Local strain energy state at the crack tip vicinity in discrete model of material




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НазваниеLocal strain energy state at the crack tip vicinity in discrete model of material
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LOCAL STRAIN ENERGY state

AT THE CRACK TIP VICINITY IN DISCRETE MODEL OF MATERIAL

Dragan B. Jovanović1


Faculty of Mechanical Engineering, University of Niš, A. Medvedeva 14, 18000 Niš, Serbia
E-mail: jdragan@masfak. ni.ac.yu

Summary

According to discrete model of material, solids may be represented as systems of discrete masses (atomic lattice), linked by interacting forces, interatomic forces or simple bonds (Fig. 1) . Not only mechanical loads are involved in crack growth, but also chemical, thermo-mechanical, electro-mechanical, acoustic and other physical phenomena are also involved. Advantage of a discrete model is ability to explain crack healing, slow subcritical crack growth, sound generation during crack propagation, effects of chemical processes at the tip of the crack, nonlinearity of stress, strain and energy distribution in the crack tip region, influence of temperature on crack propagation. All of this give great advantage to procedures based on model of discrete masses (atomic lattice). Different intrinsic interatomic force functions are used to represent mechanical interaction between the neighboring atoms (discrete masses) in lattice. Different two- and three-dimensional models of lattice and relations for total potential energy of selected models of lattice are analysed (Fig. 2).




Figure 1. Model of three-dimensional lattice Figure 2. Surface of potential energy for two-dimensional x-y lattice and possible trajectories of crack propagation


In a plate with different ruling cases of global or general stress state, different local stress, and strain states will appear at vicinity of the crack, as a result of interaction between crack geometry and global stress state. By taking in consideration of elliptically shaped crack, and by introducing different general stress states, corresponding local stress and dislocation distributions at the vicinity of such crack were obtained. Three-dimensional model of lattice with crack was applied.

Based on obtained diagrams of stress distribution, and displacements distribution, conclusions about influence of global stress state on local distribution of stresses and displacements at surrounding of the crack were drawn out.

Interactions of different physical phenomenon involved in initiation and propagation of cracks, and in the process of fracture and damage, have directed research towards analyzing processes at atomic and molecular level. There is consistency between conclusions based on discrete (atomic) and macroscopic models, related to strain energy distribution in vicinity of the crack tip. Assumed functions of interatomic forces are presented and their relations with the potential energy are analyzed. It is shown that: The site of fracture coincides with the location of minimum strain energy density, and yielding with maximum strain energy density (Sih G. C., see Ref. [13]) ( see Fig. 2).


Key words: crack, mathematical form of localized energetic structure, discrete model of material, atomic lattice, functions of interatomic forces, potential energy of atomic bond, total potential energy of lattice, activation energy, strain energy surfaces.


References

[1] Berdichevskii V., Truskinovskii L., Energy Structure of Localization, Local Effects in the Analysis of Structures, Edited by Pierre Ladeveze, Elsevier, New York, 1985

[2] Brankov G., Fracture Theory at Atomic Level, Journal of Theoretical and Applied mechanics, Year XXIV, No 2, Sofija, 1993.

[3] Gdoutos E. E., Fracture Mechanics, Kluwer Academic Publishers, Dordrecht, 1993

[4] Gdoutos E. E., Problems of mixed mode crack propagation, Martinus Nijhoff Publishers, Kluwer Academic Publishers, The Hague, 1984

[5] Goro{ko O. A., Hedrih (Stevanovi}) K., Analiti~ka dinamika (mehanika) diskretnih naslednih sistema, Izdava~ka jedinica Univerziteta u Ni{u, 2000, Ni{

[6] Gurtin E. Morton, The Nature of Configurational Forces, Arch. Rational Mech. Anal. 131 pp 67-100, Springer-Verlag, 1995

[7] Hedrih (Stevanovi}) K., Discrete Continuum Method, COMPUTATIONAL MECHANICS,

WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China,  2004 Tsinghua University Press & Springer-Verlag.

[8] Krausz A. S., Krausz K., Fracture Kinetics of Crack Growth, Kluwer Academic Publishers, Netherlands, 1988

[9] Knott J. F., Met B., Fundamentals of Fracture Mechanics, Butterworth & Co (Publishers) Ltd London, 1973

[10] Jovanovi} B. D., Jovanovi} M., Stress state and strain energy distribution at the vicinity of elliptical crack with compression forces acting on it's contour, YUSNM, Ni{ 2000, Facta Univers., Series Mechanics, Automatic Control and Robotics, Vol. 3, No. 11, 2001, pp. 223-230

[11] Jovanovi} B. D., Stress state and deformation (strain) energy distribution ahead crack tip in a plate subjected to tension, Facta Universitatis., Series Mechanics, Automatic Control and Robotics, Vol. 3, No. 12, 2002, pp. 443-455

[12] Parton V. Z., Fracture Mechanics - From Theory to Practice, Institute of Chemical Engineering, Moscow, Gordon and Breach Science Publishers, 1992

[13] Sih G. C., Prediction of crack growth characteristics, Proceedings of an International Symposium on Absorbed specific energy and/or strain energy density criterion, pp 3-16, Martinus Nijhoff Publishers the Hague, 1982

[14] Sih G. C., Thermal/Mechanical interaction associated with the micromechanisms of material behavior, Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania, 1987

[15] Theocaris P. S., The strain-energy-density criterion-investigation for its applicability, Proceedings of an International Symposium on Absorbed specific energy and/or strain energy density criterion, pp 17-32, Martinus Nijhoff Publishers the Hague, 1982






1 Faculty of Mechanical Engineering, University of Niš, A. Medvedeva 14, 18000 Niš, Serbia, E-mail: jdragan@masfak. ni.ac.yu


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