F

B

,v ,V

l,a L,A

x

A X

D

FIGURE 1.4 Single incompressible truss member.

of strain, hence the Cauchy, or true, stress σ is either,

l2 ’ L2 l

σ=E or σ = E ln (1.10a,b)

2L2 L

where E is a constitutive constant that, in ignorance, has been chosen to

be the same irrespective of the strain measure being used. Physically this is

obviously wrong, but it will be shown below that for small strains it is ac-

ceptable. Indeed, it will be seen in Chapter 4 that the Cauchy stress cannot

be simply associated with Green™s strain, but for now such complications

will be ignored.

The equation for vertical equilibrium at the sliding joint B, in nomen-

clature that will be used later, is simply,

x

R(x) = T (x) ’ F = 0; T = σa sin θ; sin θ = (1.11a,b,c)

l

where T (x) is the vertical component, at B, of the internal force in the

truss member and x gives the truss position. R(x) is the residual or out-of-

balance force, and a solution for x is achieved when R(x) = 0. In terms of

the alternative strain measures, T is,

Evx l2 ’ L2 Evx l

T= 2 or T = ln (1.12a,b)

2L2 l2

l L

Note that in this equation l is function of x as l2 = D2 + x2 and therefore

T is highly nonlinear in x.

Given a value of the external load F , the procedure that will eventually

be used to solve for the unknown position x is the Newton“Raphson method,

8 INTRODUCTION

Logarithmic

F/EA Green

0.4

0.3

q

0.2

0.1

’ X/L

0

X/L

’0.1 0

p

’0.2

p

’0.3

x/L

’0.4

’4 ’3 ’2 ’1 0 1 2 3 4

FIGURE 1.5 Truss example: load de¬‚ection behavior.

but in this one-degree-of-freedom case it is easier to choose a value for x and

¬nd the corresponding load F . Typical results are shown in Figure 1.5, where

an initial angle of 45 degrees has been assumed. It is clear from this ¬gure

that the behavior is highly nonlinear. Evidently, where ¬nite deformations

are involved it appears as though care has to be exercised in de¬ning the

constitutive relations because di¬erent strain choices will lead to di¬erent

solutions. But, at least, in the region where x is in the neighborhood of its

initial value X and strains are likely to be small, the equilibrium paths are

close.

In Figure 1.5 the local maximum and minimum forces F occur at the so-

called limit points p and q, although in reality if the truss were compressed

to point p it would experience a violent movement or snap-through behavior

from p to point p as an attempt is made to increase the compressive load

in the truss beyond the limit point.

By making the truss member initially vertical we can examine the large

strain behavior of a rod. The typical load de¬‚ection behavior is shown in

Figure 1.6, where clearly the same constant E should not have been used to

represent the same material characterized using di¬erent strain measures.

Alternatively, by making the truss member initially horizontal, the sti¬ening

e¬ect due to the development of tension in the member can be observed in

Figure 1.7.

Further insight into the nature of nonlinearity in the presence of large

deformation can be revealed by this simple example if we consider the ver-

tical sti¬ness of the truss member at joint B. This sti¬ness is the change

in the equilibrium equation, R(x) = 0, due to a change in position x and is

9

1.3 NONLINEAR STRAIN MEASURES

F/EA

0.5

0.45 Green

0.4

0.35

0.3

Logarithmic

0.25

0.2

0.15

0.1

0.05

0 x/L

1 1.5 2 2.5 3 3.5 4

FIGURE 1.6 Large strain rod: load de¬‚ection behavior.

F/EA

0.3

0.25 Green

0.2

Logarithmic

0.15

0.1

0.05

x/L

0

0 1 2

FIGURE 1.7 Horizontal truss: tension sti¬ening.

generally represented by K = dR/dx. If the load F is constant, the sti¬ness

is the change in the vertical component, T , of the internal force, which can

be obtained with the help of Equations (1.11b,c) together with the incom-

pressibility condition a = V /l as,

dT

K=

dx

d σV x

=

l2

dx

ax dσ 2σax dl σa

= ’2 +

l dl l dx l

dσ 2σ x2 σa

=a ’ + (1.13)

l l2

dl l

10 INTRODUCTION

All that remains is to ¬nd dσ/dl for each strain de¬nition, labelled G and

L for Green™s and the logarithmic strain respectively, to give,

dσ El dσ E

= and = (1.14a,b)

L2

dl dl l

G L

Hence the sti¬nesses are,

L2 x2 σa

A

KG = E ’ 2σ 2 + (1.15a)

l2

L l l

x2 σa

a

KL = (E ’ 2σ) 2 + (1.15b)

l l l

Despite the similarities in the expressions for KG and KL , the gradient of

the curves in Figure 1.5 shows that the sti¬nesses are generally not the

same. This is to be expected, again, because of the casual application of the

constitutive relations.

Finally it is instructive to attempt to rewrite the ¬nal term in (1.15a)

in an alternative form to give KG as,

x2 SA L2

A

KG = (E ’ 2S) 2 + ; S=σ (1.15c)

l2

L l L

The above expression introduces the second Piola“Kirchho¬ stress S, which

gives the force per unit undeformed area but transformed by what will be-

come known as the deformation gradient inverse, that is, (l/L)’1 . It will

be shown in Chapter 4 that the second Piola“Kirchho¬ stress is associated

with Green™s strain and not the Cauchy stress, as was erroneously assumed

in Equation (1.10a). Allowing for the local-to-global force transformation

implied by (x/l)2 , Equations (1.15c,b) illustrate that the sti¬ness can be

expressed in terms of the initial, undeformed, con¬guration or the current

deformed con¬guration.

The above sti¬ness terms shows that, in both cases, the constitutive

constant E has been modi¬ed by the current state of stress σ or S. We

can see that this is a consequence of allowing for geometry changes in the

formulation by observing that the 2σ term emerges from the derivative of

the term 1/l2 in Equation (1.13). If x is close to the initial con¬guration X

then a ≈ A, l ≈ L, and therefore KL ≈ KG .

Equations (1.15) contain a sti¬ness term σa/l (= SA/L) which is gen-

erally known as the initial stress sti¬ness. The same term can be derived

by considering the change in the equilibrating global end forces occurring

when an initially stressed rod rotates by a small amount, hence σa/l is also

called the geometric sti¬ness. This is the term that, in general, occurs in an

11

1.3 NONLINEAR STRAIN MEASURES

y

P

u Y

0

P

90“

X

x

X

’Y

FIGURE 1.8 90 degree rotation of a two-dimensional body.

instability analysis because a su¬ciently large negative value can render the

overall sti¬ness singular. The geometric sti¬ness is unrelated to the change

in cross-sectional area and is purely associated with force changes caused by

rigid body rotation.

The second Piola“Kirchho¬ stress will reappear in Chapter 4, and the

modi¬cation of the constitutive parameters by the current state of stress

will reappear in Chapter 5, which deals with constitutive behavior in the

presence of ¬nite deformation.

1.3.3 CONTINUUM STRAIN MEASURES

In linear stress“strain analysis the deformation of a continuum body is mea-

sured in terms of the small strain tensor µ. For instance, in a simple two-

dimensional case µ has components µxx , µyy , and µxy = µxy , which are

obtained in terms of the x and y components of the displacement of the

body as,

‚ux

µxx = (1.16a)

‚x

‚uy

µyy = (1.16b)

‚y

1 ‚ux ‚uy

µxy = + (1.16c)

2 ‚y ‚x

These equations rely on the assumption that the displacements ux and uy

12 INTRODUCTION