Crack growth in pyrographite under the conditions of radiation
Crecimiento de grietas en pirografito bajo condiciones de radiación
Pedro Alejandro Tamayo-Meza a, Viacheslav Alexandrovich Yermishkin b, Usiel Sandino Silva-Rivera c, Alejandro Leyva-Díaz d, Josué Osmar Trejo-Escandón e, Juan Manuel Sandoval-Pineda f & Luis Flores-Herrera g
a Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. ptamayom@ipn.mx
b Institute of Metallurgy, Russian Academy of Science, Moscow, Russian Federation.
c Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. usiel31@yahoo.com.mx
d Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. a.leyvadíaz@gmail.com ]]>
e Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. jtescandon@hotmail.com
f Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. jsandovalp@ipn.mx
g Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México, México. lafloresh@ipn.mx
Received: November 27th, 2014. Received in revised form: April 12th, 2015. Accepted: December 10th, 2015.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Abstract
The damage induced by radiation in the pyrographite is accompanied by significant plastic deformation. Microcracks arise in the edges of the radiated areas that develop radially in the direction of the unaffected matrix. The peculiarities in the formation of the tension state of the radiated area and adjacent unaffected areas are analyzed in order to explain the reasons behind the growth of cracks. The analysis is carried out on graphite disks with constant thicknesses that are exposed to radiation with high-energy electrons in an HVTEM-JEOL 1000. A differential equation for specific load conditions is obtained from the analysis of equilibrium conditions of a disc-shaped element.
Keywords: Pyrographite, crack, irradiation, HVTEM, stress state.
]]> ResumenPalabras clave: Pirografito, grieta, irradiación, HVTEM, estado de esfuerzos.
1. Introduction
Exposure of pyrographite to radiation with high-energy electrons causes a type K → A phase transition, accompanied by considerable plastic deformation. It was established that microcracks emerge at the edges of the irradiated zone, these microcracks then grow unceasingly in the radial direction within the irradiated matrix, Fig. 1.
To clarify the possible causes of the incubation and growth of cracks, it is necessary to examine the peculiarities in the formation of a stress state in the irradiated zone as well as in the conjugated matrix zone that is not affected by the radiation. This cannot be caused by the change of the temperature field in the irradiated zone, since the heating magnitude in the HVTEM column by means of the electron beam does not exceed 40ºC [1,2]. Besides, prolonged radiation experiments undoubtedly favor the balance of the temperature field gradients in both the sample thickness and the radial direction within the irradiated zone limits.
2. Theoretical approach
]]> The radiation scheme of pyrographite samples in the form of flat disks performed within the HVTEM is shown in Fig. 2.As a result of a prolonged high energy electron bombing, the amorphization is developed in the pirografites, which leads to a decrease in the material specific volume in the irradiated zone [3]. Considering that the irradiated zone is surrounded by a matrix that is not affected by radiation, a forced interaction consequently arises between it and the loosened of the irradiated zone's material [4].
From the equilibrium conditions of a circular sheet element of constant thickness analysis, it is possible to obtain a differential equation for a determined loading scheme. Following the plate theory methods [5,6], we obtain the equilibrium equation for the plate and solve it for the initial and boundary conditions that are derived from the supposed scheme of interaction between an irradiated zone wrapped by a sample matrix. The differential equation is obtained as follows:
where 
is the radial stress, and 
is the tangential stress.
The link between the components of the stress tensor , , 
and the deformations 
, 
can be expressed in terms of the Generalized Hook's Law:
where E is the Young modulus, 
is the Poisson modulus.
Supposing that in the conditions of our problem 
, it is possible to obtain the expressions for 
and from the eq. (2) as follows:
It is not difficult to prove that
where is the radial deformation, is the tangential deformation, u is the radial displacement.
After substituting eq. (4) in the eq. (3) we obtain:
After substituting eq. (5) in the equilibrium differential equation (1), the latter can be expressed as:
Finally, it can be presented in the following form:
where 
and 
are integration constants, the values of which come from the contour conditions.
In our problem, these can be expressed as r = c; 
; r=∞; 
for the matrix outside the zone of radiation, and r = c; 
; r = 0; u = 0 for the irradiated zone.
To determine the integration constants in solution (8), we inserted them in eq. (5) and consequently obtain:
Considering the contour conditions, eq. (9) for the plate placed outside the irradiated zone obtained the following form:
from which the expression for the integration constants c1 y c2 can be presented as
Similarly, for the irradiated zone, the substitution of the corresponding boundary conditions leads to the following expressions:
From which we deduce:
After substituting the values of and in eq. (8) and (9) we obtain:
3. Discussion of the results
]]> The material in the irradiated area has been under the effect of a biaxial stress state in compression, the magnitude of which can be determined by the degree of amorphization process development. The surrounding matrix is also in a state of biaxial stress. However, the compression forces decrease in the radial direction that is inversely proportional to the square of the radius measured from the center spot of radiation, but the stress forces, which are equal by modulus to compression forces, act in the tangential direction,. A physical approach to the problem is to determine the magnitudes of the contact pressures
, which act in the limits of the irradiated area. To find the value of this pressure, we use the solution proposed by Lame [5], which describes the emergence of a contact pressure when a cylindrical piece joins a sheet having a central perforation with a certain tightening Dc: 
where 
is the sheet displacement in the borders of the irradiated zone described by eq. (12), 
is the zone of displacement in the boundary described by eq. (17), 
is the tightening intensity.
In this case the tightening is a result of the swelling of the irradiated zone due to amorphization development stimulated by radiation. We determined its size from the conditions of substance mass conservation in the irradiated zone during the radiation process applied to it:
where 
is the initial crystalline pyrographite density, 
is the present value of the material density in the irradiated zone, h is the sample thickness, and c is the irradiated zone radius.
After certain transformations, eq. (20) can be represented in the following form:
Recognizing that during the pyrographite amorphization process the deformation develops isotropically, i.e., 
, and is the density of totally amorphized pyrographite 
, and the crystalline pyrographite is expressed as , then according to the mixture law:
is the amorphization degree. Eq. (21) can be expressed in the following form:
Solving eq. (23) with respect to c and substituting eq. (22) in it we obtain
Substituting eq. (24) in eq. (19), we have that for r = c
where eq. (26) is for the contact pressure:
where 
and 
are Poisson coefficients for the crystalline and amorphous pyrographite, respectively; and 
are Young moduli for the crystalline and amorphous pyrographite, respectively.
, the tightening 
. The maximum value of 
is reached when pyrographite is amorphized totally in the irradiated zone. At this moment the contact pressure reaches the value of 
When the character of stress distribution in the matrix zone not exposed to radiation described by formula 11 is analyzed, the reasons for the emergence of cracks at the border of the irradiated area and their propagation in the radial direction become clear.
For r = c, the stress forces in the tangential direction reach their maximum value. The resistance under these forces is much weaker in pyrographite compared with the effects of compression forces [7]. The fact that the cracks propagate along the crystallographic directions 
indicates that the fracture anisotropy resulting from the crystallographic structure of pyrographite expresses itself, and the characteristics of the deformation and stress state in an anisotropic approximation can be considered to be approximations. In Fig. 3, the kinetics of crack development in a matrix of pyrographite unaffected by the radiation is observed. It can be seen that the curve consists of two sections:
I. A section in which the crack grows at a rate of 1.10-5 cm/sec, and
II. A section in which the crack develops at a rate of 1.1 x 10-7 cm/sec.
4. Conclusions
1. We established that when a stream of high-energy electrons impacts pyrographite, amorphization is stimulated in it, which can be described as a transformation of the crystalline phase into the amorphous state. We studied the influence of the radiation parameters on the amorphization kinetics.
]]> 2. We proposed a phenomenological model of amorphization by radiation, which considers a possibility of a direct phase transformation of the type K → A, as well as the inverse transformation A → K. Based on this, we obtained a formula that describes the kinetics of amorphization by radiation in the following form:
3. We detected the effect of pyrographite amorphization under the influence of a high-energy electron stream inside an HVTEM. We proposed a phenomenological model that describes the kinetics of amorphization in pyrographite due to radiation using reference points put in the material surface. This was performed by means of the implantation of ionized copper atoms.
We studied the deformation and stress state of pyrographite, both in the zone irradiated by electrons as well as in the zones adjacent to the matrix unaffected by radiation.
We proposed a method to determine the energy threshold of radiation damage in pyrographite using data from the kinetics analysis of its amorphization under conditions of radiation by high-energy electrons performed inside the HVTEM column under various stress acceleration values.
Acknowledgment
The authors express their gratitude to all the staff at the Laboratory 20, Baikov Institute of Metallurgy, Russian Academy of Sciences, for their support. We also thank the Secretary of Research of the National Polytechnic Institute of Mexico, SIP, for their support for Project 20131380, 20141031.
References
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P.A. Tamayo-Meza, is a Dr. of Technology and Metallurgy. He holds a PhD. in Engineering from the Academy of Sciences of the USSR, Moscow, Russia, and an MSc. in Mechanical Engineering from the Faculty of Engineering, University of Moscow, Russia. His professional experience in research includes the development of fiber reinforced materials, mechanics of materials characterization by quasi-relaxation, transmission electron microscopy, scanning, and atomic force; the study of corrosion phenomena, solid state physics, physics, and fracture mechanics; creep and fatigue, physical mechanics of materials, and theory of heat treatment; and mechanical cryogenic treatment. ORCID: 0000-0001-8026-8928
V.A. Yermishkin, obtained his Dr. in Physical Sciences and Mathematics in Lomonosov State University. He works as chief of Laboratory of High-Voltage Electron Microscopy in the Institute of Metallurgy and Materials Science at the Russian Academy of Sciences. His professional experience in research includes physics, fracture mechanic, creep and fatigue, transmission electron microscopy, scanning and atomic force and development of fiber reinforced materials. He has published more than 132 works. ORCID: 0000-0001-7280-4516
U.S. Silva-Rivera, obtained his Dr. of Science in Mechanical Engineering, specializing in Mechanical Design (2015) and his MSc. degree in Manufacturing Engineering (2011) from the School of Mechanical and Electrical Engineering in the National Polytechnic Institute of Mexico, and his BSc. degree in Military Industrial Engineering, specializing in Chemical Engineering (2003), from Military Engineering School in the University of the Army and Air Force. His research interests are experimental ballistics, computational fluid dynamics, materials and mechanical design; he has a knowledge of production processes and quality control of small arms ammunition, as well as in design, development and production of assault rifles. ORCID: 0000-0001-5597-1638
A. Leyva-Díaz, obtained his BSc. degree from the Technological Institute of Tuxtla Gutierrez in 2011, and his MSc. in Manufacturing Engineering from the School of Mechanical and Electrical Engineering in the National Polytechnic Institute of Mexico in 2014. He has a knowledge of mechanical design, static and dynamic analysis in 3D models through finite element method software. His research interests are Mechanical Design, Finite Element Analysis, and the development and characterization of new materials. ORCID: 0000-0002-7478-9441
J.O. Trejo-Escandón received his BSc. in Mechanical Engineering in 2012, from the Technological Institute of Tuxtla Gutierrez, Mexico and a MSc. in Manufacturing Engineering in 2015, from the National Polytechnic Institute - IPN, Mexico. He currently works at the Development & Innovation Center in Schneider-Electric in Monterrey, Mexico, where he works as a Structural Simulation Specialist. His areas of research interest focus on Mechanical Design, Finite Element analysis and development and characterization of new materials. ORCID: 0000-0002-5076-7045
J.M. Sandoval-Pineda, is a professor in the National Polytechnic Institute of México. He obtained his Dr. of Science in Mechanical Engineering with honors in 2008 and his MSc. in Mechanical Engineering in 2004, from the National Polytechnic Institute, Mexico. His professional experience in research includes physics, fracture mechanic, creep and fatigue, and transmission electron microscopy. ORCID: 0000-0002-6529-7920
]]> L.A. Flores-Herrera, is a professor in the National Polytechnic Institute of México. He obtained his Dr. of Science in Mechanical Engineering with honors in 2007 and his MSc. in Mechanical Engineering in 2003 from the National Polytechnic Institute, Mexico. Hi is a specialist in Finite Element Analysis and his research area focuses on Microelectromechanical systems (MEMS). ORCID: 0000-0003-1081-5193 ]]>
