Application of rheological model of material with microdefects and nanodefects with hydrogen in the case of cyclic loading

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1 Key Engineering Materials Submitted: ISSN: , Vols , pp Revised: doi: / Accepted: Trans Tech Publications, Switzerland Online: Application of rheological model of material with microdefects and nanodefects with hydrogen in the case of cyclic loading Yuriy A Yakovlev 1,a*, Alexander K. Belyaev,1b,Vladimir A. Polyanskiy 1,2 b. 1 Institute for Problems in Mechanical Engineering, RAS, V.O., Bolshoy pr., 61, St. Petersburg, Russia 2 St.-Petersburg State Polytechnical University, Polytekhnicheskaya, 29, St. Petersburg, Russia a yura.yakovlev@gmail.com, b vice.ipme@gmail.com, c vapol@mail.ru Keywords: hydrogen, diffusion of hydrogen, two-continuum model, Mathieu equation, instability chart. Abstract. The paper deals with the example of application of the two continuum rheological model of materials with microdefects, nanodefects and solute hydrogen for calculation of stress and strain in cylindrical specimen under periodic loading. The model suggested allows one to relate the mechanical characteristics with the hydrogen concentration. The stability analysis of the system metal-hydrogen is carried out. The influence of parameters of the mechanical loading, hydrogen concentration and parameters of sorption and desorption of hydrogen from the surface of the internal defects (traps) of various nature on the system stability is performed. It is shown that the influence of hydrogen can be considered as parametric instability of a continuous medium under mechanical deformation. It can be important for forming or plastic deformation of materials and nano-materials containing hydrogen. Introduction Hard conditions of functioning of machines and mechanisms as well as tendency to decrease the weight and increase the lifetime leads to wide application of high-strength materials. Using of the high-strength materials is extremely important in aviation, aerospace and automobile industry. It is typical for these branches that the ratio of the structure mass to the payload plays the critical part. The minimum safety coefficient for strength leads to the necessity of exact calculation with account for all factors during the structure exploitation. On the other side using the high-strength materials has a hidden danger. In contrast to the common steels the high-strength steels are more sensitive to the inner material defects. In many cases the existence of the inner material defects is related to existing hydrogen in the material [1]. The initial hydrogen appear in materials during production of structural materials from water vapor, oils and others hydrogenous mixtures. As a rule, initial hydrogen concentration is small as compared with concentration of other components. In many structural materials the initial hydrogen concentration is in the range from 0,1 ppm to 100 ppm. For example, the hydrogen concentration for low-quality steel is about 1-4 ppm, but for modern steels with special mechanical properties, it can be of the order of 0,01-0,2 ppm. The hydrogen concentration increases as a result of different physical and chemical processes. As a rule, the mechanical properties of material are degenerated after the hydrogen concentration has increased. Even a small local increase in hydrogen concentration can lead to change of mode of deformation and destruction of material. For many materials the irreversible change of mechanical properties occurs with a double increase in the hydrogen concentration. Therefore the problem of the influence of low concentrations of hydrogen in metals attracts nowadays a lot of attention. The local increase in hydrogen concentration in special steels from 0,1 ppm to 0,2 ppm occurs faster than increase from 2 ppm to 4 ppm in classic steels. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-12/05/16,02:35:25)

2 Key Engineering Materials Vols It is well known [2,3] that hydrogen is not only accumulated in the metals, but it changes its binding energy. Accumulation and redistribution of the hydrogen in materials occurs faster after mechanical loading. It means the material destruction can take place even without accumulation of the hydrogen from outside as a result of redistribution of hydrogen in the material volume and the hydrogen redistribution over the levels of binding energy. As a rule, the hydrogen binding energy changes after mechanical loading. It means that the accumulation of the diffuse hydrogen occurs during mechanical loading. It is very dangerous because the hydrogen concentration strongly correlates with the mechanical properties. Therefore the effect of hydrogen degradation of mechanical properties in the calculation of resource of the material should be taken into account. Two-continuum model. The majority of papers [4,5,6,7] addresses the effect of hydrogen on the strength of materials and suggests primarily the phenomenological models or micro-models. The problem of redistribution of hydrogen over the traps in continuum is not discussed. The degradation of mechanical properties of continuum in these papers is carried out by means of some empirical dependence or by means of the simplified approximation of micro-models. In contrast, we propose a two-continuum model of solid [8,9] which enables one to describe the influence of a small concentration of hydrogen on the mechanical properties of materials. The first continuum is the structural material, whereas the second continuum describes the hydrogen dynamics. The hydrogen diluted in the structural materials can be conditionally divided into that with low bonding energy and that with higher bonding energy levels. The hydrogen with low binding energy is diffuse, and its interaction with material is very weak. The high bonded hydrogen interacts with material more intensively. We suppose that the mechanical properties of continuum degrade due to this interaction. The suggested model is capable to describe both the hydrogen diffusion and its interaction with the material. For one-dimensional case the main equation of this model is: Here the superscript (1) and (2) refers to values of the first continuum (solid body) and the second continuum (diffuse hydrogen). is number of mobile particles of hydrogen, is the number of hydrogen particles attached to the lattice, is number of particles connected by the perfect bonds. is the reactive force. is the internal force that determines the reaction of interaction between the first and second components of the considered continuum and its definition essentially depends on the character of processes in the material. is the pressure of the diffuse hydrogen, is the density of first continuum, is the mass of hydrogen particles, the is velocity, is the density of mobile particles of hydrogen, are some constants of the material, and denotes the section of the diffusion channel that depends on strain.

3 594 Material Forming ESAFORM 2015 Modeling the hydrogen diffusion in metals under cyclic loading. The most common type of loading is a cyclic loading. Consider the application of twocontinuum model in this case. Adaptation of the two-continuum model for the case of cyclic loading yields the following equation of the hydrogen balance: (1) This equation is solved using Fourier method equations:. It leads to the following (2) We consider a cyclic loading with small strain amplitude of frequency, i.e.,. Let us suppose as the first approximation that. Substituting into equation (2) enables us to put it in the form of the generalized Mathieu equation where the new parameters are:, (3). The parameter has the dimension of frequency, but it is not the natural frequency of mechanical system. We can understand it as a measure of the interaction of hydrogen with continuum. The dimensionless parameter can be understood as the intensity of the external mechanical loading expressed in terms of the hydrogen concentration. This stability of this equation is investigated. The first approximation of boundary of the main instability area is given by: (4) Substituting eq. (4) into eq.(3) and simplifying the result leads to the dependence: or (5) where:,,,

4 Key Engineering Materials Vols Part of the plot for is presented on the Fig. 1. This dependence is the border of the instability chart of equation (5). instability stability Figure 1. Instability chart It is seen, that the plot has a minimum near the point ω under the condition: This condition is realized only at. This correlation between coefficients is typical for alloys which can accumulate the hydrogen. One can see from equation (5) that the dependence is a function of parameters C 2 and. Consider how the parameters C 2 and have an influence on the instability chart. Some plots for several values of parameters C 2 and are presented in Figures 2-4. Figure 2. Instability chart for the case C 2 =

5 596 Material Forming ESAFORM 2015 Figure 3. Instability chart for the case C 2 < Figure 4. Instability chart for the case C 2 > After analysis of plots it is seen that instability chart strongly depends on parameters C 2 and. Parameters C 2 and depends on process of accumulation the hydrogen inside material. The result allows us to interpret the process of destruction as the parametric resonance instability. The results obtained allow one to calculate a safe level of load at which the fracture does not occur. This safe level is given by:

6 Key Engineering Materials Vols Conclusions The influence of parameters of the mechanical loading, hydrogen concentration and parameters of sorption and desorption of hydrogen from the surface of the internal micro- and nanodefects (traps) of various nature on the system stability is performed. Influence of hydrogen can be considered as parametric instability of a continuous medium under mechanical deformation. There is a safe level of the hydrogen concentrations and mechanical strain under which the parametric instability does not occur. These concentration and mechanical strain can be calculated from the material constants and results of the experiments with the material specimens. These results can be interpreted as a hydrogen accumulation depending on the strain. It can be important for forming or plastic deformation of materials and nano-materials containing hydrogen. Acknowledgements The financial support of the Russian Foundation for Basic Research, grants a and a, is gratefully acknowledged. References [1] Tomoki Doshida, Kenichi Takai, Dependence of hydrogen-induced lattice defects and hydrogen embrittlement of cold-drawn pearlitic steels on hydrogen trap state, temperature, strain rate and hydrogen content, Acta Materialia, Volume 79, 2014, [2] Polyanskiy A.M., Polyanskiy V.A. Determination of Hydrogen Binding Energy in Various Materials by Means of Absolute Measurements of its Concentration in Solid Probe// Hydrogen Materials Science and Chemistry of Carbon Nanomaterials, SPRINGER SCIENCE + BUSINESS MEDIA B.V p [3] Polyanskiy A.M., Polyanskiy V.A., Yakovlev Yu.A. Experimental determination of parameters of multichannel hydrogen diffusion in solid probe, Int. J. of Hydrogen Energy 30(39) 2014, p [4] Birnbaum, H.K., & Sofronis, P. Hydrogen-enhanced localized plasticity alpha mechanism for hydrogen-related fracture. Mat. Sci. and Eng., 1994, A 176(1 2), [5] Delafosse, D., & Magnin, T. Hydrogen induced plasticity in stress corrosion cracking of engineering systems. Eng. Fract. Mech.s, 2001, 68(6), [6] Sofronis, P., Liang, Y., & Aravas, N. Hydrogen induced shear localization of the plastic flow in metals and alloys. European J. of Mech., 2001, A/Solids 20(6), [7] Van Leeuwen, H.P. The kinetics of hydrogen embrittlement: A quantitative diffusion model. Eng. Fract.Mech., 1974, 6(1), [8] Belyaev, A.K. and Indeitsev, D.A. and Polyanskiy, V.A. and Sukhanov, A.A. Theoretical Model for the Hydrogen-Material Interaction as a Basis for Prediction of the Material Mechanical Properties. Sandia National Laboratory, Albuquerque, New Mexico, [9] A.K.Belyaev, V.A.Polyanskiy, Yu.A.Yakovlev. Stresses in pipeline affected by hydrogen. Acta Mechanica, vol. 224, No. 3-4, 2012 p

7 Material Forming ESAFORM / Application of Rheological Model of Material with Microdefects and Nanodefects with Hydrogen in the Case of Cyclic Loading /