By Ralph Abraham
Booklet by way of Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu, Richard Cushman
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Extra info for Foundations of Mechanics: 2nd Edition
From the point of view of thermodynamics, the damage variable must be able to indicate the irreversible process of micro-structural change within the material. Therefore, damage is an internal variable, which must be a function of material properties, loading, temperature, time, etc. However, this internal variable must be capable of prescribing damage growth and propagation [2-35 rv 36 , 2-61]. The choice of the damage variable is not simple. It can be achieved either by a micro-structural analysis or by a direct generalization of experimental data.
0 Fig. 4 Damage Evolution due to Super-Plastic Void Growth Based on the dominant damage mechanisms of super-plastic materials discussed above, two types of damage evolution equations have been developed. One assumes that the mechanism of void growth is due to stress directed vacancy diffusion. The deformation of the cavitated material is described as a cavitated cylinder, the radius of which is equal to the cavity spacing 1. For the behavior of the slab containing a spherical cavity of radius T'h, the proposed mechanism needs to account for the extent of the diffusion zone indicated by rd.
Therefore, in the simplest case, the damage variable is always to be considered as a scalar function , but in a more complex case, the damage variable must be considered as a vector or tensor function. For example, in the case of anisotropy the damage state within a material must be expressed by a set of stat e variables- tensor functions which had been first suggested by Leckie and Onat [2-61]' Cordebois and Sidoroff [2-181]' Murakami and Ohno [2-49]. As Lemaitre pointed out in [2-28"-'31]' between the micro-level and macrolevel, there exists a macro-scale level of constitutive equations for strain behavior.
Foundations of Mechanics: 2nd Edition by Ralph Abraham