Nonlinear finite elements for continua and structures.

*(English)*Zbl 0959.74001
Chichester: Wiley. xvi, 650 p. (2000).

This remarkable and comprehensive book represents the state of knowledge of nonlinear solid continua and structures, and describes the principles of construction of efficient finite element (FE) procedures. More than 200 references are listed, but the contributions of the authors to the FE technology are taken into account preferably. The treatise is intended for final year undergraduates, postgraduates, academic engineers and applied mathematicians working with sophisticated FE software, and for users of nonlinear FE programs. In addition, it is well suited for self-study.

The book is organized as follows: Chapter 1 contains a short introduction including the history of nonlinear finite elements. Chapter 2 is restricted to one-dimensional models. Three formulations are explained: total Lagrangian, updated Lagrangian, and Eulerian one. Strong and weak forms of the basic equations are given, followed by simple FE discretizations. Chapter 3 is devoted to the fundamentals of nonlinear continuum mechanics in Eulerian and Lagrangian description, using various stress and (rate of) deformation measures, including corotational formulations. Besides of the conservation equations, the authors introduce here objective rates (Jaumann, Truesdell, Green-Naghdi) needed in constitutive relations. In chapter 4 the use of Lagrangian meshes (moving with the material) is explained, starting from the weak formulation (principles of virtual power and virtual work) and introducing FE approximations as trial functions for both the updated and the total Lagrangian formulations. The first one is given with respect to the Eulerian coordinates, the second with respect to the Lagrangian ones and with restriction to hyperelastic materials. Chapter 5 deals with constitutive equations, with particular emphasis on material models that are relevant to the treatment of material nonlinearities and large deformations. Hypoelastic material laws are considered as special cases of general elastic-plastic constitutive relations, and some hyperelastic laws are discussed. The authors also consider rate-independent and rate-dependent plastic deformations, and viscoelastic behavior. An implementation in FE code is discussed, together with required mathematical backgrounds.

Chapter 6 treats solution procedures for nonlinear FE discretizations (explicit and implicit methods for transient and static problems, Newton iterations, methods for treating constraints, material stiffness, geometric stiffness and load stiffness). The statement that the load stiffness matrix is generally nonsymmetric in the presence of follower forces, however, is not correct for natural pressure fields due to the existence of load potential. The (geometric) stability of the nonlinear solutions to the discretized system is investigated by means of a linear approximation. Besides, the authors give estimates for numerical stability of time integration procedures, and, in addition, discuss the phenomenon of material instability. Chapter 7 is devoted to arbitrary Lagrangian-Eulerian formulations. It provides tools for Eulerian analysis (with fixed elements), and describes relevant hybrid techniques including stabilization procedures. An elastic-plastic wave propagation problem is presented as an example. In chapter 8 some hints are given on the design of elements for large scale calculations and for constrained media problems such that the elements have optimal properties (consistency, stability, completeness, no locking and no spurious modes). Chapter 9 deals with beams and shells, starting from continuum-based element formulations and introducing the structural assumptions (Kirchhoff, Timoshenko etc.) a posteriori. In this way, the general procedure needs only be modified. Finally, chapter 10 describes contact-impact phenomena governed by inequalities, where special attention is focussed on nonsmooth solutions. All FE procedures are carried out here on Lagrangian meshes, and Coulomb’s friction is taken into account.

Many figures, examples and exercises support the understanding of this comprehensive contents, and all relevant equations are summarized in clear boxes. The reader can find numerous hints on an appropriate choice of the FE approach, and is informed about difficulties inherent in nonlinear analysis.

The book is organized as follows: Chapter 1 contains a short introduction including the history of nonlinear finite elements. Chapter 2 is restricted to one-dimensional models. Three formulations are explained: total Lagrangian, updated Lagrangian, and Eulerian one. Strong and weak forms of the basic equations are given, followed by simple FE discretizations. Chapter 3 is devoted to the fundamentals of nonlinear continuum mechanics in Eulerian and Lagrangian description, using various stress and (rate of) deformation measures, including corotational formulations. Besides of the conservation equations, the authors introduce here objective rates (Jaumann, Truesdell, Green-Naghdi) needed in constitutive relations. In chapter 4 the use of Lagrangian meshes (moving with the material) is explained, starting from the weak formulation (principles of virtual power and virtual work) and introducing FE approximations as trial functions for both the updated and the total Lagrangian formulations. The first one is given with respect to the Eulerian coordinates, the second with respect to the Lagrangian ones and with restriction to hyperelastic materials. Chapter 5 deals with constitutive equations, with particular emphasis on material models that are relevant to the treatment of material nonlinearities and large deformations. Hypoelastic material laws are considered as special cases of general elastic-plastic constitutive relations, and some hyperelastic laws are discussed. The authors also consider rate-independent and rate-dependent plastic deformations, and viscoelastic behavior. An implementation in FE code is discussed, together with required mathematical backgrounds.

Chapter 6 treats solution procedures for nonlinear FE discretizations (explicit and implicit methods for transient and static problems, Newton iterations, methods for treating constraints, material stiffness, geometric stiffness and load stiffness). The statement that the load stiffness matrix is generally nonsymmetric in the presence of follower forces, however, is not correct for natural pressure fields due to the existence of load potential. The (geometric) stability of the nonlinear solutions to the discretized system is investigated by means of a linear approximation. Besides, the authors give estimates for numerical stability of time integration procedures, and, in addition, discuss the phenomenon of material instability. Chapter 7 is devoted to arbitrary Lagrangian-Eulerian formulations. It provides tools for Eulerian analysis (with fixed elements), and describes relevant hybrid techniques including stabilization procedures. An elastic-plastic wave propagation problem is presented as an example. In chapter 8 some hints are given on the design of elements for large scale calculations and for constrained media problems such that the elements have optimal properties (consistency, stability, completeness, no locking and no spurious modes). Chapter 9 deals with beams and shells, starting from continuum-based element formulations and introducing the structural assumptions (Kirchhoff, Timoshenko etc.) a posteriori. In this way, the general procedure needs only be modified. Finally, chapter 10 describes contact-impact phenomena governed by inequalities, where special attention is focussed on nonsmooth solutions. All FE procedures are carried out here on Lagrangian meshes, and Coulomb’s friction is taken into account.

Many figures, examples and exercises support the understanding of this comprehensive contents, and all relevant equations are summarized in clear boxes. The reader can find numerous hints on an appropriate choice of the FE approach, and is informed about difficulties inherent in nonlinear analysis.

Reviewer: Hans Bufler (GrĂ¤felfing)

##### 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 |

74Cxx | Plastic materials, materials of stress-rate and internal-variable type |

74Kxx | Thin bodies, structures |

74M15 | Contact in solid mechanics |

74M20 | Impact in solid mechanics |