DOC PREVIEW
MIT 3 11 - REVIEW

This preview shows page 1-2-3-4 out of 11 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 11 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 11 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 11 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 11 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 11 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

LECTURE #24 :3.11 MECHANICS OF MATERIALS F03INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : http://web.mit.edu/cortiz/www• REVIEW : RUBBER ELASTICITY I, II• DERIVATION OF STRESS VERSUS STRAIN LAWS FOR RUBBER ELASTICITY• INTRODUCTION TO VISCOELASTICITYSUMMARY : LJ PotentialII. 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↑disorder and entropy↓# of available configurations, Ω↓elastic restoring force, Felastic=-externally applied force, F ↑2. General Statistical Mechanical Formulas :3. Gaussian Formulas For Stretching a Single Polymer Chain :FrFelasticFelastic≈r1F1xyzB= number of chain conformationsP(r)= probability of finding a free chain end a radial distance, r, away from fixed chain end (origin)~OS(r)= configurational entropy=klnP(r)A(r)= Helmholtz free ener ΩB22gy =U(r)-TS(r)= -TklnP(r)-dA(r)f(r)= entropic elastic force= drdF(r)-dA(r)k(r)= (global) entropic chain stiffness= = drdr0322223222B22BB2cBB2cBB2c4br3P(r)exp[br]where b=2na4brS(r)=klnexp[br]3kT3kTA(r)=r=r2La2na3kT3kTf(r)=-r=rLana3kT3kTk(r)==constantLana ππ=−−=Linear ElasticitySUMMARY : Lecture # 21II. Freely-Jointed Chain (FJC) Model (Cont’d)_4. NonGaussian FormulasII. Assemble Strands into Network“Affine Deformation” :each crosslink is embedded in an elastic continuum and the r-vector of each strand transforms linearly with macroscopic deformationλ2λ1λ3macroscopicallydeform rubber cube<r2>1/2r’F1>0x1x2x3molecularlevel deformationF=0x1x2x3(r1’ r2’ r3’)(r1 r2 r3)-0.2-0.15-0.1-0.0500 100 200 300(a = 0.6 nm, n = 300)Felastic (nN)r (nm)non-GaussianFJCGaussianFJC-0.2-0.15-0.1-0.0500 100 200 300(a = 0.6 nm, n = 300)Felastic (nN)r (nm)non-GaussianFJCGaussianFJCBc357ckTrrf(r)=-L*(x)where : x==anaLL*(x)inverse Langevin function 92971539L*(x)=3x+xxx5758751L(x)COTH(x)xFor low stretches ( r<L): Gaussian formulas holdFor large str = ++=−cetches ( r> L): nonGaussian formulas holdfoL r'=extension ratio==1+=Lr10 compression1 tensionA=change in Helmholtz free energy of network on deformation/unit volume of network A=reversible work of deformation of network/unit voFFλελλ>>>∆=∆∆=∆[ ]222Bx123x3123123123lume of networkkTA32number of network strandsunit volume (m)1dAdAdAdA,,,ddddEYoung's Modulus or Modulus of GAUSSIAN CONSTANT VOLUME DEFORMATIONνλλλνλλλσσσσλλλλ∆=++−==∆∆∆∆=====0d Elasticitydεσε→=-0.5-0.4-0.3-0.2-0.100 50 100 150 200Felastic(nN)a = 0.1 nma = 0.2 nma = 0.3 nma = 0.6 nma = 1.2 nma = 3.0 nmFrFchainFchainF≈Effect of a and n in FJC-0.5-0.4-0.3-0.2-0.100 100 200 300(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)r (nm)Felastic(nN)r (nm)n=100 n=200 n=300 n=400 n=500(a)(b)Effect of Statistical Segment LengthEffect of Chain LengthStress versus Strain Equations for Uniaxial Deformationλ1λ2λ3x3x2x1[ ]222Bx123123kTA321 GAUSSIAN CONSTANT VOLUME DEFORMATIONνλλλλλλ∆=++−=Uniaxial Deformation: Comparison ofTheory with Experimentλ11/λ11/21/λ11/2x3x2x1012345671 2 3 4 5 6 7 8 9λ1σ1(MPa)crosslinksFF“strand”Uniaxial Deformation: Comparison ofTheory with Experimentλ11/λ11/21/λ11/2x3x2x1012345671 2 3 4 5 6 7 8 9λ1σ1(MPa)crosslinksFF“strand”Uniaxial Deformation: Comparison ofTheory with Experimentλ11/λ11/21/λ11/2x3x2x1012345671 2 3 4 5 6 7 8 9λ1σ1(MPa)crosslinksFF“strand”Viscoelasticity: When Does it Occur?Viscoelasticity: Basic DefinitionsBasic Mechanical Models for


View Full Document

MIT 3 11 - REVIEW

Documents in this Course
Load more
Download REVIEW
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view REVIEW and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view REVIEW 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?