lec32_clapeyron

lec32_clapeyron - Clapeyron’s formulas for 1D For the...

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Unformatted text preview: 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, totav exr l ternal v workr(lecture 27): Wd = ξ ⋅Fd +ξ d ⋅R We split this up into: (1) work due to force BCs v vr W (ξ ) = ξ ⋅ F d and (2) work due to displacement BCs r vd r W ( R) = ξ ⋅ 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: 1* (W + W ) 2 1 ψ * ( N i ) = (W * + W ) 2 ψ (δ i ) = Now we calculate the potential and complementary energy: ε pot = ψ (δ i ) − W ε com = ψ * ( N i ) − W * By using the expressions for the free energy and complementary free energy…: ε pot = 1* (W − W ) 2 ε com = 1 (W − W * ) 2 1 Summary – the following set of equations are called Clapeyron’s formulas 1* (W + W ) 2 1 ψ * ( N i ) = (W * + W ) 2 1 ε pot = (W * − W ) 2 1 ε com = (W − W * ) 2 ψ (δ i ) = Significance: Can calculate free energy, complementary free energy, potential energy and complementary energy directly from the boundary conditions (external work)! 2 ...
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lec32_clapeyron - Clapeyron’s formulas for 1D For the...

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