Boundary Integral Equation Method in the Modeling of Nonlinear Deformation and Failure of~the~3D~Inhomogeneous Media

V. A. Petushkov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 96-114.

Résumé

The method of boundary integral equations is applied for solving the nonlinear problems of thermo-elastic-plastic deformation and fracture of inhomogeneous 3D bodies of the complex form. Solution is constructed on the basis of the generalized identity of Somigliana involving method of sequential linearization in the form of initial plastic deformations. The increments of plastic deformation are determined on the basis of the flow theory of hardening elastoplastic media with the use of modifed Prandtl–Reus's relations. The cases of complex thermo mechanical loading of compound piecewise homogeneous media in contact are considered. For describing the processes of nonlinear deformation and fracture of the bodies with a complex geometry and local peculiarities a method of discrete domains (DDBIEM) is developed. The solutions of some practical significant 3D non-linear problems of the mechanics of deformation and fracture are presented.

Détail

Keywords: inhomogeneous 3D media, nonlinear deformation and fracture, boundary integral equation method, collocation approach
Mots-clés : subdomains method.

Bibliographie

[1] S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, International Series of Monographs in Pure and Applied Mathematics, 83, Pergamon Press, New York, 1965, xii+259 pp. | MR | MR | Zbl

[2] V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, New York, 1979, xix+929 pp. | MR | Zbl

[3] K. Hayami, “Variable Transformations for Nearly Singular Integrals in the Boundary Element Method”, Publ. Res. Inst. Math. Sci., 41 (2005), 821–842 | DOI | MR | Zbl

[4] V. A. Petushkov, “Numerical realization of the method of boundary integral equations as applied to nonlinear problems of mechanics of deformation and fracture of volume bodies”, Model. Mekh., 3(30):1 (1989), 133–156 (In Russian) | Zbl

[5] S. Rjasanow, O. Steinbach, The fast solution of boundary integral equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, Heidelberg, 2007, xi+279 pp. | DOI | MR | Zbl

[6] Y. J. Liu, S. Mukherjee, N. Nishimura, M. Schanz, W. Ye, A. Sutradhar, E. Pan, N. A. Dumont, A. Frangi, A. Saez, “Recent Advances and Emerging Applications of the Boundary Element Method”, Appl. Mech. Rev., 64:3 (2012), 030802, 38 pp. | DOI

[7] N. A. Makhutov, V. A. Petushkov, V. I. Zysin, “Using the method of boundary integral equations for the numerical solution of volume problems of nonlinear fracture mechanics”, Strength of Materials, 20:7 (1988), 833–841 | DOI

[8] V. A. Petushkov, V. I. Zysin, ““MEGRE-3D” software package for numerical simulation of nonlinear processes of deformation and fracture 3D bodies. Algorithms and implementation in OS ES”, Programmnoye obespecheniye matematicheskogo modelirovaniya [Software for Mathematical Simulation], Nauka, Moscow, 1992, 111–126 (In Russian)

[9] G. C. Hsiao, O. Steinbach, W. L. Wendland, “Domain decomposition methods via boundary integral equations”, J. Comp. Appl. Math., 125:1–2 (2000), 521–537 | DOI | MR | Zbl

[10] V. A. Petushkov, R. M. Shneiderovich, “Thermoelastic plastic deformation of corrugated shells of revolution at finite displacements”, Strength of Materials, 11:6 (1979), 578–585 | DOI

[11] J. Nečas, I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies — An Introduction, Studies in Applied Mechanics, 3, Elsevier Sci. Publ. Co., Amsterdam, New York, 1980, 342 pp. | DOI | MR

[12] W. L. Wendland, “On Some Mathematical Aspects of Boundary Element Methods for Elliptic Problems”, The mathematics of finite elements and applications, v. V, ed. J. R. Whiteman, Acad. Press Inc., New York, 1985, 193–227 | DOI | MR

[13] M. Costabel, “Boundary Integral Operators on Lipschitz Domains: Elementary Results”, SIAM J. Math. Anal., 19:3 (1988), 613–626 | DOI | MR | Zbl

[14] G. Strang, G. Fix, An Analysis of the Finite Element Method, 2nd edition, Wellesley-Cambridge Press, Wellesley, MA, 2008, 400 pp. ; G. Streng, D. Fiks, Teoriya metoda konechnykh elementov, Mir, M., 1977, 294 pp. | MR | Zbl | MR

[15] J. M. Crotty, “A block equation solver for large unsymmetric matrices arising in the boundary integral equation method”, Int. J. Numer. Meth. Engng., 18:7 (1982), 997–1017 | DOI | Zbl

[16] V. A. Petushkov, A. F. Anikin, “Investigation of the fracture of three-dimensional elastic bodies with cracks”, Soviet Applied Mechanics, 22:9 (1986), 815–822 | DOI

[17] A. F. Anikin, V. A. Petushkov, ““SAPR-82” software package and the computational aspects of the computer modeling of three-dimensional deformation and failure processes of structures at elevated temperatures”, Strength of Materials, 19:7 (1987), 944–951 | DOI

[18] K. Park, G. H. Paulino, “Cohesive Zone Models: A Critical Review of Traction-Separation Relationships Across Fracture Surfaces”, Appl. Mech. Rev., 64:6 (2013), 060802, 20 pp. | DOI

[19] G. C. Hsiao, W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences, 164, Springer, Berlin, 2008, xix+618 pp. | DOI | MR | Zbl

[20] I. S. Raju, J. C. Newman Jr., “Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates”, Engng. Fracture Mech., 11:4 (1979), 817–829 | DOI