3.1 General Equations

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Nonlinearities are involved in a lot of applications. Either the strains and/or rotations are

large, such that the Lagrangian strain cannot be approximated by the infinitesimal strain,

or there are discontinuities such as in contact phenomena. Another frequent source of

nonlinearities is nonlinear material behavior. Although this chapter focuses on geometric

nonlinearities, the present section treats both geometric and material nonlinearities.

Nonlinear problems are usually broken down into a repetition of linear ones. This can

best be illustrated by a one-dimensional nonlinear problem. Consider the nonlinear equation

f (x) = F. (3.1)

Both the left-hand side and the right-hand side are plotted in Figure 3.1 as a function of

x. Suppose we know a starting value x0, which is reasonably close to the solution of our

equation (or close to “a” solution, since a nonlinear equation can have multiple solutions).

To find the solution, the function f (x) is locally linearized at x = x0 by replacing it

by its tangent line. Accordingly, Equation (3.1) now reads

f (x0) + (x − x0)f

_

(x0) = F (3.2)

which can be solved using a linear equation solver. This yields a first approximation of the

solution, which we will call x1. Now the same procedure can be repeated until the relative

difference between two subsequent solutions is smaller than a specified value _:

____

xi − xi−1

xi−1

____

≤ _. (3.3)

For solutions close to zero, one sometimes has to resort to the absolute difference. This is

called the Newton–Raphson method. If the true tangent is taken, it exhibits a quadratic rate

of convergence. However, whether it converges at all largely depends on the following:

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

144 GEOMETRIC NONLINEAR EFFECTS

F

f (x)

x0 x1 x2 x

Figure 3.1 The Newton–Raphson method

1. How close the starting solution is to the final solution. The local maximum between

the starting guess and the true solution in Figure 3.2 leads to no convergence.

2. The smoothness of the nonlinear function. Because of the jump in Figure 3.3, the

Newton–Raphson procedure does not converge.

For our applications, the Newton–Raphson method will be used throughout. For other

solution methods, the reader is referred to (Zienkiewicz and Taylor 1989) and (Matthies

and Strang 1979). A nice treatise on the computability of nonlinear problems is given in

(Belytschko and Mish 2001).

How can the Newton–Raphson method be applied to the governing finite element

equations? The major equation for mechanical problems is Equation (2.1). The nonlinearities

arise twofold in the term SKLδEKL on the left-hand side:

1. For materials of mechanical grade 1 and thermal grade 1, the second Piola–Kirchhoff

stress S is generally a nonlinear function of E and its time derivatives (Equation

(1.382)).

2. The Lagrange strain E is a nonlinear function of U (Equation (1.84)), in rectangular

coordinates:

2EKL = UK,L + UL,K + UM

,KUM,L. (3.4)

For the material nonlinearity, the Newton–Raphson method is applied in a straightforward

manner. Assume that we find an intermediate solution E0 with corresponding stress

S0(E0). Linearizing S at E0 yields

SKL ≈ SKL

0

+ ∂SKL

∂EMN

____

0

(EMN − E0

MN). (3.5)

GEOMETRIC NONLINEAR EFFECTS 145

F

f (x)

x0 x1

x2 x3

x

Figure 3.2 Local maximum

F

f (x)

x0 x1

x2 x3

x

Figure 3.3 Discontinuous function

146 GEOMETRIC NONLINEAR EFFECTS

Denoting

_KLMN

0 := ∂SKL

∂EMN

____

0

(3.6)

Equation (3.5) yields

SKL ≈ SKL

0

+ _KLMN

0 (EMN − E0

MN). (3.7)

Differentiating Equation (3.4) yields an expression for the infinitesimal perturbation δEKL:

δEKL = 12

(δUK,L + δUL,K + UM

,KδUM,L + UM,LδUM

,K ). (3.8)

Accordingly,

SKLδEKL

= _SKL

0

+ 12

_KLMN

0 _(UM,N + UN,M + UR

,MUR,N) − (VM,N + VN,M + V R

,MVR,N)__

・ 12

・ (δUK,L + δUL,K + UP

,KδUP,L + UP,LδUP

,K ) (3.9)

where V is the displacement corresponding to E0. Defining the new displacement increment

W (to reduce the length of the equations V and W are used instead of the more intuitive

notation U0 and _U respectively)

W := U − V (3.10)

and replacing U in Equation (3.9) by V + W leads to

SKLδEKL

= _SKL

0

+ 12

_KLMN

0 (WM,N + WN,M + V R

,MWR,N + WR

,MVR,N + WR

,MWR,N)_

・ 12

・ _δWK,L + δWL,K + (V P

,K

+ WP

,K )δWP,L + (VP,L + WP,L)δWP

,K

_ . (3.11)

In the above equations, V is the displacement calculated thus far and known. The

unknown is the incremental displacement W. In Equation (3.11), the terms linear in W

are force contributions, the quadratic terms contribute to the stiffness and the higher-order

terms are neglected. Consequently, Equation (3.11) yields

SKLδEKL ≈ 12

SKL

0 _δWK,L + δWL,K + V P

,KδWP,L + VP,LδWP

,K

_

+ 12

SKL

0 _WM

,KδWM,L + WM,LδWM

,K

_

+ 14

_KLMN

0 _WM,N + WN,M + V R

,MWR,N + WR

,MVR,N_

・ _δWK,L + δWL,K + V P

,KδWP,L + VP,LδWP

,K

_ . (3.12)

GEOMETRIC NONLINEAR EFFECTS 147

Because of the symmetries in SKL

0 and _KLMN

0 (SKL

0

= SLK

0 and _KLMN

0

= _LKMN

0

=

_KLNM

0 ), Equation (3.12) further reduces to

SKLδEKL ≈ SKL

0 _δWK,L + V P

,KδWP,L_ + SKL

0 WP

,KδWP,L

+ _KLMN

0 _WM,N + V R

,MWR,N_ ・ _δWK,L + V P

,KδWP,L_ . (3.13)

This equation applies to linear as well as to nonlinear materials. The only difference is that

for linear materials _KLMN

0 is constant, for nonlinear materials it is a function of EKL.

By substituting Equation (3.13) into Equation (2.1), one obtains, instead of Equation (2.6),

_

V0

WM,N_KLMN

0 δWK,L dV + _

V0

SKL

0 WP

,KδWP,L dV

+ _

V0

_KLMN

0 _V R

,MWR,NδWK,L + V P

,KWM,NδWP,L + V R

,MV P

,KWR,NδWP,L_ dV

= _

A0t

T

K

(N)δWK dA + _

V0

ρ0f KδWK dV + _

V0

[βKL(θ)T − γ KL]δUK,L dV

− _

V0

SKL

0 (δWK,L + V P

,KδWP,L) dV − ρ0

_

V0

D2V K

Dt2 δWK dV

− ρ0

_

V0

D2WK

Dt2 δWK dV. (3.14)

The first term on the left-hand side is the traditional (linear) stiffness term, the second is the

stress stiffness and the third is the large deformation stiffness. The last term on the righthand

side is the mass term. By renaming indices, one can also write for Equation (3.14),

_

V0

__KLMN

0

+ SNL

0 GMK + _KLRN

0 V M

,R

+ _SLMN

0 V K

,S

+_SLRN

0 V M

,RV K

,S

_WM,NδWK,L dV

= _

A0t

T

K

(N)δWK dA + _

V0

ρ0f KδWK dV + _

V0

[βKL(θ)T − γ KL]δUK,L dV

− _

V0

_SKL

0

+ SML

0 V K

,M

_δWK,L dV − ρ0

_

V0

D2V K

Dt2 δWK dV

− ρ0

_

V0

D2WK

Dt2 δWK dV (3.15)

which has a completely similar form to Equation (2.6). Accordingly, Equation (2.27)

            K _W_ +         M

D2

Dt2 _W_ = _F_ (3.16)

148 GEOMETRIC NONLINEAR EFFECTS

also applies here together with Equations (2.28) to (2.30), where now

            Ke(iK)(jM)

= _

V0e

ϕi,Lϕj,N __KLMN

0

+ SNL

0 GMK

+_KLRNV M

,R

+ _SLMNV K

,S

+ _SLRNV M

,RV K

,S

_ dVe (3.17)

            Me(iK)(jM)

= ρ0

_

V0e

ϕiϕj dVe (3.18)

_F_e(iK)

= _F_ext

e(iK)

− _F_int

e(iK)

− _

V0e

ρ0

D2V K

Dt2 ϕi dVe (3.19)

_F_ext

e(iK)

= _

At0e

T

K

(N)ϕi dAe + _

V0e

ρ0f Kϕi dVe

+ _

V0e

[βKL(θ)T − γ KL]ϕi,L dVe. (3.20)

_F_int

e(iK)

= _

V0e

_SKL

0

+ SML

0 V K

,M

_ ϕi,L dVe. (3.21)

Consequently, each iteration (Figure 3.1) in a nonlinear calculation leads to a linear set of

equations and the same solvers can be used as in the linear case.

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