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1.3 NONLINEAR STRAIN MEASURES



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

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