Stress tracking in thin bars by eigenstrain actuation

This contribution focuses on stress tracking in slender structures. The axial stress distribution of a linear elastic bar is investigated, in particular, we seek for an answer to the following question: in which manner do we have to distribute eigenstrains, such that the axial stress in a bar is equal to a certain desired stress distribution, despite external forces or support excitations are present? In order to track a certain time- and space-dependent stress function, smart actuators, such as piezoelectric actuators, are needed to realize eigenstrains. Based on the equation of motion and the constitutive relation, which relate stress, strain, displacement and eigenstrains, an analytical solution for the stress tracking problem is derived. The starting point for the derivation of a solution for the stress tracking problem is a semi-positive definite integral depending on the error stress which is the difference between the actual stress and the desired stress. Our derived stress tracking theory is verified by two examples: first, a clamped-free bar which is harmonically excited is investigated. It is shown under which circumstances the axial stress vanishes at every location and at every time instant. The second example is a support-excited bar with end mass, where a desired stress profile is prescribed.