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lec32_clapeyron

# lec32_clapeyron - Clapeyrons formulas for 1D For the linear...

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Clapeyron’s formulas for 1D For the linear elastic case: ψ ( δ i ) = ψ * ( N i ) We will now exploit this property (specific to linear elasticity!) to derive a set of equations called “Clapeyron’s formulas”: Recall, total external work (lecture 27): v r v r W d = ξ F d + ξ d R We split this up into: (1) work due to force BCs v v r W ( ξ ) = ξ F d and (2) work due to displacement BCs r v r W * ( R ) = ξ d R Therefore ψ ( δ i ) + ψ * ( N i ) = W * + W Using the fact that the complementary free energy and free energy are equal, we arrive at: ψ ( δ i ) = 1 ( W * + W ) 2 ψ * ( N i ) = 1 ( W * + W ) 2 Now we calculate the potential and complementary energy: ε pot = ψ ( δ i ) W ε com = ψ * ( N

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