Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

∂ϕ∂ϕT= gradT=−∂d∂t( q T )(4.34)The thermodynamic forces are the components of the vectorϕ = constant surface in the space of the flux variables.grad ϕ normal to theThe Legendre-Fenchel transformation enables to define the corresponding potential*ϕ , the dual of ϕ with respect to the variables dtand− q . By definition:T⎧*Tϕ ( , T ) sup ⎪⎛⎞ ⎫grad: qTgrad = ⎨ t− ⋅ −ϕ( t, ) ⎪T d q⎜⎬⎝T ⎟d (4.35)⎪⎩⎠ T⎪⎭⎛q ⎞⎜dt,⎜⎝ T ⎟⎠It can be shown that, if the function ϕ * is differentiable, the normality property ispreserved for the variables d tandcan then be written as:− q and the complementary laws of evolutionT* *∂ϕq ∂ϕdt= − =∂TT ∂gradT(4.36)The properties that the potentials ϕ and ϕ * must possess for the automaticsatisfaction of the second principle of thermodynamics are the following: they mustbe non negative, convex functions, zero at the origin. It should be noted that thenormality rule is sufficient to ensure the satisfaction of the second principle ofthermodynamics, but it is not a necessary condition.The whole problem of modeling a phenomenon lies in the determination of theanalytical expressions for the thermodynamic potential Ψ and for the dissipationpotential ϕ or its dual*ϕ , and their identification in characteristic experiments. In70