Much of modern engineering owes its existence to the finite element method (FEM), which makes the impossible possible while also allowing us to build bigger, safer, and cheaper than ever before. Yet, as with any other tool in the engineer’s toolbox, users must have a detailed understanding in order to make full use of its capabilities. Intended for advanced undergraduate and graduate students, I expected a book as rigorous as the discipline itself. Sadly, that was not the case.
Like all first chapters, we start with an introduction to FEM and how it fits into the field of computational mechanics. The authors identify a number of limitations, a few commercial implementations, and some ancillary software components. Chapter 2 provides a very brief introduction to MATLAB. In chapter 3, the authors detail the direct stiffness method (DSM) and cover discretization error, the formulation of displacement and force matrices, the structural stiffness matrix, changes in coordinate systems, nodal reaction forces, and the computation of internal forces and stresses. Key concepts are explained using a variety of 2D planar trusses.
Chapter 4, building on the knowledge gained in the prior two chapters, is a step-by-step guide to using a finite element solver in MATLAB for 2D truss problems. Having covered the full process from a twenty thousand foot perspective, the subsequent chapters focus on individual aspects that make up the domain space. Chapter 5 looks into the design of virtual domains. It explains how to deal with different scales of analysis and simplifications of complex structures, and ends with an overview of using various software packages for designing and building a virtual model. The next chapter details mesh generation. Starting with on overview of various mesh elements and the impact that meshing choices have on the final converged solution, the authors explain how to deal with areas of stress concentration and crossing material boundaries.
In chapter 7, the reader is taken step by step through the derivation of some basic element formulations. Starting with 1D elements, and through to 3D bodies, the authors describe the use of Lagrangian polynomials as shape functions to derive a strain-displacement matrix first before building up a stiffness matrix. Jacobian matrices are discussed for the higher-dimensional examples, while both triangular and quadrilateral elements are covered for 2D cases. Chapter 8 is all about boundary conditions. Distinguishing between loads and Dirichlet, Neumann, mixed, and periodic boundary conditions, a substantial portion of the chapter is spent describing an author-provided, MATLAB-based periodic boundary condition generator for use with Abaqus finite element software. The chapter ends with explanations on how to apply periodic boundary conditions for nonperiodic meshes.
One of the longer chapters in the book (chapter 9) is on measures of stress and strain, highlighting the wide variety of mathematical definitions that exist for both once multidimensional bodies are considered. Starting with the kinematics of deformation, the authors cover deformation, rotation, stretch, and velocity gradients. This is followed by the mathematical definitions of a substantial number of strain and deformation tensors. A similar approach is applied to derive seven commonly used stress tensor definitions. The chapter then moves on to more practical formulations of stress and identifies principle stresses, both normal and shear, before finishing off with descriptions of hydrostatic stress and the von Mises stress.
Chapter 10 covers the use of constitutive models for describing how a material will respond to stress. Starting with linear elasticity models, isotropic, orthotropic, and anisotropic models are formulated. The section on plasticity models derives the constitutive mathematics of elastoplasticity for perfect plasticity, as well as incorporating isotropic and kinematic hardening that can happen to the material at the yield stress. Viscoelasticity is covered next. Both creep and the relaxation responses are described before detailing different models that can be used to model a viscoelastic material response, including temperature dependence. An even larger number of models are described for dealing with nonlinear elasticity. Finally, the chapter delves into user-defined material models, including why you may need to do so and how to go about making your first attempt inside in Abaqus.
The final chapter is a very brief look at the future of FEM and areas for improvement that can be undertaken as research projects. Overall, the book covers a substantial amount of information. There are also numerous problems at the end of each chapter to help cement the knowledge in place. Yes, some topics could have been better explained, but were it not for one fatal flaw I would be recommending this book. The problem is that this book is flooded with typographical errors: technical errors in the text, bugs in source code examples, bad grammar, spelling mistakes, mislabeled graphs, even incorrectly rotated axis labels. Even my cursory review picked up over 100 issues, which as it stands will severely impede students’ ability to grasp and understand the material. Although it may be unfair to say that this book was rushed to print without adequate technical and editorial review, at the very least it may be best to wait for the second edition.
