Hans Irschik

Considering the example of uni-axial compressive shock waves in a linear viscoelastic half space, this study demonstrates the possibility of equivalence between the stresses due to body forces and due to eigenstrain loadings in an advanced dynamic setting. The different loading cases that are compared for equivalence may be accompanied by equal or different constitutive behavior. In the latter case, initial elastic moduli are taken as equal, but the time rates of the relaxation functions may be different. A straightforward procedure for deriving the eigenstrain loadings that are necessary for stress equivalence is developed. The Green function problem of a concentrated body force of the Dirac delta function type traveling into the half-space is used as force loading, and a closed expression of the necessary eigenstrain loadings in integral form is presented, the eigenstrains acting in the absence of force loadings. Due to the linearity of the problem at hand, applying negative eigenstrains together with the force loading in case of equal constitutive behavior produces complete stress compensation, an effect that is of particular interest in smart structures and materials. Appropriateness of the method is demonstrated in two examples concerning a viscoelastic behavior of the Poynting–Thomson type in the force loading problem, the auxiliary constitutive behavior in the eigenstrain loading problem being either also of the Poynting–Thomson type or purely elastic.

Following a research strategy that dates back to Ernst Mach, we study similarities and analogies that may exist for linear elastic and viscoelastic bodies when dynamically loaded by forces and eigenstrains. Particularly, we treat equivalence of displacements in two problems with different sets of loadings and possibly different constitutive behavior. Special interest is given to the presence of singular waves. In the present first part of our treatise, we deal with generalized body force analogies, while displacement control, i.e. shape control and displacement tracking will be treated in the forthcoming second part of our study. Besides a review on previous work, with an emphasis on unification of works of our group, some novel results in the form of generalizations are presented, and perspectives for prospective future work are given.

Applications of a novel time-integration technique to the non-linear and linear dynamics of mechanical structures are presented, using an extended Picard-type iteration. Explicit discrete-mechanics approximations are taken as starting guess for the iteration. Iteration and necessary symbolic operations need to be performed only before time-stepping procedure starts. In a previous investigation, we demonstrated computational advantages for free vibrations of a hanging pendulum. In the present paper, we first study forced non-linear vibrations of a tower-like mechanical structure, modeled by a standing pendulum with a non-linear restoring moment, due to harmonic excitation in primary parametric vertical resonance, and due to excitation recordings from a real earthquake. Our technique is realized in the symbolic computer languages Mathematica and Maple, and outcomes are successfully compared against the numerical time-integration tool NDSolve of Mathematica. For out method, substantially smaller computation times, smaller also than the real observation time, are found on a standard computer. We finally present the application to free vibrations of a hanging double pendulum. Excellent accuracy with respect to the exact solution is found for comparatively large observation periods.