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strain tensor µ by the Lam´ material coe¬cients » and µ as,
e
σ = »(trµ)I + 2µµ
This equation can be rewritten in terms of the fourth-order elasticity
tensor C as,
σ = C : µ; C = »I — I + 2µI; Cijkl = »δij δkl + 2µδik δjl
Alternatively the above relationship can be inverted to give the strain
expressed in terms of the stress tensor. To achieve this, note ¬rst that
taking the trace of the above stress“strain equation gives,
trσ = (3» + 2µ)trµ
and consequently µ can be written as,
1 »trσ
µ= σ’ I
2µ 2µ(3» + 2µ)
or in terms of the Young™s modulus E and Poisson™s ratio ν as,
1 µ(3» + 2µ) »
µ= [(1 + ν)σ ’ ν(trσ)I]; E= ; µ=
E »+µ 2» + 2µ
Hence the inverse elasticity tensor can be de¬ned as,
ν 1+ν
µ = C ’1 : σ; C ’1 = ’ I — I + I
E E
25
2.3 LINEARIZATION AND THE DIRECTIONAL DERIVATIVE



f


f(x0)




x
x0 x1 x2

FIGURE 2.3 One-degree-of-freedom nonlinear problem f (x) = 0.


2.3 LINEARIZATION AND THE DIRECTIONAL
DERIVATIVE

Nonlinear problems in continuum mechanics are invariably solved by lin-
earizing the nonlinear equations and iteratively solving the resulting linear
equations until a solution to the nonlinear problem is found. The Newton“
Raphson method is the most popular example of such a technique. Correct
linearization of the nonlinear equations is fundamental for the success of
such techniques. In this section we will consolidate the concept of the direc-
tional derivative introduced in Chapter 1. The familiar Newton“Raphson
scheme will be used as the initial vehicle for exploring the ideas that will
eventually be generalized.


2.3.1 ONE DEGREE OF FREEDOM
Consider the one-degree-of-freedom nonlinear equation shown in Figure 2.3,
f (x) = 0 (2.87)
Given an initial guess of the solution, x0 , the function f (x) can be ex-
pressed in the neighborhood of x0 using a Taylor™s series as,
1 d2 f
df
(x ’ x0 )2 + · · ·
f (x) = f (x0 ) + (x ’ x0 ) + (2.88)
2 dx2
dx x0 x0

If the increment in x is expressed as u = (x’x0 ) then (2.88) can be rewritten
as,
1 d2 f
df
u2 + · · ·
f (x0 + u) = f (x0 ) + u+ (2.89)
2 dx2
dx x0 x0
26 MATHEMATICAL PRELIMINARIES



To establish the Newton“Raphson procedure for this single-degree-of-freedom
case, (2.89) is linearized by truncating the Taylor™s expression to give,
df
f (x0 + u) ≈ f (x0 ) + u (2.90)
dx x0

This is clearly a linear function in u, and the term u(df /dx)|x0 is called
the linearized increment in f (x) at x0 with respect to u. This is generally
expressed as,
df
Df (x0 )[u] = u ≈ f (x0 + u) ’ f (x0 ) (2.91)
dx x0

The symbol Df (x0 )[u] denotes a derivative, formed at x0 , that operates in
some linear manner (not necessarily multiplicative as here) on u.
Using Equation (2.90) the Newton“Raphson iterative procedure is set up
by requiring the function f (xk + u) to vanish, thus giving a linear equation
in u as,
f (xk ) + Df (xk )[u] = 0 (2.92)
from which the new iterative value xk+1 , illustrated in Figure 2.3, is obtained
as,
’1
df
u= ’ f (xk ); xk+1 = xk + u (2.93)
dx xk

This simple one-degree-of-freedom case will now be generalized in order
to further develop the concept of the directional derivative.


2.3.2 GENERAL SOLUTION TO A NONLINEAR PROBLEM
Consider a set of general nonlinear equations given as,
F (x) = 0 (2.94)
where the function F (x) can represent anything from a system of nonlinear
algebraic equations to more complex cases such as nonlinear di¬erential
equations where the unknowns x could be sets of functions. Consequently x
represents a list of unknown variables or functions.
Consider an initial guess x0 and a general change or increment u that, it
is hoped, will generate x = x0 + u closer to the solution of Equation (2.94).
In order to replicate the example given in Section 2.3.1 and because, in
general, it is not immediately obvious how to express the derivative of a
complicated function F with respect to what could also be a function x, a
single arti¬cial parameter is introduced that enables a nonlinear function
27
2.3 LINEARIZATION AND THE DIRECTIONAL DERIVATIVE



F(†)
f



F (0)
f(x0)




x

u
x0 0 1 2

0 1 2 3

FIGURE 2.4 Single DOF nonlinear problem f (x) = 0 and F ( ) = 0.



F in , (not equal to F ), to be established as,

F( ) = F (x0 + u) (2.95)

For example, in the one-degree-of-freedom case, discussed in Section 2.3.1,
F( ) becomes,

F ( ) = f (x0 + u) (2.96)

This is illustrated for the one-degree-of-freedom case in Figure 2.4.
A more general case, illustrating (2.95), involving two unknown variables
x1 and x2 is shown in Figure 2.5. Observe how changes the function F in
the direction u and that clearly F( ) = F (x).
In order to develop the Newton“Raphson method together with the as-
sociated linearized equations, a Taylor™s series expansion of the nonlinear
function F( ) about = 0, corresponding to x = x0 , gives,

1 d2 F
dF 2
F( ) = F(0) + + + ··· (2.97)
2d 2
d =0 =0

Introducing the de¬nition of F given in Equation (2.95) into the above Tay-
lor™s series yields,
2
d2
d
F (x0 + u) = F (x0 ) + F (x0 + u) + F (x0 + u) + · · ·
2d2
d =0 =0
(2.98)

Truncating this Taylor™s series gives the change, or increment, in the non-
28 MATHEMATICAL PRELIMINARIES



f
F( †)
f( x1 , x 2 )




x2


x1
0 1 2
x0
u


FIGURE 2.5 Two-degrees-of-freedom nonlinear problem f (x1 , x2 ) = 0 and F ( ) = 0.



linear function F (x) as,
d
F (x0 + u) ’ F (x0 ) ≈ F (x0 + u) (2.99)
d =0

Note that in this equation is an arti¬cial parameter that is simply being
used as a vehicle to perform the derivative. In order to eliminate from the
left-hand side of this equation, let = 1, thereby giving a linear approxima-
tion to the increment of F (x) as
d
F (x0 + u) ’ F (x0 ) ≈ 1 F (x0 + u) (2.100)
d =0

where the term on the right-hand side of the above equation is the directional
derivative of F (x) at x0 in the direction of u and is written as,
d
DF (x0 )[u] = F (x0 + u) (2.101)
d =0

Note that u could be a list of variables or functions, hence the term “in the
direction” is, at the moment, extremely general in its interpretation. With
the help of the directional derivative the value of F (x0 + u) can now be
linearized or linearly approximated as,

F (x0 + u) ≈ F (x0 ) + DF (x0 )[u] (2.102)

Returning to the nonlinear Equation (2.94), setting F(x0 + u) = 0 in
(2.102) gives,

F (x0 ) + DF (x0 )[u] = 0 (2.103)
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2.3 LINEARIZATION AND THE DIRECTIONAL DERIVATIVE



which is a linear equation with respect to u* . Assuming that (2.103) can
be solved for u, then a general Newton“Raphson procedure can be re-
established as,


DF (xk )[u] = ’F (xk ) ; xk+1 = xk + u (2.104)



2.3.3 PROPERTIES OF THE DIRECTIONAL DERIVATIVE
The directional derivative de¬ned above satis¬es the usual properties of the
derivative. These are listed for completeness below,


(a) If F (x) = F 1 (x) + F 2 (x) then,


DF (x0 )[u] = DF 1 (x0 )[u] + DF 2 (x0 )[u] (2.105a)


(b) The product rule: if F (x) = F 1 (x) · F 2 (x), where “ · ” means any type
of product, then,


DF (x0 )[u] = DF 1 (x0 )[u] · F 2 (x0 ) + F 1 (x0 ) · DF 2 (x0 )[u]


(c) The chain rule: if F (x) = F 1 (F 2 (x)) then,

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