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the deformation is su¬ciently small to enable the e¬ect of changes in the
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

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