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I came across the following constitutive equation relating entropy with the change of Helmholtz free energy with respect to change in temperature while holding deformation tensor fixed.

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Source of equation: Nonlinear Solid Mechanics A Continuum Approach for Engineering

So, since there's a negative sign, it means that if Helmholtz free energy (or stored energy) increases with the increase in temperature, then the entropy will decrease. I don't understand why?

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  • $\begingroup$ It doesn't say anything about the entropy decreasing. $\endgroup$ Commented Feb 5, 2022 at 15:13

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Lets assume entropy is a positive quantity (and we can always define our zero of entropy such that this is true). Then this equation is saying that an increase in temperature will decrease the Helmholtz free energy, and that it will decrease faster in a higher entropy system.

The Helmholtz free energy is the amount of energy available to do useful work. As the system becomes "more disordered" the less energy is available to do work.

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This starts out basically from the conventional $$A=U-TS$$ combined with dU=TdS-PdV, such that $$dA=-SdT-PdV$$If there is no deformation, dV is zero (which is equivalent to constant deformation gradient tensor). So you are left with $$dA=-SdT$$which is microscopically equivalent to $$d\psi=-\eta dT$$ at constant deformation gradient tensor.

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  • $\begingroup$ Thank you, can you please define the quantities introduced? $\endgroup$
    – user134613
    Commented Feb 16, 2022 at 9:55

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