Numerical modeling and calculation of compliance for centrally cracked specimen based on graph model of elastic solid

Аuthors

Tyrymov A. A.

Volgograd State technical University, 28, Lenin avenue, Volgograd, 400005, Russia

e-mail: tyrymov2010@yandex.ru

Abstract

The method of numerical analysis of the mechanical fields in the deformable body, based on graph model of an elastic medium in the form of directed graph, is considered. A new type of elementary cell is introduced to describe the features that appear near a crack tip in isotropic elastic material. An elementary cell is considered as a subgraph that corresponds to a single element of continuum. The configuration of a cell is defined by installing hypothetical devices upon an element of a solid. The equations of elementary cell are derived using the deformation energy as invariant that remains constant under the transformation of an element into a cell. A procedure determining the parameters of the elementary cell is described. The graph of a whole body is constructed the same way as for an elementary cell. Equations of state of the original solid body are derived by using the transformation of generalized coordinates of a decomposed solid body elements into a system of generalized coordinates of entire solid body. This transformation is performed using nonsingular and mutually inverse matrices. The specific nature of the graph discrete solid body model is such that allow you to construct nonsingular matrices without numerical inversion. Kirchhoff graph laws were used for derivation of a system of defining equations. Graph rules (apex and contour) have mechanical interpretation and their application cause the equations of equilibrium and strain compatibility to be satisfied when the net dimensions are reduced. The results of numerical calculation of compliance and stress intensity factor in a centrally cracked tensile plate obtained by graph method are presented.

Keywords:

mathematical modeling, elasticity, directed graph, crack, stress singularity, stress intensity factor, compliance

References

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