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Extracta Mathematicae

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Geometrical models, Curves on surfaces -- Mathematical models


Material aging is understood as changes of material properties with time. The aging is usually observed as an improvement of some properties and a deterioration of others. For example an increase of rigidity and strength and reduction in toughness with time are commonly observed in engineering materials. In an attempt to model aging phenomena on a continuum (macroscopical) level one faces three major tasks. The first is to identify an adequate age parameter that represents, on a macroscopic scale, the micro and sub microscopical features, underlying the aging phenomena such as nucleation, growth and coalescence of microdefects, physico-chemical transformations etc. The age parameter should be considered as a parameter of state, in addition to the conventional parameters such as stress tensor and temperature.

The second task consists of formulation of a constitutive equation of aging, i.e., equations of age parameter evolution expressed in terms of controlling factors, e.g., load and temperature. It is expected that at common circumstances a small variation of controlling factors results in a small variation of age parameter. However, at certain conditions, a sudden large variation of age parameter may result from a small perturbation of controlling factors. Experimental examination, classification and analysis of the condition that lead to such a catastrophic behavior, constitute the third task of the modeling. Formulation of local failure criteria within the scope of continuum mechanics is an example of this task.

In many engineering materials the aging is manifested in variations of mass density as well as in the spectrum of relaxation time. Thus in a macroscopic test the aging can be detected in variations of intrinsic (material) length and time scales. Following this notion, in the present paper we employ the material metric tensor G as an age parameter. An evolution of Gin 4D -material space-time determines in our approach an inelastic behavior and time dependent material properties recorded by an external observer.

The objective of the present work is to derive the constitutive equations of aging based on Extremal Action Principle. The variational approach seems to be most promising in view of complexity of the problem and lack of experimental data. It provides with a guide line for the experimental examination of the basic assumptions and modifications, if necessary.


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