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| Mechanical Properties | |||||||||||
| ·
For
a material with N0 sub-chains per unit volume, statistical thermo-dynamics
gives the entropy change associated with a mean length change from L0
in the stress free state to L under stress as:
ΔS = (S0 - S) = (N0 k/2){(L/L0)2 + 2(L0/L) - 3} · Differentiating with respect to L and multiplying by T gives: Fs = (N0 kT/L0 ){(L/ L0) - (L0 /L)2 } · The stress, σ, due to this entropy-induced force is obtained by dividing by the sample area, A, and defining: n = (N0 /AL) = ((N0 /AL0)( L0/L), where n is the number of cross-links per unit volume of the sample. · This gives: σ = nkT{( L/ L0)2 - (L0 /L)} · The contribution to the elastic modulus from this term is: ES = (dσ/dε)T = L0(dσ /dL)T = nkT {(L0 /L) + 2 ( L/ L0)2} · For zero applied stress, L = L0 and: ES = 3nkT |
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