1998 Fiscal Year Final Research Report Summary
Mathematical Modeling and Mathematical Analysis of Fracture
| Project/Area Number |
09640275
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Yamaguchi University |
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Principal Investigator |
MIYOSHI Tetsuhiko Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (60040101)
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| Co-Investigator(Kenkyū-buntansha) |
KAMINISHI Ken Faculty of Engineering, Yamaguchi University Associate Professor, 工学部, 助教授 (50177581)
OKADA Mari Faculty of Engineering, Yamaguchi University Associate Professor, 工学部, 助教授 (40201389)
KURIYAMA Ken Faculty of Engineering, Yamaguchi University Professor, 工学部, 教授 (10116717)
HATAYA Yasushi Faculty of Science, Yamaguchi University Research Assistant, 理学部, 助手 (20294621)
NAKAUCHI Nobumitsu Faculty of Science, Yamaguchi University Associate Professor, 理学部, 助教授 (50180237)
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| Project Period (FY) |
1997 – 1998
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| Keywords | 2-dimensional crack / fracture mechanics / singular solution / finite element method / numerical analysis / partial differential equation |
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| Research Abstract |
The mathematical modeling of 2-dimensional crack propagation is the touchstone whether Mathematics can take" fracture" into its research object. From the view point of continuum mechanics there was no . satisfactory model. The results of this research are summarized as follows.
(1) Miyoshi[1] found that the curvature at the crack tip satisfies a functional equation (formally, an ordinally differential equation of Riccati type) from the stress free condition on the crack. The character- istics of this model are (1) the "higer order stress factors" appear as the coefficients of the equation, as well as the usual stress intensity factors (2) the quadratic nonlinearity may suggest the branching of the crack (3)the model is useful for stability analysis of the cracks.
(2) A formula to compute numerically the "higer order stress factors" is given at the same time. It is impor- tant that only the displacements are used in this formula, since in practical computation the displacements are easy to treat.
All the models used by engineers to compute the crack path are based on some mechanical assumptions. The present model, however, is derived without any such assumtion except that the deformation is gov- erned by linear elasticity. Therefore, this should be regarded as the first rigorous "crack model" which is not approximate, although there might any essential difference between ours and the others in practical engineering applications.
In order to justify our modeling we tried numerical computation by using finite element method and boundary element method. Unfortunately, such conventional methods were not useful for our aim due to the low accuracy of such methods. We intend to try an analytic approach instead of numerical justification.
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