Already in 1967 the first author, one of the founders of the finite element (FE) method, had published withY. K. Cheung the first book about this subject, with less than 300 pages [for review seeZbl 0189.24902]. Due to rapidly growing applications of this method to more and more complex problems (non-elastic material behaviour, large deformations etc.), the subsequent editions took into account this development which led – coauthored by R. L. Taylor – to the three contiguous volume format of the 5th edition (2000) [see for reviewZbl 0991.74002;Zbl 0991.74003;Zbl 0991.74004] with nearly 1500 pages. In the present 6th edition both authors intend, to treat the whole work as an assembly of three separate ones, capable of being used independently of each other. According to the preface, the (present) first volume of the 6th edition is to be renamed “The Finite Element Method: Ist Basis and Fundamentals”. Both authors contributed essentially to the FE method which enables the user to solve complex problems of engineering mechanics, arising likewise in technical practice and research.
The book is organized as follows: In chapter 1 the equations of classical elasticity are presented in its strong and weak (variational) forms, and in chapter 2 the FE displacement approximation is explained, including numerical integration. Further, transient problems are formulated via the additional discretization of time, and processes with material and geometric nonlinearities are discussed. Chapter 3 treats solution strategies for a system of nonlinear algebraic equations (Newton methods, line search procedures). Chapter 4 deals with the more complex material behaviour (linear viscoelasticity, classical plasticity, explicit and implicit methods, isotropic and generalized plasticity resp., nonlinear creep models, viscoplasticity, creep of metals, soil mechanics, no-tension materials). Chapter 5 is devoted to the theory of a hyperelastic material under finite deformation and to the corresponding FE approximations which are based on displacement and mixed variational formulations. In addition, the authors consider pressure loads and state that the tangent load-stiffness matrix is unsymmetric in general, with the exception of a constant pressure loading which acts on a closed surface. This statement, however, is incorrect: If there exists a pressure potential, the pressure load operator is conservative and, as a consequence, the tangent stiffness is symmetric. In chapter 6 the authors present, using a rate form, some more general methods for the modelling of elastic, viscoelastic and elastoplastic behaviours. Chapter 7 introduces into the modelling of contact of deformable bodies under frictional conditions using the FE discretization. Chapter 8 contains the elements of analytical mechanics and multibody systems, and chapter 9 (written by Nenat Bicanic) is concerned with the so-called discrete element method for the simulation of dynamic behaviour of systems of rigid, pseudorigid or deformable separated or contacting bodies.
In the remaining part of the book mainly structural mechanics problems are discussed. Chapter 10 contains the linear rod and beam equations (Bernoulli, Timoshenko), including FE solutions. Further in chapter 11 the Kirchhoff plate equations are established and various discretization procedures are explained. In the following chapter 12 thick plates (Reissner, Mindlin) are studied including mixed formulation, collocation constraints and inelastic behaviour. According to the historical development, thin shells are considered at first as an assembly of flat elements (chapter 13). In chapter 14 the special case of unsymmetric shells is investigated, using straight and curved elements respectively. The more general case of thick shells is treated in chapter 15. The finite strip method is given a special attention in chapter 16. Large displacement problems of elastic beams, plates and shells are discussed in chapter 17, together with the stability of equilibrium. Chapter 18 is contributed by B. A. Schrefler; it deals with multiscale material modelling for the evaluation of effective properties of materials with internal structure, for instance composites. In the final chapter 19 and in the appendix the authors present additional and useful hints for effective FE procedures, for the solution of nonlinear and eigenproblems, for the use of the isoparametric concept and for the invariants of second-order tensors.
Summarizing, it can be stated that the book, written by two well-known pioneers of the FE method, represents the state of the art of this method. It is suited mainly for graduates, but also for academic engineers and applied mathematicians, and can be used for self-study as well. The list of references contains more than 1000 authors.

Reviewer: Hans Bufler (Gräfelfing)