INTRODUCTION

The equilibrium equations derived in the previous chapter are written in terms of the stresses inside the body. These stresses result from the deformation of the material, and it is now necessary to express them in terms of some measure of this deformation such as, for instance, the strain. These relationships, known as constitutive equations, obviously depend on the type of material under consideration and may be dependent on or independent of time. For example, the classical small strain linear elasticity equations involving Young modulus and Poisson ratio are time-independent, whereas viscous fluids are clearly entirely dependent on strain rate.

Generally, constitutive equations must satisfy certain physical principles. For example, the equations must obviously be objective, that is, frame-invariant. In this chapter the constitutive equations will be established in the context of a hyperelastic material, whereby stresses are derived from a stored elastic energy function. Although there are a number of alternative material descriptions that could be introduced, hyperelasticity is a particularly convenient constitutive equation given its simplicity and it constitutes the basis for more complex material models such as elastoplasticity, viscoplasticity, and viscoelasticity.

HYPERELASTICITY

Materials for which the constitutive behavior is only a function of the current state of deformation are generally known as elastic. Under such conditions, any stress measure at a particle X is a function of the current deformation gradient F associated with that particle. Instead of using any of the alternative strain measures given in Chapter 4, the deformation gradient F, together with its conjugate first Piola–Kirchhoff stress measure P, will be retained in order to define the basic material relationships.

INTRODUCTION

This chapter considers the uniaxial (one-dimensional) large-displacement, largestrain, rate-independent elasto-plastic behavior applicable to structural analysis of pin-jointed trusses. The motivation is to expand and reinforce previous material and to introduce some topics that will reappear later when elasto-plastic behavior of continua is considered. For example, various nonlinear geometrical descriptors will be linearized providing further examples of the use of the directional derivative.

Formulations start with the kinematic description of the motion in three dimensional space of a truss member (axial rod) that undergoes large displacements and rotations leading to large or small strain that causes stress which may reach the limit or yield stress of the material. For simplicity, it will be assumed that the strain in the truss member is uniform. Consequently, the fundamental measure of deformation in the axial rod is the stretch λ = l/L, which is the ratio of the deformed length to the undeformed length, see Figure 3.1.

The internal forces in the truss are easily determined from simple strength of- material considerations involving the true (or Cauchy) stress, σ, defined for a truss as the internal axial force, T, divided by the deformed cross-sectional area, a. However, for large deformation the elasto-plastic behavior is best characterized using an alternative stress known as the Kirchhoff stress, τ, defined as συ/V, see Figure 3.1. In preparation for Chapter 6 it will be shown how the Kirchhoff stress can be derived from a hyperelastic energy function involving the natural logarithm of the stretch.

INTRODUCTION

We have seen in the previous chapters that the solution to the nonlinear equilibrium equations is basically achieved using the Newton–Raphson iterative method. In addition, in a finite element context it is advisable to apply the external forces in a series of increments. This has the advantage of enhancing the converging properties of the solution and, where appropriate, provides possible intermediate equilibrium states. Moreover, for path dependent materials such as those exhibiting plasticity, these intermediate states represent the loading path which needs to be accurately followed. Furthermore, it is clear that the two fundamental quantities that facilitate the Newton–Raphson solution are the evaluation of the residual force and the tangent matrix. In this chapter we shall describe the FORTRAN implementation of the solution procedure in the teaching program FLagSHyP, (Finite element Large Strain Hyperelasto-plastic Program).

It is expected that the reader already has some familiarity with the computer implementation of the finite element method in the linear context. Consequently, this chapter will emphasize those aspects of the implementation that are of particular relevance in the nonlinear finite deformation context. In this respect, it is essential to understand two crucial routines. Firstly, the master routine that controls the overall organization of the program and, secondly, the subroutine elemtk. This latter routine computes the equivalent nodal forces due to internal stress and the main components of the tangent stiffness matrix. Consequently, it provides a vehicle for examining those aspects of the computation that are particular to finite deformation analysis.

INTRODUCTION

In order to make this book sufficiently self-contained, it is necessary to include this chapter dealing with the mathematical tools that are needed to achieve a complete understanding of the topics discussed in the remaining chapters. Vector and tensor algebra is discussed, as is the important concept of the general directional derivative associated with the linearization of various nonlinear quantities that will appear throughout the book.

Readers, especially with engineering backgrounds, are often tempted to skip these mathematical preliminaries and move on directly to the main text. This temptation need not be resisted, as most readers will be able to follow most of the concepts presented even when they are unable to understand the details of the accompanying mathematical derivations. It is only when one needs to understand such derivations that this chapter may need to be consulted in detail. In this way, this chapter should, perhaps, be approached like an instruction manual, only to be referred to when absolutely necessary. The subjects have been presented without the excessive rigors of mathematical language and with a number of examples that should make the text more bearable.

VECTOR AND TENSOR ALGEBRA

Most quantities used in nonlinear continuum mechanics can only be described in terms of vectors or tensors. The purpose of this section, however, is not so much to give a rigorous mathematical description of tensor algebra, which can be found elsewhere, but to introduce some basic concepts and notation that will be used throughout the book.

NONLINEAR COMPUTATIONAL MECHANICS

Two sources of nonlinearity exist in the analysis of solid continua, namely, material and geometric nonlinearity. The former occurs when, for whatever reason, the stress strain behavior given by the constitutive relation is nonlinear, whereas the latter is important when changes in geometry, however large or small, have a significant effect on the load deformation behavior. Material nonlinearity can be considered to encompass contact friction, whereas geometric nonlinearity includes deformation-dependent boundary conditions and loading.

Despite the obvious success of the assumption of linearity in engineering analysis, it is equally obvious that many situations demand consideration of nonlinear behavior. For example, ultimate load analysis of structures involves material nonlinearity and perhaps geometric nonlinearity, and any metal-forming analysis such as forging or crash-worthiness must include both aspects of nonlinearity. Structural instability is inherently a geometric nonlinear phenomenon, as is the behavior of tension structures. Indeed the mechanical behavior of the human body itself, say in impact analysis, involves both types of nonlinearity. Nonlinear and linear continuum mechanics deal with the same subjects such as kinematics, stress and equilibrium, and constitutive behavior. But in the linear case an assumption is made that the deformation is sufficiently small to enable the effect of changes in the geometrical configuration of the solid to be ignored, whereas in the nonlinear case the magnitude of the deformation is unrestricted.

Practical stress analysis of solids and structures is unlikely to be served by classical methods, and currently numerical analysis, predominately in the form of the finite element method, is the only route by which the behavior of a complex component subject to complex loading can be successfully simulated.

In the preceding chapter, a nonlinear finite element formulation for the large-deformation analysis was presented. This formulation, which is consistent with the motion description used in the theory of continuum mechanics and can be used to correctly describe an arbitrary rigid-body motion, leads to a constant mass matrix and nonlinear vector of elastic forces. The formulation imposes no restrictions on the amount of rotation or deformation within the element, except for the restriction imposed by the order of the interpolating polynomials used. In large-deformation problems, in general, the shape of deformation of the bodies can be complex and this, in turn, necessitates the use of a large number of finite element nodal coordinates in order to be able to correctly capture the geometry of deformation. Therefore, in the analysis of the large deformation problem using the absolute nodal coordinate formulation discussed in the preceding chapter, one simply selects an adequate number of finite elements and formulates the equations of motion in terms of the element nodal coordinates. There is no need to introduce another reference frame or be concerned with the use of coordinate reduction techniques. The results published in the literature on the absolute nodal coordinate formulation demonstrated that this formulation can be used in modeling very large deformations with relatively small number of finite elements compared to other existing nonlinear finite element formulations.

The kinematic and force equations developed in the preceding two chapters are general and applicable to all types of materials. The mechanics of solids and fluids is governed by the same equations, which do not distinguish between different materials. The definitions of the strain and stress tensors, however, are not sufficient for describing the behavior of continuous bodies. The force–displacement relationship or equivalently the stress–strain relationship is required in order to be able to distinguish between different materials and solve the equilibrium equations. The continuum displacements depend on the applied forces, and the force–displacement relationship depends on the material of the continuum. To complete the specification of the mechanical properties of a material, one needs additional set of equations called the constitutive equations, which serve to distinguish one material from another. The form of the constitutive equations of a material should not be altered in the case of a pure rigid-body motion. These equations, therefore, must be objective, and should not lead to change in the work and energy of the stresses under an arbitrary rigid-body motion. Using the constitutive equations, the partial differential equations of equilibrium obtained in the preceding chapter can be expressed in terms of the strains. Using the strain–displacement relationships, these equilibrium equations can be expressed in terms of displacements or position coordinates and their time and spatial derivatives.

Matrix, vector, and tensor algebras are often used in the theory of continuum mechanics in order to have a simpler and more tractable presentation of the subject. In this chapter, the mathematical preliminaries required to understand the matrix, vector, and tensor operations used repeatedly in this book are presented. Principles of mechanics and approximation methods that represent the basis for the formulation of the kinematic and dynamic equations developed in this book are also reviewed in this chapter. In the first two sections of this chapter, matrix and vector notations are introduced and some of their important identities are presented. Some of the vector and matrix results are presented without proofs because it is assumed that the reader has some familiarity with matrix and vector notations. In Section 3, the summation convention, which is widely used in continuum mechanics texts, is introduced. This introduction is made despite the fact that the summation convention is rarely used in this book. Tensor notations, on the other hand, are frequently used in this book and, for this reason, tensors are discussed in Section 4. In Section 5, the polar decomposition theorem, which is fundamental in continuum mechanics, is presented. This theorem states that any nonsingular square matrix can be decomposed as the product of an orthogonal matrix and a symmetric matrix. Other matrix decompositions that are used in computational mechanics are also discussed. In Section 6, D'Alembert's principle is introduced, while Section 7 discusses the virtual work principle.

In this chapter, the general kinematic equations for the continuum are developed. The kinematic analysis presented in this chapter is purely geometric and does not involve any force analysis. The continuum is assumed to undergo an arbitrary displacement and no simplifying assumptions are made except when special cases are discussed. Recall that in the special case of an unconstrained three-dimensional rigid-body motion, six independent coordinates are required in order to describe arbitrary rigid-body translation and rotation displacements. The general displacement of an infinitesimal material volume on a deformable body, on the other hand, can be described in terms of twelve independent variables; three translation parameters, three rigid-body rotation parameters, and six deformation parameters. One can visualize these modes of displacements by considering a cube that may undergo an arbitrary displacement. The cube can be translated in three independent orthogonal directions (translation degrees of freedom), it can be rotated as a rigid body about three orthogonal axes, and it can experience six independent modes of deformation. These deformation modes are elongations or contractions in three different directions and three shear deformation modes. It is shown in this chapter that the rotations and the deformations can be completely described using the matrix of the position vector gradients, which in general has nine independent elements. This fact can be mathematically proven using the polar decomposition theorem discussed in the preceding chapter. The deformation at the material points on the body can be described in terms of six independent strain components.

The analysis of plastic deformation is important in many engineering applications including crashworthiness, impact analysis, manufacturing problems, among many others. When materials undergo plastic deformations, permanent strains are developed when the load is removed. Many materials exhibit elastic–plastic behaviors, that is, the material exhibits elastic behavior up to a certain stress limit called the yield strength after which plastic deformation occurs. If the stress of elastic-plastic materials depends on the strain rate, one has a rate-dependent material, otherwise the material is called rate independent. In the classical plasticity analysis of solids, a nonunique stress–strain relationship that is independent of the rate of loading but does depend on the loading sequence is used (Zienkiewicz and Taylor, 2000). In rate-dependent plasticity, on the other hand, the stress–strain relationship depends on the rate of the loading.

The yield strength of elastic–plastic materials can increase after the initial yield. This phenomenon is known as strain hardening. In the theory of plasticity, there are two types of strain hardening, isotropic and kinematic hardening. In the case of isotropic hardening, the yield strength changes as the result of the plastic deformation. In the case of kinematic hardening, on the other hand, the center of the yield surface experiences a motion in the direction of the plastic flow. The kinematic hardening behavior is closely related to a phenomenon known as the Bauschinger effect, which is the result of a reduction in the compressive yield strength following an initial tensile yield.

In the preceding chapters, the general nonlinear continuum mechanics theory was presented. In order to make use of this theory in many practical applications, a finite dimensional model must be developed. In this model, the partial differential equations of equilibrium are written using approximation methods as a finite set of ordinary differential equations. One of the most popular approximation methods that can be used to achieve this goal is the finite element method. In this method, the spatial domain of the body is divided into small regions called elements. Each element has a set of nodes, called nodal points that are used to connect this element with other elements used in the discretization of the body. The displacement of the material points of an element is approximated using a set of shape functions and the displacements of the nodes and possibly their derivatives with respect to the spatial coordinates. In this case, the dimension of the problem depends on the number of nodes and number and type of the nodal coordinates used.

In the literature, there are many finite element formulations that are developed for the deformation analysis of mechanical, aerospace, structural, and biological systems. Some of these formulations are developed for small-deformation and small-rotation linear problems, some for large-deformation and large-rotation nonlinear analysis, and the others for large-rotation and small-deformation nonlinear problems. Several numerical solution procedures and computational algorithms are also proposed for solving the resulting system of finite element differential equations.

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.

In the theory of continuum mechanics, stresses are used as measures of the forces and pressures. As in the case of strains, different definitions can be used for the stresses; some of these definitions are associated with the reference configuration, whereas the others are associated with the current deformed configuration. The effect of the forces on the body dynamics can only be taken into consideration by using both stresses and strains. These stress and strain components must be defined in the same coordinate system in order to have a consistent formulation. In this chapter, several stress measures are introduced and the relationship between them is discussed. The Cauchy stress formula is presented and used to develop the partial differential equations of equilibrium of the continuous body. The equations of equilibrium are used to develop an expression for the virtual work of the stress forces expressed in terms of the stress and strain components. The objectivity of the stress rate and the energy balance equations are also among the topics that will be discussed in this chapter.

EQUILIBRIUM OF FORCES

In this section, the equilibrium of forces acting in the interior of a continuous body is considered. Let P be a point on the surface of the body, n be a unit vector directed along the outward normal to the surface at P, and be the area of an element of the surface that contains P in the current configuration.