On the notion of curvature and its mechanical meaning in a geometrically exact plane beam theory

摘要

With the aim to answer to the question about what is the correct notion for curvature to adopt in constitutive relationships, this paper considers a geometrically exact beam theory built on very basic kinematic assumptions. The theory is developed in a consistent way, by deducing equations of motion from the Principle of Virtual Work. Further, by stipulating a relation between internal work for one- and three-dimensional beams, relationships among internal forces and stresses are found. Constitutive equations, which end up to be strongly coupled and nonlinear, are written in explicit form for a specific material model with linear behavior. The role of different notions of curvature to adopt in analysis of beams is investigated and linear, linearized and nonlinear equations for bending moment are considered. In particular, uncoupled linear approximation provides indication about the more suitable curvature definition. Further, a mechanical interpretation of generalized internal stresses is also given and benchmark numerical examples enlighten some features of the model.