Closed-form solution in bi-Helmholtz kernel based two-phase nonlocal integral models for functionally graded Timoshenko beams

Abstract

Static bending behavior of functionally graded (FG) Timoshenko beams is studied via both strain- and stress-driven two-phase local/nonlocal mixed integral models based on the bi-Helmholtz kernel. The differential governing equations and boundary conditions are derived via the principle of minimum potential energy, the relations between strain and nonlocal stress are expressed as integral equations with the bi-Helmholtz kernel. The Laplace transform technique is utilized to obtain closed-form solutions for the first time. The numerical results are validated by reverting to the local case, and it is concluded that the present model shows well-posedness for exploring bounded beams in contrast to Eringen's pure nonlocal model: for all boundary and loading conditions, the increasing nonlocal length-scale parameter shows consistently softening or stiffening effect on the beam behaviour, according to the strain- or the stress-driven model, respectively. Moreover, the effects of shear-deformation and FG characteristics are also investigated. As the FG parameter increases, the non-dimensional deflection increases, while the softening effect caused by the strain-driven nonlocal phase seems to be more significant.