Nonlinear continuum mechanics for finite element analysis.

*(English)*Zbl 0891.73001
Cambridge: Cambridge University Press. xvii, 248 p. (1997).

The book offers modern topics in nonlinear computational mechanics. It provides a unified introduction to nonlinear continuum mechanics, nonlinear finite element schemes, and to solution techniques employed in the software. The monogaph can be recommended to postgraduate students and to researchers from mechanical, aerospace and civil engineering areas.

The book contains 8 chapters and an appendix. In the first chapter, the authors introduce basic computational concepts of nonlinear mechanics, e.g. nonlinear strain measures, linearization, and directional derivatives. An example of a nonlinear truss is solved by using a short FORTRAN program which is a prototype of the main finite element program developed further in the book.

After a chapter dealing with mathematical preliminaries, (vector and tensor algebra, linearization of directional derivatives and some elements of tensor analysis), the authors present the kinematics of finite deformations. The central concept is the deformation gradient tensor, which describes the relationship between vectors from undeformed and deformed body configurations. Lagrangian and Eulerian descriptions of motion are discussed together with the linearized kinematics which is used later in the Newton-Raphson solution process.

Chapter 4 introduces stress and equilibrium concepts for a deformable body undergoing a finite motion. The stress is first defined in the current configuration in a standard way, leading to the Cauchy stress tensor used in linear analysis. In contrast to linear small displacement theory, the stress quantities can be defined in the initial body configuration. This leads to Piola-Kirchhoff stress tensors and to alternative equilibrium equations. Finally, the objectivity of several stress rates is discussed.

In chapter 5 the authors write out the constitutive equations of hyperelastic materials. Isotropic hyperelasticity, both in Lagrangian and in Eulerian descriptions, and for compressible and incompressible materials, is considered. Then the equations are extended to a general description in principal directions for the cases of plane strain, plane stress, and for the uniaxial behaviour. Chapter 6 deals with the linearized equilibrium equations. In order to develop a Newton-Raphson solution procedure, the virtual work principle is linearized before the discretization. A large part of this chapter is devoted to incompressible hyperelastic materials and to the development of a mean dilatation procedure by using the Hu-Washizu variational principle.

All previous chapters provide a foundation for the discretized and linearized equilibrium equations considered in chapter 7. The linearization leads to finite element formulation, whereas the discretization of the linearized equilibrium equations introduces the tangent matrix. The mean discrete dilatation technique is presented in detail, together with applications of the Newton-Raphson procedure enhanced with line search and arc length methods. In the final chapter 8, the authors present a nonlinear finite element FORTRAN program for the solution of finite deformation problems for neo-Hookean hyperelastic compressible and incompressible materials and discuss the key subroutines of the program.

The book contains 8 chapters and an appendix. In the first chapter, the authors introduce basic computational concepts of nonlinear mechanics, e.g. nonlinear strain measures, linearization, and directional derivatives. An example of a nonlinear truss is solved by using a short FORTRAN program which is a prototype of the main finite element program developed further in the book.

After a chapter dealing with mathematical preliminaries, (vector and tensor algebra, linearization of directional derivatives and some elements of tensor analysis), the authors present the kinematics of finite deformations. The central concept is the deformation gradient tensor, which describes the relationship between vectors from undeformed and deformed body configurations. Lagrangian and Eulerian descriptions of motion are discussed together with the linearized kinematics which is used later in the Newton-Raphson solution process.

Chapter 4 introduces stress and equilibrium concepts for a deformable body undergoing a finite motion. The stress is first defined in the current configuration in a standard way, leading to the Cauchy stress tensor used in linear analysis. In contrast to linear small displacement theory, the stress quantities can be defined in the initial body configuration. This leads to Piola-Kirchhoff stress tensors and to alternative equilibrium equations. Finally, the objectivity of several stress rates is discussed.

In chapter 5 the authors write out the constitutive equations of hyperelastic materials. Isotropic hyperelasticity, both in Lagrangian and in Eulerian descriptions, and for compressible and incompressible materials, is considered. Then the equations are extended to a general description in principal directions for the cases of plane strain, plane stress, and for the uniaxial behaviour. Chapter 6 deals with the linearized equilibrium equations. In order to develop a Newton-Raphson solution procedure, the virtual work principle is linearized before the discretization. A large part of this chapter is devoted to incompressible hyperelastic materials and to the development of a mean dilatation procedure by using the Hu-Washizu variational principle.

All previous chapters provide a foundation for the discretized and linearized equilibrium equations considered in chapter 7. The linearization leads to finite element formulation, whereas the discretization of the linearized equilibrium equations introduces the tangent matrix. The mean discrete dilatation technique is presented in detail, together with applications of the Newton-Raphson procedure enhanced with line search and arc length methods. In the final chapter 8, the authors present a nonlinear finite element FORTRAN program for the solution of finite deformation problems for neo-Hookean hyperelastic compressible and incompressible materials and discuss the key subroutines of the program.

Reviewer: O.Simionescu (Bucureşti)

##### MSC:

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

74S05 | Finite element methods applied to problems in solid mechanics |

74B20 | Nonlinear elasticity |

74-04 | Software, source code, etc. for problems pertaining to mechanics of deformable solids |