geometrical con¬guration 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

1

2 INTRODUCTION

form of the ¬nite element method, is the only route by which the behav-

ior of a complex component subject to complex loading can be successfully

simulated. The study of the numerical analysis of nonlinear continua using

a computer is called nonlinear computational mechanics, which, when ap-

plied speci¬cally to the investigation of solid continua, comprises nonlinear

continuum mechanics together with the numerical schemes for solving the

resulting governing equations.

The ¬nite element method may be summarized as follows. It is a pro-

cedure whereby the continuum behavior described at an in¬nity of points

is approximated in terms of a ¬nite number of points, called nodes, located

at speci¬c points in the continuum. These nodes are used to de¬ne regions,

called ¬nite elements, over which both the geometry and the primary vari-

ables in the governing equations are approximated. For example, in the

stress analysis of a solid the ¬nite element could be a tetrahedra de¬ned by

four nodes and the primary variables the three displacements in the Carte-

sian directions. The governing equations describing the nonlinear behavior

of the solid are usually recast in a so-called weak integral form using, for

example, the principle of virtual work or the principle of stationary total

potential energy. The ¬nite element approximations are then introduced

into these integral equations, and a standard textbook manipulation yields

a ¬nite set of nonlinear algebraic equations in the primary variable. These

equations are then usually solved using the Newton“Raphson iterative tech-

nique.

The topic of this book can succinctly be stated as the exposition of the

nonlinear continuum mechanics necessary to develop the governing equations

in continuous and discrete form and the formulation of the Jacobian or

tangent matrix used in the Newton“Raphson solution of the resulting ¬nite

set of nonlinear algebraic equations.

1.2 SIMPLE EXAMPLES OF NONLINEAR STRUCTURAL

BEHAVIOR

It is often the case that nonlinear behavior concurs with one™s intuitive

expectation of the behavior and that it is linear analysis that can yield the

nonsensical result. The following simple examples illustrate this point and

provide a gentle introduction to some aspects of nonlinear behavior. These

two examples consider rigid materials, but the structures undergo ¬nite

displacements; consequently, they are classi¬ed as geometrically nonlinear

problems.

3

1.2 SIMPLE EXAMPLES OF NONLINEAR STRUCTURAL BEHAVIOR

L

FL/K

20

K

π/2

F

15

M=K

nonlinear

10

linear

F 5

F

0

Free body diagram 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(a) (b)

FIGURE 1.1 Simple cantilever.

1.2.1 CANTILEVER

Consider the weightless rigid bar-linear elastic torsion spring model of a

cantilever shown in Figure 1.1(a). Taking moments about the hinge gives

the equilibrium equation as,

FL cos θ = M (1.1)

If K is the torsional sti¬ness of the spring, then M = Kθ and we obtain the

following nonlinear relationship between F and θ,

FL θ

= (1.2)

K cos θ

If the angle θ ’ 0, then cos θ ’ 1, and the linear equilibrium equation

is recovered as,

K

F= θ (1.3)

L

The exact nonlinear equilibrium path is shown in Figure 1.1(b), where clearly

the nonlinear solution makes physical sense because θ < π/2.

1.2.2 COLUMN

The same bar-spring system is now positioned vertically (see Figure 1.2(a)),

and again moment equilibrium about the hinge gives,

PL θ

PL sin θ = M or = (1.4)

K sin θ

4 INTRODUCTION

PL/K

P P 5

4.5

nonlinear

4

3.5

3

linear

L

2.5

2

1.5

K 1

0.5

M=K bifurcation

0

P 0 1 2 3

’3 ’2 ’1

Free body diagram

(a) (b)

FIGURE 1.2 Simple column.

The above equilibrium equation can have two solutions: ¬rstly if θ = 0, then

sin θ = 0, M = 0, and equilibrium is satis¬ed; and secondly, if θ = 0, then

PL/K = θ/ sin θ. These two solutions are shown in Figure 1.2(b), where

the vertical axis is the equilibrium path for θ = 0 and the horseshoe-shaped

equilibrium path is the second solution. The intersection of the two solutions

is called a bifurcation point. Observe that for PL/K < 1 there is only one

solution, namely θ = 0 but for PL/K > 1 there are three solutions. For

instance, when PL/K ≈ 1.57, either θ = 0 or ±π/2.

For very small values of θ, sin θ ’ θ and (1.4) reduces to the linear (in

θ) equation,

(K ’ PL)θ = 0 (1.5)

Again there are two solutions: θ = 0 or PL/K = 1 for any value of θ, the

latter solution being the horizontal path shown in Figure 1.2(b). Equation

(1.5) is a typical linear stability analysis where P = K/L is the elastic

critical (buckling) load. Applied to a beam column such a geometrically

nonlinear analysis would yield the Euler buckling load. In a ¬nite element

context for, say, plates and shells this would result in an eigenvalue analysis,

the eigenvalues being the buckling loads and the eigenvectors being the

corresponding buckling modes.

Observe in these two cases that it is only by considering the ¬nite dis-

placement of the structures that a complete nonlinear solution has been

achieved.

5

1.3 NONLINEAR STRAIN MEASURES

L

A

a

l

FIGURE 1.3 One-dimensional strain.

1.3 NONLINEAR STRAIN MEASURES

In the examples presented in the previous section, the beam or column re-

mained rigid during the deformation. In general structural components or

continuum bodies will exhibit large strains when undergoing a geometrically

nonlinear deformation process. As an introduction to the di¬erent ways in

which these large strains can be measured we consider ¬rst a one-dimensional

truss element and a simple example involving this type of structural com-

ponent undergoing large displacements and large strains. We will then give

a brief introduction to the di¬culties involved in the de¬nition of correct

large strain measures in continuum situations.

1.3.1 ONE-DIMENSIONAL STRAIN MEASURES

Imagine that we have a truss member of initial length L and area A that is

stretched to a ¬nal length l and area a as shown in Figure 1.3. The simplest

possible quantity that we can use to measure the strain in the bar is the

so-called engineering strain µE de¬ned as,

l’L

µE = (1.6)

L

Clearly di¬erent measures of strain could be used. For instance, the change

in length ∆l = l ’ L could be divided by the ¬nal length rather than the

initial length. Whichever de¬nition is used, if l ≈ L the small strain quantity

µ = ∆l/l is recovered.

An alternative large strain measure can be obtained by adding up all

the small strain increments that take place when the rod is continuously

stretched from its original length L to its ¬nal length l. This integration

6 INTRODUCTION

process leads to the de¬nition of the natural or logarithmic strain µL as,

l

dl l

µL = = ln (1.7)

l L

L

Although the above strain de¬nitions can in fact be extrapolated to

the deformation of a three-dimensional continuum body, this generalization

process is complex and computationally costly. Strain measures that are

much more readily generalized to continuum cases are the so-called Green

strain µG and Almansi strain µA de¬ned as,

l2 ’ L2

µG = (1.8a)

2L2

l2 ’ L2

µA = (1.8b)

2l2

Irrespective of which strain de¬nition is used, a simple Taylor series anal-

ysis shows that for the case where l ≈ L, all the above quantities converge

to the small strain de¬nition ∆l/l. For instance, in the Green strain case,

we have,

(l + ∆l)2 ’ l2

µG (l ≈ L) ≈

2l2

1 l2 + ∆l2 + 2l∆l ’ l2

=

l2

2

∆l

≈ (1.9)

l

1.3.2 NONLINEAR TRUSS EXAMPLE

This example is included in order to introduce a number of features as-

sociated with ¬nite deformation analysis. Later, in Section 1.4, a small

FORTRAN program will be given to solve the nonlinear equilibrium equa-

tion that results from the truss analysis. The structure of this program is,

in e¬ect, a prototype of the general ¬nite element program presented later

in this book.

We consider the truss member shown in Figure 1.4 with initial and

loaded lengths, cross-sectional areas and volumes: L, A, V and l, a, v re-

spectively. For simplicity we assume that the material is incompressible and

hence V = v or AL = al. Two constitutive equations are chosen based,

without explanation at the moment, on Green™s and a logarithmic de¬nition

7