LECTURE 24 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www REVIEW RUBBER ELASTICITY I II DERIVATION OF STRESS VERSUS STRAIN LAWS FOR RUBBER ELASTICITY INTRODUCTION TO VISCOELASTICITY SUMMARY LJ Potential II Freely Jointed Chain FJC Model Cont d 1 Qualitative Description of Single Chain Stretching links rotate so as to uncoil and extend polymer chain along stretching axis F disorder and entropy of available configurations elastic restoring force Felastic externally applied force F Felastic r Felastic 2 General Statistical Mechanical Formulas number of chain conformations P r probability of finding a free chain end a radial distance r away from fixed chain end origin O S r configurational entropy kB lnP r z A r Helmholtz free energy U r TS r Tk B lnP r dA r 0 f r entropic elastic force dr dF r d 2 A r k r global entropic chain stiffness dr dr 2 r1 y 3 Gaussian Formulas For Stretching a Single Polymer Chain x 4b 3r 2 3 exp b 2r 2 where b 2na 2 4b3 r 2 S r kB ln exp b 2 r 2 3k T 3k T A r B 2 r 2 B r 2 2na 2L c a P r 3k T 3k T f r B2 r B r na Lc a 3k T 3k T k r B2 B constant na Lc a F1 Linear Elasticity SUMMARY Lecture 21 II Freely Jointed Chain FJC Model Cont d 4 NonGaussian Formulas a 0 6 nm n 300 0 k T r r f r B L x where x na Lc a L x inverse Langevin function Felastic nN 0 05 9 297 5 1539 7 L x 3 x x3 x x 5 75 875 1 L x COTH x x For low stretches r Lc Gaussian formulas hold For large stretches r Lc nonGaussian formulas hold Gaussian FJC 0 1 non Gaussian FJC 0 15 0 2 0 100 II Assemble Strands into Network 200 300 r nm Affine Deformation each crosslink is embedded in an elastic continuum and the r vector of each strand transforms linearly with macroscopic deformation extension ratio Lf r 1 Lo r 1 0 compression 1 tension A F change in Helmholtz free energy of network on deformation unit volume of network A F reversible work of deformation of network unit volume of network k T 2 2 2 A B x 1 2 3 3 GAUSSIAN 2 number of network strands x unit volume m 3 1 2 3 1 F1 0 r 1 r2 r3 x3 r1 r2 r3 F 0 r r2 1 2 x2 x3 x1 3 macroscopically deform rubber cube CONSTANT VOLUME DEFORMATION d A d A d A d A 1 2 3 d d 1 d 2 d 3 d E Young s Modulus or Modulus of Elasticity d 0 molecular level deformation 2 x1 1 x2 Effect of a and n in FJC F F r Fchain Fchain Effect of Statistical Segment Length 0 a Felastic nN 0 1 0 2 a 0 1 nm a 0 2 nm a 0 3 nm a 0 6 nm a 1 2 nm a 3 0 nm 0 3 0 4 0 5 0 50 Felastic nN 150 200 r nm Effect of Chain Length 0 b 100 0 1 0 2 0 3 n 100 n 200 n 300 n 400 n 500 0 4 0 5 0 100 200 300 r nm a Elastic force versus displacement as a function of the statistical segment length a for the non Gaussian FJC model Lcontour 200 nm and b elastic force versus displacement as a function of the number of chain segments n for the non Gaussian FJC model a 0 6 nm Stress versus Strain Equations for Uniaxial Deformation k B T x 12 22 32 3 GAUSSIAN 2 1 2 3 1 CONSTANT VOLUME DEFORMATION A x1 1 2 x3 3 x2 Uniaxial Deformation Comparison of Theory with Experiment x1 7 1 1 MPa 6 1 11 2 x2 5 x3 1 11 2 4 3 2 crosslinks strand F F 1 0 1 2 3 4 5 1 6 7 8 9 Uniaxial Deformation Comparison of Theory with Experiment x1 7 1 1 MPa 6 1 11 2 x2 5 x3 1 11 2 4 3 2 crosslinks strand F F 1 0 1 2 3 4 5 1 6 7 8 9 Uniaxial Deformation Comparison of Theory with Experiment x1 7 1 1 MPa 6 1 11 2 x2 5 x3 1 11 2 4 3 2 crosslinks strand F F 1 0 1 2 3 4 5 1 6 7 8 9 Viscoelasticity When Does it Occur Viscoelasticity Basic Definitions Basic Mechanical Models for Viscoelasticity k
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