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**Nonlinear mechanics of structures. New approach and methods of non-incremental computation.
(Mécanique non linéaire des structures. Nouvelle approche et méthodes de calcul non incrémentales.)**
*(French)*
Zbl 0855.73003

Paris: Hermes. 304 p. (1996).

It is not easy to give in a short review of all the ideas containing in a book which relates to so many new findings in computational mechanics. To summarize, the general aim is to describe new methods of solutions of nonlinear mechanical problems in a non-incremental way and to estimate the numerical errors compared with exact solutions. The proposed method, called the large time increment method, is an iterative one stated on the whole evolution time. Based on an error bracketting, the method works like some lower and upper energy bounds.

After the first chapter, where Drucker’s inequality is recalled in view of numerical stability purposes, the next two chapters describe the so-called “error in constitutive law” which is recognized as the most tolerable one, compared to other equations of continuum mechanics. This error is written by using nonlinear dissipative terms, expressed through conjugated pseudo-functionals of dissipation for “standard” materials, in a very modern style employing the Legendre-Fenchel transform and Moreau’s and Nguyen’s ideas. This set of variables, together with the free energy functional, comprises kinematically admissible terms and statically admissible (S.A.) stresses.

The general iterative method is a double search direction method, one step local and nonlinear by resolution of a first order differential equation, the second global and linear, and both on the whole time interval. These two steps alternatively use two kinds of approximation operators, upward and downward, respectively, up to a given error. Many details, properties and theorems are given concerning uniqueness, numerical stability or convergence. The general formulation written in terms of stresses should deserve some complements (e.g. proposition 1, chapter 2, or obtaining S.A. stresses).

After an illustrative and welcome simple example (chapter 5), chapter 6 deals with the global stage again in all the space-time interval, which uses a discretized process in terms of a product of scalar functions of the time by a set of space functions.

Chapter 7 describes cyclic loadings with an efficient double time scale procedure. An industrial application illustrates the method. Chapter 8 deals with parallelism possibilities through a domain decomposition method, comprising elementary structures and interfaces, each with their own laws.

The last chapter 9 discusses two approaches to geometrically nonlinear problems. The first method proposes to insert nonlinear terms into the constitutive dissipative law, the global step being still linear, but S.A. stresses are difficult to obtain using this approach. This difficulty is shown to be overcome by use of a primal procedure. A theory is developed with the introduction of the associated rotation and with the definition of quantities called \(\Omega_0\)-material (P. Rougée), and also with the notion of turned velocity and the deformation in terms of Euler variables by use of Jaumann derivatives. Some tedious algebra seems unavoidable. Another method employs nonlinear equilibrium equations in case of soft nonlinearities. An example illustrates this kind of problems.

The book is extremely rich, as well in the ideas as in the applications treated by the author and his competent team of researchers. The proposed method, somewhat revolutionary, seems to be more efficient than many others. The reading is often agreeable, sometimes difficult, because of details and completeness presented. However, the reviewer points out that the important problem of structural stability still deserves a research effort in order to provide users a similar efficiency.

After the first chapter, where Drucker’s inequality is recalled in view of numerical stability purposes, the next two chapters describe the so-called “error in constitutive law” which is recognized as the most tolerable one, compared to other equations of continuum mechanics. This error is written by using nonlinear dissipative terms, expressed through conjugated pseudo-functionals of dissipation for “standard” materials, in a very modern style employing the Legendre-Fenchel transform and Moreau’s and Nguyen’s ideas. This set of variables, together with the free energy functional, comprises kinematically admissible terms and statically admissible (S.A.) stresses.

The general iterative method is a double search direction method, one step local and nonlinear by resolution of a first order differential equation, the second global and linear, and both on the whole time interval. These two steps alternatively use two kinds of approximation operators, upward and downward, respectively, up to a given error. Many details, properties and theorems are given concerning uniqueness, numerical stability or convergence. The general formulation written in terms of stresses should deserve some complements (e.g. proposition 1, chapter 2, or obtaining S.A. stresses).

After an illustrative and welcome simple example (chapter 5), chapter 6 deals with the global stage again in all the space-time interval, which uses a discretized process in terms of a product of scalar functions of the time by a set of space functions.

Chapter 7 describes cyclic loadings with an efficient double time scale procedure. An industrial application illustrates the method. Chapter 8 deals with parallelism possibilities through a domain decomposition method, comprising elementary structures and interfaces, each with their own laws.

The last chapter 9 discusses two approaches to geometrically nonlinear problems. The first method proposes to insert nonlinear terms into the constitutive dissipative law, the global step being still linear, but S.A. stresses are difficult to obtain using this approach. This difficulty is shown to be overcome by use of a primal procedure. A theory is developed with the introduction of the associated rotation and with the definition of quantities called \(\Omega_0\)-material (P. Rougée), and also with the notion of turned velocity and the deformation in terms of Euler variables by use of Jaumann derivatives. Some tedious algebra seems unavoidable. Another method employs nonlinear equilibrium equations in case of soft nonlinearities. An example illustrates this kind of problems.

The book is extremely rich, as well in the ideas as in the applications treated by the author and his competent team of researchers. The proposed method, somewhat revolutionary, seems to be more efficient than many others. The reading is often agreeable, sometimes difficult, because of details and completeness presented. However, the reviewer points out that the important problem of structural stability still deserves a research effort in order to provide users a similar efficiency.

Reviewer: R.Valid (Paris)

### MSC:

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

74Kxx | Thin bodies, structures |

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