2.094 — Finite Element Analysis of Solids and Fluids
Fall ‘08
Lecture 10 - F.E. large deformation/general nonlinear analysis
Prof. K.J. Bathe
MIT OpenCourseWare
We developed
�
t
τij teij d tV = tR
Reading:
Ch. 6
(10.1)
tV
1
teij =
2
�
�
∂ui
∂uj
+ t
∂ txj
∂ xi
�
(10.2)
τij δt eij d tV = tR
t
tV
1
δt eij =
2
�
∂(δui ) ∂(δuj )
+
∂ t xj
∂ t xi
(10.3)
�
(≡ t eij )
(10.4)
In FEA:
t
F = tR
(10.5)
In linear analysis
t
F = K t U ⇒ KU = R
(10.6)
In general nonlinear analysis, we need to iterate. Assume the solution is known “at time t”
t
x=
0
x + tu
(10.7)
Hence t F is known. Then we consider
t+Δt
F =
t+Δt
R
(10.8)
Consider the loads (applied external loads) to be deformation-independent, e.g.
41
MIT 2.094
10. F.E. large deformation/general nonlinear analysis
Then we can write
t+Δt
F = tF + F
(10.9)
t+Δt
U = tU + U
(10.10)
where only t F and t U are known.
F ∼
= tK ΔU ,
t
K = tangent stiffness matrix at time t
(10.11)
From (10.8),
t
K ΔU =
t+Δt
R − tF
(10.12)
We use this to obtain an approximation to U . We obtain a more accurate solution for U (i.e.
using
t+Δt
K (i−1) ΔU (i) =
t+Δt
U
(i)
=
t+Δt
R − t+ΔtF (i−1)
t+Δt
U
(i−1)
+ ΔU
t+Δt
F)
(10.13)
(i)
(10.14)
Also,
t+Δt
F (0) = tF
t+Δt
K
(0)
(10.15)
t
= K
(10.16)
U (0) = tU
(10.17)
t+Δt
Iterate for i = 1, 2, 3 . . . until convergence. Convergence is reached when
�
�
�
�
�ΔU (i) � < �D
�
�2
� t+Δt
t+Δt (i−1) �
R−
F
�
� < �F
2
Note:
�a�2 =
��
2
(ai )
i
�
ΔU
(i)
=U
i=1,2,3...
ΔU (1) in (10.13) is ΔU in (10.12).
(10.13) is the full Newton-Raphson iteration.
How we could (in principle) calculate t K
Process
• Increase the displacement t Ui by �, with no increment for all t Uj , j �= i
• calculate
t+�
F
• the i-th column in tK = ( t+�F − tF ) /� =
∂ tF
.
∂ tUi
42
(10.18)
(10.19)
MIT 2.094
10. F.E. large deformation/general nonlinear analysis
So, perform
Pictorially,
⎡
..
⎢ .
⎢
t
K = ⎢ ...
⎣
..
.
this process for i = 1, 2, 3, . . . , n, where n is the total number of degrees of freedom.
..
.
..
. ···
..
.
⎤
⎥
⎥
⎥
⎦
A general difficulty: we cannot “simply” increment Cauchy stresses.
•
t+Δt
τij referred to area at time t + Δt
• t τij referred to area at time t.
We define a new stress measure, 2nd Piola - Kirchhoff stress,
refers to original configuration. Then,
t+Δt
0 Sij
= 0tSij + 0 Sij
t+Δt
0 Sij ,
where 0 in the leading subscript
(10.20)
The strain measure energy-conjugate to the 2nd P-K stress 0tSij is the Green-Lagrange strain 0t�ij
Then,
�
0V
t
0 Sij
Also,
�
0V
δ 0t�ij d 0V = tR
t+Δt
0 Sij
δ t+Δt0�ij d 0V =
(10.21)
t+Δt
R
(10.22)
Example
43
MIT 2.094
10. F.E. large deformation/general nonlinear analysis
t
F = tR
t+Δt
F =
(10.23)
t+Δt
R
(10.24)
(every time it is in equilibrium)
(10.13) and (10.14) give:
i = 1,
t+Δt
K (0) ΔU (1) =
t+Δt
R − t+ΔtF (0) ≡ fn( tU )
=
t+Δt
K (1) ΔU (2) =
t+Δt
U
(1)
t+Δt
(10.25)
(1)
(10.26)
R − t+ΔtF (1)
(10.27)
U
(0)
+ ΔU
i = 2,
t+Δt
t+Δt
U
(2)
=
t+Δt
U
(1)
+ ΔU
(2)
(10.28)
44
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http://ocw.mit.edu
2.094 Finite Element Analysis of Solids and Fluids II
Spring 2011
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