Size-dependent Elasticity in Materials

In this work, we combine the nonlocal theory of Eringen into the E-B beam bending together with nonlinear kinematics [3]. We briefly present the derivation and key equations of this nonlinearnonlocal beam theory and investigate the role of nonlinearity and nonlocality for simply supported nanoscaled beams.

💡 Research Summary

This paper presents a unified theoretical framework that merges Eringen’s nonlocal elasticity theory with the Euler‑Bernoulli beam formulation while incorporating geometric nonlinearity through von Kármán strain‑displacement relations. The motivation is to capture size‑dependent mechanical behavior observed in nanoscale beams, where classical local linear models either over‑predict stiffness or fail to account for large‑deflection effects.

The authors begin by reviewing the fundamentals of nonlocal elasticity. In Eringen’s model, stress at a point is expressed as a spatial integral of strain over the body, introducing an internal length scale (e_{0}a) that quantifies the strength of long‑range interatomic interactions. By applying a differential approximation, the constitutive relation acquires a higher‑order operator ((1-(e_{0}a)^{2}\partial^{2}/\partial x^{2})) acting on the classical stress‑strain law. This operator reduces the effective bending rigidity as the structural dimensions approach the internal length scale, thereby reproducing the experimentally observed softening of nanobeams.

Next, the paper incorporates geometric nonlinearity. Using the von Kármán strain expression (\varepsilon_{xx}=u_{,x}+ \frac{1}{2}w_{,x}^{2}), the axial force (N = EA,u_{,x}) becomes a function of the transverse slope, providing a stiffening contribution when deflections are large. The combination of the nonlocal operator with this nonlinear axial term yields the governing equation for a simply supported beam:

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