**Modern Mechanical Engineering** Vol.2 No.3(2012), Article ID:22198,14 pages DOI:10.4236/mme.2012.23012

Quasi-Relaxation Transforms, Meromorphic Curves and Hereditary Integrals of the Stress-Deformation Tensor to Metallic Specimens

^{1}Department of Research in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Mexico, Russia

^{2}Baikov Institute of Research in Metallurgy and Material Sciences, Mexico, Russia

Email: francisco.bulnes@tesch.edu.mx, stropoy@baikov.institute.ru, yermishkinv@baikov.institute.ru

Received April 23, 2012; revised May 31, 2012; accepted June 11, 2012

**Keywords:** Hereditary Integrals; Meromorphic Curves; Meta-Stability State, Plastic Energy; Quasi-Relaxation Transforms

ABSTRACT

Into the study of quasi-relaxation, in the past researches it has been concluded that the condition of meta-stability in the metallic specimen is given by the plasticity explained by the plastic energy in the process of the quasi-relaxation. It is calculated through quasi-relaxation functional of this energy to obtain a spectra in the space D(σ – ε; t), that induces the existence of functions φ(t), and Ψ(t), related with the fundamental curves of quasi-relaxation given by σ(t), with their poles in, which is got in the maximum of stress given by σ_{0} = σ_{1}. Also the tensor of plastic deformation that represents the plastic load during the application of specimen machine, cannot be obtained without poles in the space D(σ; t), corresponding the curves calculated into the space D(σ – ε; t), by curves that in the kinetic process of quasi-relaxation are represented by experimental curves in coordinates log σ – t. This situation cannot be eluded, since in this phenomena exist dislocations that go conform fatigue in the nano-crystalline structure of metals. From this point of view, is necessary to obtain a spectral study related to the energy using functions that permits the modeling and compute the states of quasi-relaxation included in the poles in the deformation problem to complete the solutions in the space D(σ – ε; t), and try a new method of solution of the differential equations of the quasi-relaxation analysis. In a nearly future development, the information obtained by this spectral study (by our integral transforms), will be able to give place to the programming through the spectral encoding of the materials in the meta-stability state, which is propitious to a nano-technological transformation of materials, concrete case, some metals.

1. Introduction

In the last 30 years, the experimental technique to the characterization of materials with the use of testing machines has experimented a big heyday. In the conventional machines of essays, where the specimen previously is loaded up to an initial level of the stress, after that which the motorize system of the machine is disconnected, it is a observed a spontaneous fall of stress. The kinetic of the fall of the stress is registered during all the process of the essays [1]. Similar experiment must be executed in a programmed specially machine, in which during the essay of automatic manage stays constant the longitude of the specimen that; is to say, the condition of the essay in regime of quasi-relaxation can be expressed in the following form

(1)

or well,

(2)

This condition define the meta-stability as a state of constant deformation only in their plastic characteristics in the initial process of dislocations [2,3], where the energy of the nano-crystals accumulate the enough energy to maintain the specimen in a stable range of recovering to original state, in a very short time interval [2,4]. In this respect, it is necessary to realise a deep study of traces of deformation tensor in function of the stress tensor corresponding plastic deformation and use a functional of energy [5], that measures this recover energy due the nanocrystals [6].

2. Constitutive Equations to Stress-Deformation Tensor

Considering a material M, like the defined space by the limit surface specimen-machine; (the surface or zone of work of the machine on specimen [5]):

(3)

where is the differentiable surface in ×[7]. Here, is the density of the dislocations, is the relaxation function, and, is the velocity of applied stress to specimen., is the stress space [3,7,8]

(4)

, with k, the state tensor of media. By stress-deformation theory [3], the rate of the stress tensor, comes give by the expression in all region all space of the material M,

(5)

to all, with k, the state tensor of media and, the deformation tensor. Then [9]:

(6)

where, is the conformal elastic tensor of, and, is conformal inelastic tensor of the tensor. Note that b, is an element belonging to tensor space, (tensor of range 4, in the ordinary space) [6,10,11], which participates in the symmetrization of the elastic module tensor of the deformation tensor, and analogously, to the elastic part of. This helps us to understand that the realised transforms on the material subject to the stresses, are plastic transformations and produces only dislocations or laminar displacements in the material [4,7,12], (staying invariant the structure), thus the actions of the contributions of the inelastic and elastic conformal components of the rate of stress, and, are linear and these are superposes [8]. Then the stress space, satisfies the orthogonal relations to, and, respectively [9]:

As consequence of it, the set of solutions or integrals that we obtain to method of quasi-relaxation is included in this orthogonal decomposition to their more general integral, which is determined through the corresponding hereditary integrals in solid deformations [2,7], and whose quasi-relaxation can be study in the space, like the sum of two components of the tensor, as the produced by the external stresses, and other by the inner stresses due to the dislocation. The similar thing to the deformations. Now we precise the necessary roll of the analysis of the deformations.

Consider the deformation tensor, which is given by the function [1]

to all E-differentiable surface in, [3,7]. Deriving with respect to t, result the rate of analogous deformation like given to the tensor, to know

(7)

where

(8)

where, is the elastic modulus tensor defined by the map [6,8,11]

(9)

with correspondence rule

(10)

and, is the inelastic modulus tensor of the tensor ε, which is a map of k(depends only on the specific properties of k).

In general, in the deformation analysis the tensors, and, is a dependence on, and k. Then through their components to a transformation system of coordinates in the space, belonging to a group of orthogonal transformations of range 2 [6], takes the form

(11)

Since, and, are symmetrical, the tensor. Then the differentiable function, must be invertible in, the tensor, must be non singular and the differentiable equivalence of the limit surfaces of the material M, submitted to stress-deformation of the tensors, and, come given by the differentiable equivalence between the deformations space, [5], and the stresses space, under the differentiable maps, and E [5,7,11].

Given that the elastic energy that is got in a specimen under stress-deformation comes measured for the elastics fields of the dynamical dislocations [4], the contribution of the inelastic modulus tensor to the given deformations for the inner stresses, that is to say, due to the dislocations, contribute to the velocity of total deformation in the quasi-relaxation equation, thus

(12)

Note: k, represent the state tensor (consider variables like density, mass, temperature, pressure, etc),

Then the total deformation by the tensor, comes given by, which clearly contributes to value of the external stresses to material, which results of relevance to the deformation in quasi-relaxation (stress of the machine on the specimen).

Lemma 2.1. The velocity of total deformation given by the tensor, in the quasi-relaxation equation come given by

(13)

Proof. If we write the tensor of deformation ε, like the composition of functions [5,13,14]

(14)

and using the rule of the chain of derivation we have the tensors

(15)

Since C, is not singular then

(16)

Using the elements of inelastic transformation of the second component of the space of the Equ 1976, pp. 138-154.