Introduction - Thermo-Elastic Behavior . Statistical Mechanical Approach — Uniaxial Deformation — Biaxial Deformation - SwollenState ( MWJ W W" - Deviations from Classical Theories — Affine Deformation — Phantom Behavior — Mooney-Rivlin Model EMA 61 65 Polymer Physics - AB Brennan 2 Materials \ Chain Statistics 5m“ "“ stag-xxx“ Statistical Mechanics - A - u Engmeenng - Objective: predict mechanical properties from first principles, i.e., equilibrium thermodynamics EMA 61 65 Polymer Physics - AB Brennan 3 Materials \ é Chain Statistics 5m“ ' kcfi-u‘aflflkk Statistical Mechanics - A - u Engmeenng - Objective: predict mechanical properties from first principles, i.e., equilibrium thermodynamics 0 Two Approaches -Gibbs Free Energy 4: AG = AH — TAS o Helmholtz Free Energy f AA=AE—TAS EMA 61 65 Polymer Physics - AB Brennan 3 Consider: d A << 1 ° mm dA = dE—TdS—SdT One knows that: dE : dq + dW for an Idea! Gas: dW = —PdV + fdl EMA 61 65 Polymer Physics - AB Brennan 4 _ M I Ii 1 '\ Thermodynamics Sail? ,é So let’s evaluate the critical values: WW [ 5’2 A j 0" f Requires constant 2 volume a” lo” T V T W EMA 61 65 Polymer Physics - AB Brennan 6 Thermodynamics 3%é So let’s evaluate the critical values: WW Requires constant [52A] _af é’lé’TV 3T” volume Also, one can show that: 52 A a S Requires é’lé’T _ 6, _l constant temp V V,T EMA 61 65 Polymer Physics - AB Brennan 6 — Thermodynamics JR Using Maxwell’s equation. 62 A __ _ 6S 62 A 6T 6! 6T 13V 6—1 W =616T One can write: [5 f1] [ 5 S] W 1 VJ“ ‘P §or at him NM Lm‘HA T . . .. (j EMA 61 65 Polymer Physics - AB Brennan 7 — Thermodynamicsim1 Using Maxwell’s equation. 62 A __ _ 6S 62 A 6T 6! 6T 13V 6—1 W =616T One can write: (2—51,; 4% f T EMA 61 65 Polymer Physics - AB Brennan 7 V,T Thus: wee [5A] [5E] [58] _ : _ —T _ a] W a! W 0"] V, And since thermal reversibility is a boundary condition, dq = TdS a} i—JO [er W 91 A) wow—0L, c hM§ s} an £52 0.53:1 I; Qua r65 eéfi‘l‘liflgtiégPolyrl‘toe}:)Phyt='.ics; - AB Brennan 8 T Substitution provides: quation of State This Equation of State, using the identity given previously, can be rewritten: — 3—15 fl 1— [ally—Tb?) Which is a second Equation of State. EMA 61 65 Polymer Physics - AB Brennan 9 Hence, one can measure ‘°°°° the internal energy of the polymer through equilibrium stress measurements. INHIIII n-‘L' IEEIIIII II II II INHIIIII I I I I H I I ‘HIIHIIII I I I I O 50 IOO ISO 200 250 300 350 400 ELONQQQN, 1, FIG. 86.——-The force of retraction f and its components (BE/8L)r,p (curve A) and —- NBS/6mm» (curve B), as obtained from force-temperature curves at fixed length such as are shown in Fig. 85, plotted against the per- cent elongation at 20°C. (Anthony, Gaston, and Guth.“) EMA 61 65 Polymer Physics - AB Brennan 10 ...
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