Computation of singular solutions in elliptic problems and elasticity.

*(English)*Zbl 0647.73010
Recherches en Mathématiques Appliquées, 5. Paris etc.: Masson. 200 p. (1987); Chichester etc.: John Wiley & Sons; £19.95 (1987).

This interesting and clearly written monograph is mainly focused on the explicit (as far as is possible) computation of the singular solutions near a corner and at interfaces for plane elliptic problems and plane elasticity problems with piecewise constant coefficients.

In Lecture II the authors recall some applications of the method of separation of variables to the solution of the Laplace problem and in Lecture III they give an illuminating analysis of the singularity of the solution of the Laplace equation \(-\Delta u=f\) at a corner when f vanishes in a neighbourhood of the corner. The general theory of singular solution is given e.g. by P. Grisvard [Elliptic problems in nonsmooth domains. Pitman, London (1985)] and shows that the solution of this problem behaves as \(r^{\alpha}u(\theta)\) \(+\) more regular terms (in polar coordinates where r is the distance to the corner). The analysis is extended in Lecture V to the transmission problem for a general 2nd order operator (while Lecture IV is devoted to some classical regularity results).

The crucial exponent \(\alpha\) is a solution of a complicated transcendental equation. The authors propose two numerical methods in order to compute the exponent \(\alpha\) : the first in Lecture V and the second in Lecture VI (devoted to the still more complicated elasticity problem in two independent variables and which is the core of the monograph). They are based on the fact that the above function u(\(\theta)\) is a solution of a 2nd order boundary value problem of a system of two ordinary differential equations that depend quadratically on \(\alpha\). At the same time the authors obtain also an approximation of the stress intensity factor. Many numerical results are presented. Different extensions are given in Lectures VII, VIII and IX and some commelastic systems) is divided into 5 chapters, i.e.: I. Instability in case of neighbouring equilibrium forms; II. Instability in case of nonneighbouring equilibrium forms; III. Instability in case of loose of equilibrium; IV. Instability in case of loose of a stable form of equilibrium; V. Bending of not entirely elastic bars. The second part (Oscillations of elastic systems) contains 4 chapters, i.e.: VI. Some oscillation problems of linear systems; VII. Dynamic action of moving loads; VIII. Aeroelastic oscillations; IX. Some oscillation problems of nonlinear systems.

Because of its new concepts, of the paradoxes and fallacies presented, the book is a valuable contribution to the field.

In Lecture II the authors recall some applications of the method of separation of variables to the solution of the Laplace problem and in Lecture III they give an illuminating analysis of the singularity of the solution of the Laplace equation \(-\Delta u=f\) at a corner when f vanishes in a neighbourhood of the corner. The general theory of singular solution is given e.g. by P. Grisvard [Elliptic problems in nonsmooth domains. Pitman, London (1985)] and shows that the solution of this problem behaves as \(r^{\alpha}u(\theta)\) \(+\) more regular terms (in polar coordinates where r is the distance to the corner). The analysis is extended in Lecture V to the transmission problem for a general 2nd order operator (while Lecture IV is devoted to some classical regularity results).

The crucial exponent \(\alpha\) is a solution of a complicated transcendental equation. The authors propose two numerical methods in order to compute the exponent \(\alpha\) : the first in Lecture V and the second in Lecture VI (devoted to the still more complicated elasticity problem in two independent variables and which is the core of the monograph). They are based on the fact that the above function u(\(\theta)\) is a solution of a 2nd order boundary value problem of a system of two ordinary differential equations that depend quadratically on \(\alpha\). At the same time the authors obtain also an approximation of the stress intensity factor. Many numerical results are presented. Different extensions are given in Lectures VII, VIII and IX and some commelastic systems) is divided into 5 chapters, i.e.: I. Instability in case of neighbouring equilibrium forms; II. Instability in case of nonneighbouring equilibrium forms; III. Instability in case of loose of equilibrium; IV. Instability in case of loose of a stable form of equilibrium; V. Bending of not entirely elastic bars. The second part (Oscillations of elastic systems) contains 4 chapters, i.e.: VI. Some oscillation problems of linear systems; VII. Dynamic action of moving loads; VIII. Aeroelastic oscillations; IX. Some oscillation problems of nonlinear systems.

Because of its new concepts, of the paradoxes and fallacies presented, the book is a valuable contribution to the field.

Reviewer: P.P.Teodorescu

##### MSC:

74B99 | Elastic materials |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74E30 | Composite and mixture properties |

65K10 | Numerical optimization and variational techniques |

35B40 | Asymptotic behavior of solutions to PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35E99 | Partial differential equations and systems of partial differential equations with constant coefficients |

74G70 | Stress concentrations, singularities in solid mechanics |