Is constitutive equation for piezo wrong?

[athosanian]

the linear constitutive equation describes piezo behavior. See wiki. But I can find an example in which linear relationship may not satisfy. Given a piezo disk wih polarization in thickness direction and upper and bottom surface as electrodes. I just short circuit the up and bottom surface and put it freely on the table. In such a condition, E=0 in the piezo no doubtly, stress T=0 also, because I do not put any force on it. Then from constitutive equation P=0. But not forget there is spontaneous polarization in disk, so P should not be zero, it has a initial value!
Ok, can we say the linear constitutive equation is wrong?

 

[Valerie Park]

No, because it is true in general condition. The linear constitutive equation only say the relation between E and P, it dose not tell us what is the initial value of P. So here we can use nonlinear constitutive equation to include the initial value of P. So, we can say that in piezo materials, the linear constitutive equation describes the piezo behavior in general condition, and nonlinear constitutive equation may be needed when we consider the initial value of polarization.