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

Affiliation(s)

Department of Research in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Mexico, Russia.

Baikov Institute of Research in Metallurgy and Material Sciences, Mexico, Russia.

Department of Research in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Mexico, Russia.

Baikov Institute of Research in Metallurgy and Material Sciences, Mexico, Russia.

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* σ0* = *σ1*. 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.

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

KEYWORDS

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

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

Cite this paper

F. Bulnes, Y. Stropovsvky and V. Yermishkin, "Quasi-Relaxation Transforms, Meromorphic Curves and Hereditary Integrals of the Stress-Deformation Tensor to Metallic Specimens,"*Modern Mechanical Engineering*, Vol. 2 No. 3, 2012, pp. 92-105. doi: 10.4236/mme.2012.23012.

F. Bulnes, Y. Stropovsvky and V. Yermishkin, "Quasi-Relaxation Transforms, Meromorphic Curves and Hereditary Integrals of the Stress-Deformation Tensor to Metallic Specimens,"

References

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[17] F. Bulnes, V. Yermishkin and E. Toledano, “Constitutive Equations of the Stress-Strain Tensor for a Metal Speci- men Rehearsal in Quasi-Relaxation Regime and Their Generalized Functional of Energy,” Proceedings of the 2nd CIMM, Vol. III, Department of Mechanical Engi- neering, Universidad Nacional de Colombia, Bogota, 8-9 October 2009, pp. 12-0001-1-0010.

[18] A. I. Markusevich, “Theory of Analytic Functions. Vol. I, and II,” Mir, Moscu, 1980.

[19] F. Bulnes and M. Shapiro, “On General Theory of Integral Operators to Analysis and Geometry,” IM/UNAM, SEPI/IPN Monograph in Mathematics, 1st Edition, P. Cladwell, Mexico, 2007.

[20] F. Bulnes, “Analysis of Prospective and Development of Effective Technologies through Integral Synergic Operators of the Mechanics,” Proceedings in Mechanical Engineering of 14a. CCIA-CIM ISPJAE, Vol. 3, Habana, 1-5 December 2008, pp. 1021-1029.

[21] F. Bulnes, V. Yermishkin and P. Tamayo, “New Method of the Characterization of Materials of Engineering: Quasi-Relaxation, Science of Materials,” Proceedings of Appliedmath I, Mexico, 22-24 September 2005, pp. 240-254.

[22] V. N. Geminov and V. Yermishkin, “Quasirelaxation as a Powerful Method of a Steady-State Creep Characteristics Prediction,” Proceeding of II International Conference on Mechanical Behavior of Materials, Vol. 1, Boston, 16-20 August 1976, pp. 138-154.

[23] B. West, M. Bologna and P. Grigolini, ”Physics of Fractal Operators,” Institute for Non-Linear Science, Springer, New York, 2003.

[1] V. S. Ivanova and V. Yermishkin, “Prochnost i Plastichnost Tugoplavkij Metallov i Monokristallov,” Metallurgiya, Vol. 1, Moscow, 1976, pp. 80-101.

[2] B. Gross, “Mathematical Structure of the Theories of Viscoelasticity,” Hermann and Cie, Paris, 1953.

[3] J. Casey and P. M. Naghdi., “Constitutive Results for Finitely Deforming Elastic-Plastic Materials,” In: K. J. Willam, Ed., Constitutive Equations: Macro and Computational Aspects, Cap I and II ASME, New Orleans, 1984.

[4] T. Imura, “Dynamic Study of the Dislocation Progress of the Plastic Deformation and Fracture by High Voltage Electron Microscopy,” Academic Press, London-New York, 1974.

[5] C. Truesdell, “Rational Mechanics,” Academic Press, New York, 1983.

[6] J. E. Marsden and R. Abraham, “Manifolds, Tensor Analysis and Applications,” Addison-Wesley, Massachusetts, 1983.

[7] L. Landau and E. M. Lifshitz, “Mechanics, Theoretical Physics Vol. I,” Addison-Wesley, New York, 1960.

[8] C. Truesdell and R. A. Toupin, “The Classical Field Theories,” In: R. G. Lerner, Ed., Encyclopedia of Physics, Chapter 1, Springer-Verlag, Berlin, 1960.

[9] F. Bulnes, “Treatise of Advanced Mathematics: Analysis of Systems and Sign,” Monograph Vol. 1, 1st Edition, Faculty of Sciences, Universidad Nacional Autónoma de México, Mexico, 1998.

[10] B. Simon and M. Reed, “Mathematical Methods for Physics: Functional Analysis,” Vol. 1, Academic Press, New York, 1972.

[11] J. Dieudonnè, “Tratise on Analyse,” Vol. 4, Academic Press, New York, 1978.

[12] J. P. Hirth and J. Lothe, “Theory of Dislocations,” McGraw-Hill Book Company, Institute of Physics, Oslo University, New York, 1972.

[13] W. Rudin, “Real and Complex Analysis,” Academic Press, New York, 1969.

[14] R. Jonikomb, “Plasticheskaya Deformaciya,” Mir Metallov, Moscu, 1972.

[15] V. Yermishkin and V. Polin, “Monokristally Tugoplavkij i Redkij Metallov, Splavov i Soyedininii,” Nauka, Moscow, 1977, pp. 157-159.

[16] E. M. Savitsky and V. S. Ivanova, “Struktura i Svoystva Monokristallov Tugoplavkij Metallov,” Nauka Academy of Sciences Publisher, Moscow, 1973, pp. 139-143.

[17] F. Bulnes, V. Yermishkin and E. Toledano, “Constitutive Equations of the Stress-Strain Tensor for a Metal Speci- men Rehearsal in Quasi-Relaxation Regime and Their Generalized Functional of Energy,” Proceedings of the 2nd CIMM, Vol. III, Department of Mechanical Engi- neering, Universidad Nacional de Colombia, Bogota, 8-9 October 2009, pp. 12-0001-1-0010.

[18] A. I. Markusevich, “Theory of Analytic Functions. Vol. I, and II,” Mir, Moscu, 1980.

[19] F. Bulnes and M. Shapiro, “On General Theory of Integral Operators to Analysis and Geometry,” IM/UNAM, SEPI/IPN Monograph in Mathematics, 1st Edition, P. Cladwell, Mexico, 2007.

[20] F. Bulnes, “Analysis of Prospective and Development of Effective Technologies through Integral Synergic Operators of the Mechanics,” Proceedings in Mechanical Engineering of 14a. CCIA-CIM ISPJAE, Vol. 3, Habana, 1-5 December 2008, pp. 1021-1029.

[21] F. Bulnes, V. Yermishkin and P. Tamayo, “New Method of the Characterization of Materials of Engineering: Quasi-Relaxation, Science of Materials,” Proceedings of Appliedmath I, Mexico, 22-24 September 2005, pp. 240-254.

[22] V. N. Geminov and V. Yermishkin, “Quasirelaxation as a Powerful Method of a Steady-State Creep Characteristics Prediction,” Proceeding of II International Conference on Mechanical Behavior of Materials, Vol. 1, Boston, 16-20 August 1976, pp. 138-154.

[23] B. West, M. Bologna and P. Grigolini, ”Physics of Fractal Operators,” Institute for Non-Linear Science, Springer, New York, 2003.