Energy principle application to response of viscoelastic bars

A wide range of practical engineering problems exists for which obtaining exact solutions directly is challenging. This is because of the complex nature of the governing differential equations or the difficulties arising from the boundary and initial conditions of the problem. To address these problems, scalar quantities, such as work and energy, are used as an alternative approach. The virtual work principle constitutes the basis for the energy and variational formulations. This study uses energy concepts to formulate viscoelastic structures and discuss the statically indeterminate axially loaded viscoelastic bar problem. A simple and efficient energy-based formulation for analysis is proposed. The total potential energy (TPE) expression in terms of the displacements of the nodes was obtained in Laplace space. The solutions that minimise the TPE expression are real displacements, and the inverse Laplace transform method is applied to transform the function back into the time domain. Different examples were considered to ensure accuracy and demonstrate the potential of the proposed solution technique. This method is convenient for obtaining a solution directly by following a few simple process steps, regardless of the change in the viscoelastic material model, the number of elements in the system, and the type of loading.