Nonlinear continuum mechanics is one of the fundamental subjects that form the foundation of modern computational mechanics. The study of the motion and behavior of materials under different loading conditions requires understanding of basic, general, and nonlinear kinematic and dynamic relationships that are covered in continuum mechanics courses. The finite element method, on the other hand, has emerged as a powerful tool for solving many problems in engineering and physics. The finite element method became a popular and widely used computational approach because of its versatility and generality in solving large-scale and complex physics and engineering problems. Nonetheless, the success of using the continuum-mechanics-based finite element method in the analysis of the motion of bodies that experience general displacements, including arbitrary large rotations, has been limited. The solution to this problem requires resorting to some of the basic concepts in continuum mechanics and putting the emphasis on developing sound formulations that satisfy the principles of mechanics. Some researchers, however, have tried to solve fundamental formulation problems using numerical techniques that lead to approximations. Although numerical methods are an integral part of modern computational algorithms and can be effectively used in some applications to obtain efficient and accurate solutions, it is the opinion of many researchers that numerical methods should only be used as a last resort to fix formulation problems. Sound formulations must be first developed and tested to make sure that these formulations satisfy the basic principles of mechanics. The equations that result from the use of the analytically correct formulations can then be solved using numerical methods.

This book is focused on presenting the nonlinear theory of continuum mechanics and demonstrating its use in developing nonlinear computer formulations that can be used in the large displacement dynamic analysis. To this end, the basic concepts used in continuum mechanics are first presented and then used to develop nonlinear general finite element formulations for the large displacement analysis. Two nonlinear finite element dynamic formulations will be considered in this book. The first is a general large-deformation finite element formulation, whereas the second is a formulation that can be used efficiently to solve small-deformation problems that characterize very and moderately stiff structures. In this latter case, an elaborate method for eliminating the unnecessary degrees of freedom must be used in order to be able to efficiently obtain a numerical solution. An attempt has been made to present the materials in a clear and systematic manner with the assumption that the reader has only basic knowledge in matrix and vector algebra as well as basic knowledge of dynamics. The book is designed for a course at the senior undergraduate and first-year graduate level. It can also be used as a reference for researchers and practicing engineers and scientists who are working in the areas of computational mechanics, biomechanics, computational biology, multibody system dynamics, and other fields of science and engineering that are based on the general continuum mechanics theory.

In Chapter 1, matrix, vector, and tensor notations are introduced. These notations will be repeatedly used in all chapters of the book, and therefore, it is necessary that the reader reviews this chapter in order to be able to follow the presentation in subsequent chapters. The polar decomposition theorem, which is fundamental in continuum and computational mechanics, is also presented in this chapter. D'Alembert's principle and the principle of virtual work can be used to systematically derive the equations of motion of physical systems. These two important principles are discussed and the relationship between them is explained. The use of a finite dimensional model to describe the continuum motion is also discussed and the procedure for developing the discrete equations of motion is outlined. The principles of momentum and principle of work and energy are presented, and the problems associated with some of the finite element formulations that violate these analytical mechanics principles are discussed. Chapter 1 also provides a discussion on the definitions of the gradient vectors that are used in continuum mechanics to define the strain components.

In Chapter 2, the general kinematic displacement equations of a continuum are developed and used to define the strain components. The Green–Lagrange strains and the Almansi or Eulerian strains are introduced. The Green–Lagrange strains are defined in the reference configuration, whereas the Almansi or Eulerian strains are defined in the current deformed configuration. The relationships between these strain components are established and used to shed light on the physical meaning of the strain components. Other deformation measures as well as the velocity and acceleration equations are also defined in this chapter. The important issue of objectivity that must be considered when large deformations and inelastic formulations are used is discussed. The equations that govern the change of volume and area, the conservation of mass, and examples of deformation modes are also presented in this chapter.

Forces and stresses are discussed in Chapter 3. Equilibrium of forces acting on an infinitesimal material element is used to define the Cauchy stresses, which are used to develop the partial differential equations of equilibrium. The transformation of the stress components and the symmetry of the Cauchy stress tensor are among the topics discussed in this chapter. The virtual work of the forces due to the change of the shape of the continuum is defined. The deviatoric stresses, stress objectivity, and energy balance equations are also discussed in Chapter 3.

The definition of the strain and stress components is not sufficient to describe the motion of a continuum. One must define the relationship between the stresses and strains using the constitutive equations that are discussed in Chapter 4. In Chapter 4, the generalized Hooke's law is introduced and the assumptions used in the definition of homogeneous isotropic materials are outlined. The principal strain invariants and special large-deformation material models are discussed. The linear and nonlinear viscoelastic material behavior is also discussed in Chapter 4.

Nonlinear finite element formulations are discussed in Chapters 5 and 6. Two formulations are discussed in these two chapters. The first is a large-deformation finite element formulation, which is discussed in Chapter 5. This formulation, called the absolute nodal coordinate formulation (ANCF), is based on a continuum mechanics theory and employs position gradients as coordinates. It leads to a unique displacement and rotation fields and imposes no restrictions on the amount of rotation or deformation within the finite element. The absolute nodal coordinate formulation has some unique features that distinguish it from other existing large-deformation finite element formulations: it leads to a constant mass matrix; it leads to zero centrifugal and Coriolis forces; it automatically satisfies the principles of mechanics; it correctly describes an arbitrary rigid-body motion including finite rotations; and it can be used to develop several beam, plate, and shell elements that relax many of the assumptions used in classical theorems. When using ANCF finite elements, no distinction is made between plate and shell elements since shell geometry can be systematically obtained using the nodal coordinates in the reference configuration.

Clearly, large-deformation finite element formulations can also be used to solve small-deformation problems. However, it is not recommended to use a large-deformation finite element formulation to solve a small-deformation problem. Large-deformation formulations do not exploit some particular features of small-deformation problems, and therefore, such formulations can be very inefficient in the solution of stiff and moderately stiff systems. The development of an efficient small-deformation finite element formulation that correctly describes an arbitrary rigid-body motion requires the use of more elaborate techniques in order to define a local linear problem without compromising the ability of the method to describe large-displacement, small-deformation behavior. The finite element floating frame of reference (FFR) formulation, widely used in the analysis of small deformations, is discussed in Chapter 6. This formulation allows eliminating high-frequency modes that do not have a significant effect on the solution, thereby leading to a lower-dimension dynamic model that can be efficiently solved using numerical and computer methods.

Although finite element (FE) formulations are based on polynomial representations, the polynomial-based geometric representation used in computer-aided design (CAD) methods cannot be converted exactly to the kinematic description used in many existing FE formulations. For this reason, converting a CAD model to an FE mesh can be costly and time-consuming. CAD software systems use computational geometry methods such as B-spline and Non-Uniform Rational B-Splines (NURBS). These methods can describe accurately complex geometry. The relationship between these CAD geometry methods and the FE formulations presented in this book are discussed in Chapter 7. As explained in Chapter 7, modeling modern engineering and physics systems requires the successful integration of computer-aided design and analysis (I-CAD-A) by developing an efficient interface between CAD systems and analysis tools or by developing a new mechanics based CAD/analysis system.

In many engineering applications, plastic deformations occur due to excessive forces and impact as well as thermal loads. Several plasticity formulations are presented in Chapter 8. First, a one-dimensional theory is used in order to discuss the main concepts and solution procedures used in the plasticity analysis. The theory is then generalized to the three-dimensional analysis for the case of small strains. Large strain nonlinear plasticity formulations as well as the J₂ flow theory are among the topics discussed in Chapter 8.

I would like to thank many students and colleagues with whom I worked for several years on the subject of flexible body dynamics. I was fortunate to collaborate with excellent students and colleagues who educated me in this important field of computational mechanics. In particular, I would like to thank my doctorate students, Bassam Hussein, Luis Maqueda, Mohil Patel, Brian Tinsley, and Liang Wang, who provided solutions for several of examples and figures presented in several chapters of the book. I would also like to thank my family for their help, patience, and understanding during the time of preparing this book.

Ahmed A. Shabana
Chicago, IL
2016