# Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
## 1 Introduction  
The classical or local theory of elasticity is scale-free. As a result, it cannot cope with situations where an internal characteristic length of material becomes comparable to an external geometric length. This is the case with micro- and nano-scale structures [1]. A remedy for these situations is the use of generalized continuity theories such as the higher-order continuum theories, namely the Cosserat theory [2], the couple stress theory [3], the micropolar theory [4], the strain gradient theory [5, 6, 7], and the nonlocal theories [8, 9, 10]. They incorporate additional material parameters in the constitutive equations that increase the complexity, and therefore, numerical methods, such as the finite element method, are employed to solve the governing equations, see for example in [11, 12, 13, 14, 15, 16, 17]. For similar developments in the field of thermoelasticity, one can see the very recent publications [18, 19, 20]. In the nonlocal continuum theory of elasticity developed by [PERSON] [10], the stress state at a point does not depend only on the strain at that particular point as in classical elasticity (local model) but is defined as an integrated average of the strain field at every point in the body (integral model). A variant of this integral model is the two-phase integral model (integro-differential model) that combines the local model and the nonlocal integral model [21, 22, 23]. Both integral models are associated with governing equations involving integral or integro-differential equations that are difficult to solve. A simplified form of nonlocality is the differential model, which includes a degenerated differential form of the integral model [10]. Due to its simplicity, the nonlocal differential model has widely been used to analyze various micro- and nano-structures including one-dimensional structures such as rods, tubes, and beams [24, 25, 26]. In particular, for the beam bending analysis, the interestedreader can look at, among others, [27; 28; 29; 30; 31]. However, several authors have reported that the nonlocal differential model for certain types of loading gives inconsistent results compared to those obtained from other types of loading and boundary conditions [27; 32]. This paradox was recently explained in [33] where it is shown that, in general, the nonlocal differential model is not equivalent to its integral counterpart unless certain conditions are met, as defined in [34]. This development has revived the interest in the nonlocal integral models, and therefore, there is a need to develop effective methods for producing exact analytical solutions. A closed-form solution of the nonlocal integral model for the bending of Euler-Bernoulli beams was recently obtained in [35]. Moreover, an analytical solution for the two-phase nonlocal integral model was obtained in [36] through a reduction to a differential equation with mixed boundary conditions as proposed by [34]. In general, integro-differential equations are usually difficult to solve directly. In the last few years, the authors have developed a direct operator technique for solving exactly Fredholm-type integro-differential equations (FIDE) with all kinds of boundary conditions, including nonlocal ones [37; 38]. In [39], a method for solving in closed form boundary value problems for a class of \(n\)th order linear Volterra-Fredholm integro-differential equations (VFIDE) of convolution type was proposed. The technique was used to construct the closed-form solution of the boundary value problem for the two-phase nonlocal integral model of Euler-Bernoulli beams under a uniformly transverse distributed load and in the case of simply supported boundary conditions. In this article, we provide the closed-form solution for three more boundary value problems corresponding to three practical cases of boundary conditions, specifically a cantilever beam, a clamped pinned beam, and a clamped beam.