Equilibrium Paths of Polygonally Damaging Structures Part One: The Nonsmooth Nonconvex Stability Problem

Damage is a broad term describing a decrease of material stiffness or strength. Damage has a strong destabilizing influence and in itself can cause instability of a structure, independent of the geometric nonlinearity. However, not all damage causes instability. Instability depends on the structural tangential stiffness matrix. If this matrix is positive definite, damage causes no instability. When this matrix ceases to be positive definite, bifurcations and instabilities can arise. They generally consist of localization of damage strain into a zone of minimum possible size, while the neighbouring domain of the structure, elastic unloading occurs. The effect of released elastic energy due to the unloading outside the localization domain leads to further stable postbifurcational equilibrium paths. In this paper, the discrete structural model needs to concentrate the material behaviour to certain points of the structure. Thus, the material change of state at these fixed points, the actual state of simultaneous material loading-unloading yields to stable postbifurcational equilibrium paths even in the case of damaging parts of the structure. The stability analysis of strain-softening structures needs to handle the continuously changing material state prescribed by a nonlinear stress-strain function. Since this leads to enormous mathematical difficulties, for the material functions, a polygonal approximation seems to be appropriate. On the other hand, some classic and modern materials such as reinforced concrete or any fiber-reinforced or composite material exhibit sawtooth type polygonal characteristics with a rehardening that follows a strain softening drop of stress. Thus, to follow exactly the polygonal behaviour of materials seems to be an important problem. However, due to the polygonal form of the constitutive law, the strain energy functionals are nonsmooth and because of the strain softening, they are also nonconvex. To handle these problems, nonsmooth analysis is applied for the nondifferentiability, and incremental analysis is used for the nonconvexity. The basis of the nonsmooth stability analysis has originally been laid down in the papers of M. Kurutz (1989, 1993a, 1993b). In this paper, the nonsmooth damage and localization instabilities due to one-dimensional polygonal stress-strain diagrams are presented. The investigations are based on the following fundamental works. In aspect of nonsmoothness and nonconvexity, the results of P. D. Panagiotopoulos (1983, 1985, 1988) are used. Related to the elastic stability, the works of J. M. T. Thompson and G. W. Hunt (1973, 1984) are followed. Concerning the inelastic stability, the works of Z. P. Bažant (1971, 1988, 1989) and the monograph of Z. P. Bažant and L. Cedolin (1991) are used. The paper is divided into two parts. In the first part, the theoretical bases of the nonsmooth nonconvex damage instabilities are detailed, while the second part shows an example illustrating the complicated problem even in the case of one kinematical degree of freedom.