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In this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

On the consistency of two-phase local/nonlocal piezoelectric integral model