Statistical models of elasticity in main chain and smectic liquid ...

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134 CHAPTER 5. SOFT ELASTICITY OF SMECTIC ELASTOMERS0.40.20-0.2λ xzλ yz-0.4λ yx1 1.02 1.04 1.06 1.08λ yyFigure 5.7: Sympathetic shears λ yx , λ yz and λ xz in responseto an imposed λ yy . Again r = 2 and θ = 30 ◦ .fixed ends near the clamps mean that any soft mode will be accompanied bymicrostructure, as is frequently the case in nematic elastomers.
5.A. DETAILS OF THE DEFORMATION MATRIX 1355.A Details of the deformation matrixHere details of the calculation of the deformation tensor required for a softmode when the λ yy component is imposed are given. The following vectorsare requiredk 0 = (0,0,1) (5.36)c 0 = (0,1,0) (5.37)w 0 = l 1/20 ·k 0 (5.38)c = (cosφ,sinφ,0) (5.39)k = k 0 = R (5.40)From these vectors we can calculate the rotation matricesUsing the expressionresults in the following⎛⎜⎝W k0 (φ) = δcosφ+(1−cosφ)k 0 k T 0 +(sinφ)k 0 ∧ (5.41)W w0 (ξ) = δcosξ + (1−cosξ)w02 w 0 w0 T + sinξ w 0 ∧ . (5.42)w 0λ = l 1/2n ·W k0 (φ)·W w0 (ξ)·l −1/20 ,√cosξcosφ−√ρr sinξsinφ − rρcosφsinξ −cosξsinφ ···√ √ ρr cosφsinξ +cosξsinφ cosξcosφ− rρsinξsinφ ···0 0 ···( √ ) ⎞(r−1)sin2θ···2ρcosφ(1−cosξ)+ ρr( √ sinξsinφ )··· − ρr sinξcosφ+(1−cosξ)sinφ ⎟⎠(r−1)sin2θ2ρ··· 1To determine ξ simply demand that the λ xy component is zero and obtain thematrix given in the text.
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iStatistical models of elasticity i
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AcknowledgementsDuring the course o
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viiiCONTENTS5 Soft elasticity of sm
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2 CHAPTER 1. INTRODUCTION1.2 Neo-cl
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4 CHAPTER 1. INTRODUCTIONticular in
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Chapter TwoThe elasticity of hairpi
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2.1. INTRODUCTION 9Fig. 2.1. This b
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2.2. MODEL OF HAIRPIN CHAINS 11nFig
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2.2. MODEL OF HAIRPIN CHAINS 13za)
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2.2. MODEL OF HAIRPIN CHAINS 15in
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2.2. MODEL OF HAIRPIN CHAINS 17The
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2.2. MODEL OF HAIRPIN CHAINS 1910.8
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2.2. MODEL OF HAIRPIN CHAINS 21This
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2.2. MODEL OF HAIRPIN CHAINS 23f g
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2.3. NON-AFFINE DEFORMATION MODEL 2
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2.4. HAIRPIN CHAIN NETWORK FREE ENE
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2.4. HAIRPIN CHAIN NETWORK FREE ENE
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2.5. CONCLUSIONS 35to their limit
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2.A. GEOMETRY OF A HAIRPIN AT ZERO
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2.B. UNDULATING CHAIN STATISTICS 39
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2.C. COUNTING CONFIGURATIONS BY IND
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2.C. COUNTING CONFIGURATIONS BY IND
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2.D. INTERPRETATION OF FN 452.D Int
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2.E. SPRING CONSTANT OF AN EXTENDED
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50 CHAPTER 3. POLARISATION OF CHIRA
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Chapter FourThe elasticity of smect
Page 93 and 94: 4.1. INTRODUCTION 83wherer 12 is th
Page 95 and 96: 4.1. INTRODUCTION 85HzyxGlobal rota
Page 97 and 98: 4.1. INTRODUCTION 87Figure 4.3: The
Page 99 and 100: 4.1. INTRODUCTION 89−D 2[(v (a)xz
Page 101 and 102: 4.2. MICROSCOPIC, FINITE DEFORMATIO
Page 107 and 108: 4.3. FINITE DEFORMATION EXAMPLES 97
Page 123 and 124: 4.4. COMPARISON WITH EXPERIMENT 113
Page 125 and 126: 4.4. COMPARISON WITH EXPERIMENT 115
Page 127: 4.5. CONCLUSIONS 117formation in th
Page 130 and 131: 120 CHAPTER 5. SOFT ELASTICITY OF S
Page 147 and 148: Bibliography[1] P. J. Flory. Statis
Page 149 and 150: 139[27] J-.C. Meiners and S. R. Qua
Page 151 and 152: 141[58] W. L. McMillan. Measurement
Page 153 and 154: 143[86] L. Golubovic and T. C. Lube
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