Elements of Polymer Structure and Viscoelasticity David M. Parks Mechanics and Materials II 2.002 February 18, 2004Outline •Elements of polymer structure•Linear vs. branched; •Vinyl polymers and substitutions•Packing of polymer chains•Random/amorphous•Glass transition temperature, TG •Semi-crystalline•Crystalline volume fraction; melting temperature•Amorphous TG •Elements of linear viscoelasticity•Creep and relaxation•Analogue modelsIdealized Linear Elastic Response Linear elasticity: [NO] Creep [NO] Relaxation INPUT: INPUT: OUTPUT: OUTPUT: Note: units of E are stress.Idealized Linear Viscous Response Linear viscosity: Creep: Relaxation: INPUT: Note: units of η are stress × time becomes arbitrarily large during e.g., sec × MPa an infinitesimal time interval, and then, like the strain-rate, goes to zero INPUT: OUTPUT: OUTPUT: The strain rate (and therefore stress)Maxwell Model: an Idealized Linear Viscoelastic Response Creep: INPUT: OUTPUT: •For times near the finite stress jump, all strain occurs in the elastic element; •During the hold period, allstrain occurs in the viscous elementMaxwell Model: an Idealized Linear Viscoelastic Response Relaxation: INPUT: •For times > 0, d ε(t)/dt=0 •During the hold period, elasticstrain is traded for viscous strain, and stress drops: OUTPUT: Characteristic relaxation time: τ = η / E“Real” Polymer Relaxation (an Idealization) INPUT: OUTPUT: Testing of real polymers under relaxation can be used to extract a time-dependent relaxation modulus, Er(t); Short-term response: Erg; Long-term response: Ere NOTE: provided |ε0| is sufficiently small (typically, less than 0.01), the relaxation modulus, Er(t), is approximately independent of ε0.“Real” Polymer Creep: (an Idealization) Testing of real polymers under suddenly-applied constant stress can be used to extract a time-dependent creep function, Jc(t); INPUT: Short-term response: Jcg; Long-term response: Jce OUTPUT: Note: units of Jc(t) : 1/ stress NOTE: provided |ε(t)| remains sufficiently small (typically, less than 0.01), the creep function, Jc(t), is approximately independent of σ0.Relaxation Modulus, Er(t) and Creep Function, Jc(t): Inverse Functions of Time? QUESTION: Are these inverse functions? Is Jc(t) × Er(t) ≡ 1 for all times? •ANSWERS: In general, they are not precise inverses. However, •Equilibrium and glassyvalues are nearly inverse: •For intermediate times, t.the error in assuming that they are inverse is typically only a few per cent at most…Linearity of Response (an Idealization) NOTE: similar linear scaling of stress relaxation response applies.Superposition of Loading Suppose that the stress history input consists of a sequence of stress jumps, ∆σ, applied at successive times t , with t0=0:i i THEN, the resulting strain history is given bySpecial Case: Load/Unload INPUT:Correspondence Principle •Suppose that a given load, P, produces displacement vector u(x) in a linear elastic body having Young’s modulus E. •The displacement vector depends on the position vectorx = x ex+ y ey+ z e .z•The magnitudes of the displacement and strain components areproportional to P and inversely proportional to E. EXAMPLE: Three-point Mid-span bending:Correspondence Principle •Now suppose that a given load jump, P(t), is applied to a geometrically identical linear viscoelastic body having creep function Jc(t). •All stress components in the body are time-independent, and spatiallyvary precisely as they do in an identical linear elastic body subject to the same load. •The loading produces time-dependent displacement vector u(x,t) and corresponding strain components. •The magnitudes of the displacement and strain components areproportional to both P and Jc(t). EXAMPLE: Three-point Mid-span bending: For suddenly-applied load, replace “1/E” with “Jc(t)” in an elastic solution.Correspondence Principle •Suppose that a given displacement jump, ∆(t), is applied to a geometrically identical linear viscoelastic body having stress relaxation modulus Er(t). •All displacement and strain components everywhere in the body are time-independent, and precisely equal those in an identical linear elastic body subject to the same applied displacement. •These boundary conditions produce time-dependent loads, P(t), and stresses •The magnitudes of the time-dependent load and of thestress components are proportional to both ∆ and to Er(t). EXAMPLE: Three-point Mid-span bending: For suddenly-applied displacement, replace “E” with “Er(t)” in an elastic