Issue 33

A. Winkler et alii, Frattura ed Integrità Strutturale, 33 (2015) 262-288; DOI: 10.3221/IGF-ESIS.33.32 278 applying a load for a sufficient amount of time, irreversible changes occur in the micro structure. It is our contention that the accumulation of small amounts of viscoplastic deformation is the primary cause of fatigue of plastics. After discussing all the morphological issues in play, we note that there is indeed a relatively large amount of energy dissipation occurring in plastics under fatigue loading which does not diffuse away as quickly as it could in metals. The local dissipation causes local heating, which in essence simply constitutes local vibrations of the molecules. The increased vibrations enable the internal structure to gain access to relatively more space, and thus the sliding behaviour of the chains becomes easier. This is from a macroscopic viewpoint perceived as a reduction in the effective viscosity at an increased temperature. Lower effective viscosities have lower dissipation rates, which causes the local temperature rise to slow down with increasing temperature. To visualise the effect of viscosity on the behaviour of the material we will next conduct an exemplary analysis on a purposely simplified plastic material. A (generalized) Maxwell model of a plastic The overall behaviour discussed up to this point can be strongly simplified into a Maxwell model [6] as illustrated in Fig. 11. Figure 22. Simplified idealised behaviour of a plastic as a Maxwell model In this figure we have simplified the behaviour of the plastic so far as to assume that it is viscoelastic. We generalize immediately though into stating that neither of the springs as shown in the network would need to be linear, and that the damper is not linear by any means. We should be content by noting that there is a damper. We have also omitted the viscoplastic part of the behaviour which we assume is responsible for the fatigue behaviour. With that in mind, let us perform a quick analysis of what happens when we do assume linearity, and merely introduce temperature dependence. This is to address the additional complexity that arises when dealing with a temperature dependent plastic. Also, for sake of argument to understand the mechanics, we are going to assume uniaxial behaviour. Thus, given the Maxwell model as in Fig. 22, we will call the upper part of the network the elastic layer, and the lower part of the network the viscous layer. We now have some balance equations that need to be satisfied: 1. The strain as observed by the elastic layer and the viscous layer are the same, it is the total strain over the network. 2. The stress over the network is equal to the stress of the elastic layer plus the stress of the viscous layer. 3. The stress in the spring in the viscous layer is equal to the stress in the damper in the viscous layer, they are in equilibrium. 4. The strain in the spring in the viscous layer plus the strain in the damper is equal to the total strain over the network. Let us now introduce some terminology to put these observations into equations. The stiffness of the upper spring is given as 0 E , the stiffness of the lower spring is given as 1 E and the viscosity of the damper is given by  . Also denote by  the total strain applied over the network and  e the strain in the spring that goes into the viscous layer and  v the strain that goes into the damper in the viscous layer. Finally, let us call the stress in the elastic layer 0  and refer to the stress in the viscous layer as 1  . The total stress over the network is given as  . Viscous Elastic