Infinitesimal deformations and stability of rods made of nonlocal elastic materials

Aim of the paper is the formulation of a criterion of infinitesimal stability for a class of rods made of nonlocal elastic materials. To that end, the nonlinear equilibrium equations of naturally straight, inextensible rods subject to terminal loads are written, and the constitutive equation assumed to represent the material response in rods of finite length is discussed. Then, the equations describing the infinitesimal deformations superimposed upon a finite one are deduced. The expression of the work done by the increments of the external loads associated with an infinitesimal deformation is employed to formulate, for the considered rods, the criterion of infinitesimal stability which does not require the existence of a stored-energy function. The criterion is applied to study the stability of simply supported rods subject to axial forces, of rods with one end clamped and the other one constrained to have the tangent parallel to the undeformed rod axis, subject to axial forces and twisting couples, and of annular rods formed from naturally straight rods with the addition of twist. The results show that rods made of nonlocal elastic materials exhibit a reduction in rigidity with respect to rods having the same geometry and made of usual elastic materials with the same tensile and shear moduli.