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
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