1 Internal Length Gradient Mechanics: From Strength of Materials and Elasticity to Plasticity and Failure K. Michos, V. Dimosthenis, K. Parisis, L. Kouris, A. Konstantinidis and E.C. Aifantis Laboratory of Mechanics and Materials, School of Engineering, Aristotle University of Thessaloniki, GR 54124 Thessaloniki, Greece Abstract This is a summary of the talk presented by the last author within a conference at Khazar University, Baku, Azerbaijan. It proposes a framework for revisiting classical approaches of strength of materials and elasticity theory (Hooke’s Law) for capturing nonlocality and inhomogeneity, as well as size effects, that classical approaches cannot describe. This is established by introducing an extra Laplacian term ∇ ଶ multiplied by an internal length parameter ℓ ଶ accounting for the effect of inhomogeneity and weak nonlocality. Higher-order differential equations and associated boundary conditions are obtained. The solutions show that beams can become stiffer, and singularities can be eliminated from dislocation lines, as well as crack tips. These examples are concluded with a similar proposition for modifying classical gravitational theory (Newton’s Law). This provides a new possibility for interpreting the ‘’strong force’’ which keeps matter together. 1. Introduction A summary of the talk delivered virtually by the last author (ECA) in the Int. Hazar Sci. Res. Conference-II on 12 April with the assistance of the 2 nd and 4 th authors is presented. Some of the results have been reported in the Master’s thesis of the 1 st author under the supervision of ECA and further discussed in the diploma thesis of the 2 nd author and the supervision of the last three authors, as well as in joint articles by the 3 rd and 5 th author with ECA. The main purpose of the presentation is to alert the civil and mechanical engineering communities that the usual strength of materials approach can be revisited by incorporating into the standard elasticity law (Hooke) an extra Laplacian term, capable of interpreting size effects exhibited by structural and material elements containing inhomogeneities and, thus, characterizing nonlocality. We begin by considering a simple beam under point or continuous load and show that this can become stiffer than its classical counterpart by varying the internal length parameter in relation to its length (Section 2). In Section 3 we discuss briefly the Laplacian term and list the gradient modification of Hooke’s Law, resulting to the so-called GradEla model which can be used to eliminate the stress/strain singularities predicted by classical elasticity for dislocation lines and crack tips. Finally, Section 4 presents a gradient- enhanced modification of Newton’s gravitational Law, which may enable a direct interpretation of the strong force that holds nuclei together. This provides a new possibility for interpreting phenomena across the scale spectrum, where engineers and physics employ Newton’s Law for their analyses.