Technical Note GKSS/WMS/00/03
interner Bericht
Finite Element Modelling
Lecture Notes
Faculty of Engineering Kiel
W. Brocks
Februar 2000
Institut für Werkstofforschung
GKSS-Forschungszentrum Geesthacht
Inhalte
Wiederholung von Grundbegriffen der Festigkeitslehre und Einführung in die
Kontinuumsmechanik fester Körper: Spannungs- und Verzerrungstensor,
Gleichgewichtsbedingungen, elastisches Stoffgesetz,
Formulierung des Festigkeitsproblems als Randwertaufgabe und als
Variationsaufgabe (Prinzip der virtuellen Arbeiten)
Einführung in die Methode der finiten Elemente (FEM): Diskretisierung des
Kontinuums, Verschiebungsansätze in den Elementen, (elastisches) Stoffgesetz,
Steifigkeitsmatrix und Lastvektoren, Lösung des Gleichungssystems
Aufbau eines FE-Programmes: erforderliche Eingaben, Programmablauf,
Ergebnisdaten
Einführung in die Anwendung des FE-Programmsystems ANSYS mit Übungen
am Rechner: Erstellung von FE-Netzen und sonstigen Eingabedaten,
Durchführung
von
FE-Rechnungen,
Ergebnisdarstellung
und
Ergebnisauswertung.
Selbständige Bearbeitung einfacher Probleme: Stab unter Zugbelastung,
Biegebelastung einer eingespannten Scheibe mit Rechteckquerschnitt, gelochte
Scheibe unter zweiachsialem Zug.
empfohlene Literatur (deutsch):
J. ALTENBACH und H. ALTENBACH: "Einführung in die Kontinuumsmechanik",
Teubner Studienbücher Mechanik, Stuttgart 1994.
K. KNOTHE, H. WESSELS: "Finite Elemente, Einführung für Ingenieure", SpringerVerlag, Berlin, 1991.
K.J. BATHE: "Finite-Elemente-Methoden", Springer-Verlag, Berlin, 1986.
O.C. ZIENKIEWICZ: "Methode der Finiten Elemente", Carl-Hanser-Verlag,
München, 1984.
========================================================
Contents
repetition of fundamentals of strength of materials and introduction into
continuum mechanics of solids: stress and strain tensors, balance equantions,
constitutive equations of linear elasticity,
formulation as boundary value problem and variational problem (principle of
virtual work)
introduction into the finite element method (FEM): discretization of the
continuum, shape and displacement functions, isoparametric elements, (elastic)
constitutive equations, local and global stiffnes matrices and load vectors, solution
of equations
general structure of a FE programme: input data, flow diagram of problem
solution, results
application of the FE programme ANSYS with computer exercises: meshes and
other input data, FE simulations, presentation, evaluation and interpretation of
results,
individual working on case and parameter studies: tensile bars, bending of a
clamped rectangular panel, panel with hole under biaxial tension
recommended literature (english):
W.M. LAI, D. RUBIN, and E. KREMPL: "Introduction to Continuum Mechanics",
Pergamon Press, 1993.
M.E. GURTIN: "An introduction to Continuum Mechanics", Academic Press, New
York, 1981.
K.J. BATHE, "Finite Element Procedures", Prentice Hall, New Jersey 07632, 1996.
O.C. ZIENKIEWICZ: "The Finite Elemente Method", McGraw-Hill Book Comp.,
Maidenhead (UK), 1977.
WMSReport0003.doc, Brocks, 27.08.02 - - 2
Contents
Introduction to continuum mechanics
• kinematics
• forces and stresses
• balance equations
• constitutive equations, HOOKE's law of elasticity
Variational principles
• variational calculus
• principle of virtual work
FE modelling and computational procedure
Shape functions:
• orthogonal elements
• general isoparametric elements
Nonlinear FE Analyses
• geometrical nonlinearity
• material nonlinearity
Examples
• plane panel under bending load
• plane panel with hole under biaxial tension
Appendix
• Notation
• Physical units (SI units)
WMSReport0003.doc, Brocks, 27.08.02 - - 3
Continuum Mechanics
Continuum mechanics is a
phenomenological field theory.
Basing on observed phenomena, mathematical models
for the mechanical behaviour of matter are formulated.
As everybody knows, the behaviour of matter is
determined by interactions of atoms and molecules.
However, an engineering modeling cannot be done on
this level.
The discretely structured matter is hence represented by
a phenomenological model, i.e. the continuum;
this is done by averaging its properties in space.
For describing the mechanical behaviour of
heterogeneous materials with strong local gradients of
the microstructure a phenomenological theory is not
sufficient, in general. For this purpose,
micromechanical models on a "meso level"
between a microstructural and a phenomenological
modeling are developed and applied.
WMSReport0003.doc, Brocks, 27.08.02 - - 4
Equations of Continuum Mechanics
1.
material independent principles
kinematics
•
•
•
•
body
configuration
motion
deformation
kinetics
• external actions:
• internal reactions:
forces
stresses
balance equations (conservation laws)
•
•
•
•
•
2.
matter
momentum
moment of momentum
energy / work / power
entropy (2nd principle of thermodynamics)
material dependent equations
constitutive equations: solids, fluids, gases
reversible, irreversible processes
The combination of all principles leads to the
formulation of initial boundary value problems.
WMSReport0003.doc, Brocks, 27.08.02 - - 5
Kinematics
Body
A body B is a three-dimensional differentiable
manifold, the elements of which are called particles
x ∈ B.
A body B is endowed with a non-negative scalar
measure, m, which is called the mass of the body.
Configuration
A configuration χ of a body B is a smooth
homeomorphism of B onto a region, B ⊂ E3, of
three-dimensional EUKLIDean space E3, x = χ(x),
called the region occupied by the body B in the
configuration χ.
Motion
A motion of a body B is a one-parameter family tχ
of configurations, x = tχ(x) = χ(x,t). The real
parameter t is the time.
Deformation
The "local" deformation results from observing an
infinitesimal vicinity of a particle in its present
configuration with respect to a reference
configuration.
WMSReport0003.doc, Brocks, 27.08.02 - - 6
Deformation
Any general motion of a body includes a deformation
which can be described as follows. A particle, x, which is
identified by its place 0 x = 0 χ( x) in a reference
configuration at time t=0, occupies the place
t
x = t χ( x) = χ( 0 χ −1 ( 0 x ), t ) = φ( 0 x, t ) = t φ( 0 x ) = 0 x + t0 u
at the time t.
The mapping tφ is called deformation,
and t0 u is the displacement of the particle.
Observing an infinitesimal vicinity of the particle in the
course of motion,
t
∂ tφ 0
x + d x = φ( x + d x ) ≈ φ( x ) + 0 ⋅ d x
∂ x
t
t
0
0
t
0
leads to the definition of the deformation gradient,
t
0
∂ tφ 0 t
F = 0 = ∇ 0 u + I = t0 H + I
∂ x
Various tensor-valued measures of the local deformation,
commonly addressed as strains, can be derived from
t
0 F.
WMSReport0003.doc, Brocks, 27.08.02 - - 7
Strain Tensors
polar decomposition
t
0
F = t0 R ⋅ t0 U = t0 V ⋅ t0 R
t
0
with
R ⋅ t0 R T = I
right CAUCHY-GREEN Tensor
t
0C
= t0 F T ⋅ t0 F = t0 U 2 = I + H + H T + H T ⋅ H
• GREEN's quadratic strain tensor
t (G)
0E
= 12 ( t0 U 2 − I)
[
) +( ∇ )⋅(
T
= t0 E + 12 ( 0 ∇ t0 u) ⋅ ( 0 ∇ t0 u)
=
t
1 0
∇
0u +
2
t
0E
(
0
∇
T
t
0u
t
0u
0
0
∇
)
T
t
0u
]
strain tensor for small deformations
• HENCKY's logarithmic ("true") strain tensor
t
0
E ( H ) = ln( t0 U)
tensor of deformation rates
t
D=
t
1
2
[ ∇ v + ( ∇ v) ]
t
t
t
t
T
˙ ( G ) ⋅ t0 F −1
D = t0 F − T ⋅ t E
WMSReport0003.doc, Brocks, 27.08.02 - - 8
Forces and Stresses
The concept describes the action of the outside world
on the body B in motion and the interaction between
the different parts P of the body.
(a) external body force
Fb (P ) = ∫ b( x, t )ρ dV , P ⊂ B
P
over the part P of B in the configuration χ.
(b) contact force
Fc (P ) = ∫ t( x; P ) dS
∂P
extended over the boundary ∂P of P .
(c) total resultant force
F(P ) = Fb (P ) + Fc (P )
(d) stress principle
t ( x; P ) = t ( x, n )
where n is the exteriour unit normal vector at the
point x on the boundary ∂P of P
This implies the existence of a stress-tensor field
S(x) such that the stress vector t(x,n) may be
expressed by
t( x, n) = n ⋅ S( x )
WMSReport0003.doc, Brocks, 27.08.02 - - 9
Principal Stresses and Invariants
S⋅n = σ n
"eigenvalue" problem
condition
det(S − σ I) = 0
characteristic equation
σ 3 − J1σ 2 − J2σ − J3 = 0
invariants of stress tensor
J1 (S) = tr(S) = σ ii
J2 (S) = 12 [ tr(S 2 ) − tr 2 (S)] = 12 (σ ijσ ij − σ iiσ jj )
J3 (S) = det(S) = 13 σ ijσ jkσ ki
three real solutions ("eigenvalues") = principal stresses
σ I ≥ σ II ≥ σ III
with corresponding principal directions
S ⋅ ni = σ i ni
σ I
S= 0

0
0
σ II
0
n I , n II , n III
(i = I , II , III )
0 
0  ni n j

σ III 
invariants in terms of principal stresses
J1 (S) = σ I + σ II + σ III
J2 (S) = −(σ I σ II + σ II σ III + σ I σ III )
J3 (S) = σ I σ II σ III
WMSReport0003.doc, Brocks, 27.08.02 - - 10
stress deviator
S′ = S − 13 σ ii I
hydrostatic stress
σ h = 13 σ ii
invariants of stress deviator
J1 (S′) = 0
J2 (S′) = 12 σ ij′ σ ij′
[
= 16 (σ 11 − σ 22 ) + (σ 22 − σ 33 ) + (σ 33 − σ 11 ) +
2
2
2
2
+σ 122 + σ 23
+ σ 132 ]
J3 (S′) = 13 σ ij′ σ ′jkσ ki′ = 13 σ ijσ jkσ ki
VON MISES
equivalent (effective) stress
σ e = 3 J 2 (S ′ )
VON MISES
yield condition
σ e ≤ R0
upper limit of purely elastic deformation
beginning plastic deformation
R0 = yield strength
WMSReport0003.doc, Brocks, 27.08.02 - - 11
plane stress
 σ xx
S =  σ xy

 0
σ xy
σ yy
0
0
0 e i e j

0
principal stresses
σI  1
2
=
+
σ
σ
 2 ( xx
yy ) ±
σ II 
1
4
(σ xx − σ yy ) + σ xy2
2
direction of first (maximum) principal stress
tan 2ϕ 0 =
2σ xy
σ yy − σ xx
maximum shear stress
τ max =
1
4
(σ xx − σ yy ) + σ xy2 = 12 (σ I − σ II )
2
direction of maximum shear stress
π
4
VON MISES equivalent (effective) stress
ϕ1 = ϕ 0 ±
2
2
2
σ e = σ xx
+ σ xx
− σ xxσ yy + 3σ xy
= σ I2 + σ II2 − σ I σ II
WMSReport0003.doc, Brocks, 27.08.02 - - 12
Balance Equations
conservation of mass
global: any finite part P of B
d
m˙ (P ) = ∫ ρ dV = 0
dt P
local (equation of continuity)
ρ˙ + ρ div x˙ = 0
balance of momentum
global
d
ρ x˙ dV
∫
dt P
local (CAUCHY's law of motion)
div S + ρ b = ρ ˙˙
x
F( P ) =
balance of moment of momentum
for non-polar media
global
d
∫ x × bρ dV + ∫ x × t dS = dt ∫ ρ x × x˙ dV
P
∂P
P
local (symmetry of stress tensor)
S = ST
WMSReport0003.doc, Brocks, 27.08.02 - - 13
General Principles
governing the mechanical behaviour of materials
1. material frame-indifference
Constitutive equations must be invariant under
changes of frame reference. If a constitutive
equation is satisfied for a process with a motion and
a symmetric stress tensor given by
x = χ( x, t ), S = S( x, t )
then it must be satisfied also for the motion and
stress tensor given by
˜ x, t˜ ) = c(t ) + Q(t ) ⋅ χ( x, t ) ,
x˜ = χ(
S˜ = S˜ ( x, t˜ ) = Q(t ) ⋅ S( x, t ) ⋅ Q T (t )
t˜ = t − τ
2. determinism
The stress in a body is determined by the history of
the motion of that body.
3. local action
In determining the stress at a given particle x, the
motion outside an arbitrary neighbourhood of xmay
be disregarded.
WMSReport0003.doc, Brocks, 27.08.02 - - 14
Constitutive Equations
principle of determinism:
t
S( x, t ) = f {χ( y, τ ), x}
τ =−∞
transformation of time:
x, y ∈ B
τ =t−s
∞
S( x, t ) = f{χ( y, t − s), x}
s=0
principle of material frame-indifference:
Q ⋅ f⋅ Q T = f{Q ⋅ χ}
∞
S( x, t ) = f{χ( y, t − s) − χ( x, t − s)}
s=0
principle of local action:
Ω : y − x ≤ δ ; x = χ( x), y = χ( y)
∞
S( x, t ) = f
s=0
{
t −s
−∞
}
−s
F( x), t − s ∇( t−∞
F( x)),...
∞
"simple materials"
−s
S( x, t ) = f{ t−∞
F( x)}
s=0
principle of fading memory:
the memory of a simple material fades in time
WMSReport0003.doc, Brocks, 27.08.02 - - 15
Hooke's Law of Elasticity
general:
< 4>
S = C ⋅⋅E
< 4 > −1
E =  C  ⋅⋅S
<4>
C
4>
 <C

 
isotropic:
,
σ ij = Cijkl ε kl
,
−1
ε ij = Cijkl
σ kl
stiffness tensor (4th order)
−1
compliance tensor (4th order)
Cijkl = 2Gδ ik δ jl + ( K − 23 G)δ ijδ kl
or
ν

S = 2G E +
(tr E)I 


1 − 2ν
ν
σ ij = 2G ε ij +
(ε kk ) δ ij 


1 − 2ν
1 
ν
E=
S−
(tr S)I 

2G  1 + ν
ν
1 
ε ij =
σ ij −
(σ kk ) δ ij 

2G 
1+ν
G = shear modulus
K = bulk modulus
ν = POISSON's ratio
WMSReport0003.doc, Brocks, 27.08.02 - - 16
Material Parameters for Linear Elasticity
Elastizitätskonstanten
λ=
µ=
E=
ν=
K=
G=
λ, µ
λ
µ
µ (3λ + 2 µ )
λ+µ
λ
2( λ + µ )
λ + 23 µ
µ
G, K
K − 23 G
G
3K ⋅ G
3K + G
3K − 2G
6 K + 2G
K
G
E
2(1 + ν )
E
ν
E
3(1 − 2ν )
E
2(1 + ν )
E, ν
Eν
(1 + ν )(1 − 2υ )
λ, µ
G
K
E
ν
LAMÉ's coefficients
shear modulus
bulk modulus
YOUNG's modulus
POISSON's ratio
LAMÉsche Konstanten
Schubmodul
Kompressionsmodul
Elastizitätsmodul
Querkontraktionszahl
WMSReport0003.doc, Brocks, 27.08.02 - - 17
Variational Principles
variational principles in mechanics
replace the (differential) equations of motion
or equilibrium, as there are
• balance of momentum,
• balance of angular momentum,
• CAUCHY's field equations;
are extremum principles for energy type quantities,
like
• work,
• kinetic energy,
• potential energy.
Differential equations of motion can be established by
methods of variational calculus.
WMSReport0003.doc, Brocks, 27.08.02 - - 18
Variational Calculus
problem:
find a set of functions
for which the integral
xi(t), i = 1, ..., n
t1
I = ∫ F(t, x1 ,..., x n , x˙1 ,..., x˙ n )dt
t0
becomes an extremum under given
"boundary" conditions
xi (t0 ) = xi0 ; xi (t1 ) = xi1
(b.c.)
definition of "varied" functions
xi (t ) = xi (t ) + εξi (t ) with ξi (t0 ) = ξi (t1 ) = 0
ξi ( t )
are arbitrary, differentiable functions
meeting the b.c.
t1
(
)
⇒ I (ε ) = ∫ F t, x1 + εξ1 ,..., x˙1 + εξ˙1 ,... dt
t0
the condition for I becoming an extremum
is
 ∂I  = 0
 ∂ε  ε =0
δxi = ξi (t ) ; δx˙ i = ξ˙i (t )
are variations of xi ; ẋi
∂I 

δI =
 ∂ε  ε =0
is the (first) variation of I
δI = 0
is the variational problem
WMSReport0003.doc, Brocks, 27.08.02 - - 19
the variational problem leads to
 ∂F
∂I
∂F ˙ 
δI =   = ∫  ξi +
ξi  dt = 0
 ∂ε  ε =0
∂x˙ i 
 ∂xi
t
t1
0
partial integration yields
t1
t1
t1
0
0
0
 ˙ ∂F 
 ∂F 
 d ∂F 
ξ
ξ
dt
=
−
∫  i ∂x˙i   i ∂x˙i  ∫  ξi dt ∂x˙i  dt
t
t
t
where [..] = 0 due to b.c.
 ∂F d ∂F 
δI = ∫ ξi 
−
 dt = 0
 ∂xi dt ∂x˙ i 
t
t1
0
and as ξi (t ) are arbitrary ("test functions")
⇒ EULER's differential equation
of the variational problem
∂F d ∂F
−
=0
∂xi dt ∂x˙ i
WMSReport0003.doc, Brocks, 27.08.02 - - 20
Variational Problem in Mechanics
u = x(t ) − x(t0 ) displacement vektor
w
arbitrary "virtual" displacement,
• independent of t
• w(t0 ) = 0 in the reference configuration
Z
energy functional
δZ = lim
ε →0 
Z (u + εw ) − Z (u) ∂Z (u + εw )
=

ε
∂ε
ε =0
variation of Z at u in w direction
(GÂTEAUX derivative)
δu = δx =
∂ (u + εw )
=w
∂ε
ε =0
virtual displacement
(1) δZ (u, αw ) = αδZ (u, w )
(2) δZ (u, w1 + w 2 ) = δZ (u, w1 ) + δZ (u, w 2 )
example:
Z = 12 u ⋅ u − b ⋅ u
1 1
1
[
2 ( u + εw ) ⋅ ( u + εw ) − b ⋅ ( u + εw ) − 2 u ⋅ u + b ⋅ u]
ε →0 ε
δZ = lim
= lim[u ⋅ w + 12 εw ⋅ w − b ⋅ w ]
ε →0
= (u − b) ⋅ δu
δZ is linear in δu
WMSReport0003.doc, Brocks, 27.08.02 - - 21
Principle of Virtual Work
CAUCHY's field equations of motion
∂σ ij
+ ρb j = ρ u˙˙j
∂xi
,
j = 1, 2, 3
multiply by virtual displacement δuj and integrate over
the volume V:
∂σ ij
∫ ∂xi δu j dV + ∫ ρb jδu j dV = ∫ ρ u˙˙jδu j dV
V
V
V
∂ (δu j )
∂σ ij
∂
∫ ∂xi δu j dV = ∫ ∂xi (σ ij δu j ) dV − ∫ σ ij ∂xi dV
V
V
V
GAUß' theorem
∂
∫ ∂xi (σ ij δu j ) dV = ∫ ni σ ij δu j dA = ∫ t j δu j dA
∂V
∂V
V
∫ σ ij
V
∂ (δu j )
∂xi
∂u j
dV = ∫ σ ij δ
dV = ∫ σ ij δε ij dV
∂
x
i
V
V
∫ ρ u˙˙jδu j dV =
V
d
ρ u˙ jδu j dV − δ ∫ 12 ρ u˙ j u˙ j dV
∫
dt V
V
δA − δW = δP − δE
WMSReport0003.doc, Brocks, 27.08.02 - - 22
virtual work of external forces
δA = ∫ t j δu j dA + ∫ ρb jδu j dV
∂V
V
virtual work of stresses (virtual strain energy)
δW = ∫ σ ij δε ij dV
V
virtual power of momentum
δP =
d
ρ u˙ jδu j dV
∫
dt V
virtual kinetic energy
δE = δ ∫ 12 ρ u˙ j u˙ j dV
V
virtual work of mass acceleration
δB = δP − δE = ∫ ρ u˙˙jδu j dV
V
WMSReport0003.doc, Brocks, 27.08.02 - - 23
δ ( A − W − P + E) = 0
The variation of the energy functional
(A - W - P + E) vanishes.
or
The energy functional (A - W - P + E) becomes an
extremum (minimum) among all admissible
states (virtual displacements).
special cases:
• rigid body
δW = 0
• elastic body
δW = ∫ Cijkl ε klδε ij dV
V
W = 12 ∫ Cijkl ε ij ε kl dV elastic strain energy
V
• static problem (equilibrium)
δB = δP − δE = 0
WMSReport0003.doc, Brocks, 27.08.02 - - 24
Example:
plane panel
p
H
L
thickness = 1
 t x ( x, y)
 0
= 
{t} = 

t
(
x
,
y
)
 y
 boundary  − p 0≤ x ≤ L, y= H
displacement field
 u x ( x, y)
{u} = 

 uy ( x, y)
boundary conditions (b.c.)
u x (0, y) = uy (0, y) = 0
dimensionless coordinates
ξ=
x
y
; η=
L
H
global shape functions, ϕ i (ξ , η ), fulfilling b. c.
ϕ1 = ξ
ϕ4 = ξ3
 ϕ1 ϕ 2
{Φ} = 
0 0
, ϕ2 = ξ 2
, ϕ 3 = ξη
, ϕ 5 = ξ 2η , ϕ 6 = ξη 2
ϕ3 ϕ4 ϕ5 ϕ6 0 0 0 0 0 0 

0 0 0 0 ϕ1 ϕ 2 ϕ 3 ϕ 4 ϕ 5 ϕ 6 
WMSReport0003.doc, Brocks, 27.08.02 - - 25
series expansion of displacement field
 6

α
ϕ
i i
 ∑

{u} =  6i =1
 = {Φ}{α}
 ∑ α i +6ϕ i 
 i =1

{α} is
 α1 
 . 
{α} =  
 . 
 
 α 12 
with
(12×1) matrix of unknowns
strain matrix
 ε xx   ∂∂ux 
∂u
{ε} =  ε yy  =  ∂y 
   ∂u ∂u 
 γ xy   ∂y + ∂x 
x
y
y
x
= {D}{u} = {D}{Φ}{α} = {B}{α}
differential operator
 ∂∂x
{D} =  0

 ∂∂y
 1L ξL
{B} =  0 0

0 0

0
∂ 
H∂η

∂ 
L∂ξ 
0   L∂∂ξ
 0
∂
∂y =
 
∂
 H∂∂η
∂x 
2η
L
3ξ 2
L
2 ξη
L
η2
L
0
0
0
0
0
0
0
0
ξ
H
0
ξ2
H
2 ξη
H
1
L
2ξ
L
0
ξ
H
η
L
0
0
3ξ 2
L
0
ξ2
H
2 ξη
L
0
2 ξη 
H

η2 
L 
WMSReport0003.doc, Brocks, 27.08.02 - - 26
stress matrix and HOOKE's law
 σ xx 
{σ } =  σ yy  = {C}{ε}
 
 σ xy 
{C} stiffness matrix
for plane stress: σ zz = 0
1 ν
E 
ν 1
{C} =
2
1−ν 
0 0

0  = {C}T

1
2 (1 − ν )
0
for plane strain: ε zz = 0
ν
1 − ν
E
 ν
1−ν
{C} =
(1 + ν )(1 − 2ν ) 
 0
0

 = {C}T

1
1
2
ν
)
2( −
0
0
WMSReport0003.doc, Brocks, 27.08.02 - - 27
virtual work of external forces
1
δA = ∫ {t}T [δ {u}]η=1 Ldξ = {f}T δ {α}
ξ =0
virtual work of stresses
1
δW =
1
∫ ∫ {σ} δ {ε}Ldξ Hdη = {α} {K}δ {α}
T
T
ξ =0 η=0
"generalized" forces
1
{f} =
T
∫ {t(ξ )} {Φ (ξ )}η
T
1
=1
Ldξ = L{t}
T
ξ =0
∫ {Φ (ξ )}η
=1
dξ
ξ =0
= − pL(0 0 0 0 0 0
1
2
1
3
1
2
1
4
1
3
1
2
)
global stiffness matrix
1
{K} = LH
∫
1
T
T
{
B
}
{
C
}{
B
}
d
ξ
d
η
=
{
K
}
∫
ξ =0 η=0
principle of virtual work
δA − δW = ({f}T − {α}T {K}) δ {α}
as δ {α} is arbitrary
{K}{α} = {f}
linear system of equations for unknowns αi
WMSReport0003.doc, Brocks, 27.08.02 - - 28
FE Modelling and Computational Procedure
Finite element methods (FEM) base on variational principles for minimizing some potential like,
e.g., the potential energy of a mechanical systems.
Variational methods replace the solution of the corresponding boundary value problem.
For the so-called "deformation methods", FEM is based upon the
Principle of Virtual Work
δΠ = − ∫ S ⋅⋅δ (grad u) dV + ∫ ρb ⋅ δu dV + ∫ t ⋅ δu dA = 0
B
∂B
B
• stress-tensor field T(x),
• external body forces, b(x), defined for any x ∈ B,
• contact forces or tractions t(x), defined for any x on the boundary ∂B.
• The continuous body B is separated by imaginary lines or surfaces into a number, K, of
(finite) elements Bk.
The union
K
B̃ = U B k
k=1
is the finite model of the body.
• The elements are assumed to be interconnected at a discrete number, N, of nodal points, xn,
situated on their boundaries.
The displacements of these nodal points
u n = u( x n ), n = 1,... N
are the basic unknown parameters of the problem.
• A set of functions, ϕ i( k ) (ξ) , i = 1, ...Nk, ξ being local coordinates, is chosen to
WMSReport0003.doc, Brocks, 27.08.02 - - 29
define uniquely the state of displacement, u˜ ( k ) (ξ) , within each element, k, in terms of its nodal
displacements, u(i k ) ,
Nk
˜ ( k ) (ξ) = ∑ ϕ i( k ) (ξ) u(i k )
u
i =1
( )
with u(i k ) = u˜ ( k ) (ξ (i k ) ), ϕ i( k ) ξ (jk ) = δ ij .
• The displacement functions uniquely define the state of deformation, i.e. some strain tensor,
˜ ( k ) (ξ) , within each element in terms of the nodal displacements, e.g., small strain
E
T
Nk 
(k )
(k )

˜E( k ) (ξ) = 1  ∂ϕ i ⋅ ∂ξ ⋅ u( k ) +  ∂ϕ i ⋅ ∂ξ ⋅ u( k )   ,
∑
i
i
2 i =1  ∂ξ ∂x
 
 ∂ξ ∂x
with
∂ξ
= JACOBIAN matrix.
∂x
• These strains, together with the constitutive properties of the material, determine the state of
stress, S˜ ( k ) (ξ) , throughout the element and also on its boundaries.
• A system of forces, t (i k ) concentrated at the nodes equilibrating the boundary stresses,
˜t ( k ) = n( k ) ⋅ S˜ ( k ) , is determined, resulting in a force-displacement or "stiffness" relationship for
each element.
Nk
t (i k ) = ∑ C(ijk ) u(jk ) .
i =1
• Nodal displacements, u(i k ) , nodal forces, t (i k ) , and element stiffnesses, C(ijk ) , are
N
assembled according to the conditions of connectivity, u(i k ) = ∑ Ain( k ) u n ,
n =1
for all elements to compose the system of equations ensuring the conditions of compatibility
and equilibrium throughout.
K
N
k =1
n =1
u˜ ( x ) = ∑ u˜ ( k ) ( x ) = ∑ψ n ( x ) u n
where
 Nk (k ) (k )
 ϕ i ui
u˜ ( x ) = ∑
i =1
 0
(k )
x ∈B k
else
K
,
Nk
ψ n ( x ) = ∑ ∑ ϕ i( k ) Ain( k )
k =1 i =1
• Any system of nodal displacements listed for the whole structure in which all the elements
participate, automatically satisfies the condition of compatibility.
WMSReport0003.doc, Brocks, 27.08.02 - - 30
As the equilibrium condition has already been satisfied within each element all that is
necessary is to establish equilibrium at the nodes of the structure. This is done by the
principle of virtual work.
• The resulting equations governing the mechanical behaviour of the entire structure contain the
nodal displacements as unknowns.
s n + fn + pn = 0 , n = 1,..., N
sn = − ∫ S ⋅
B
∂ψ n
dV ; fn = ∫ ρb ψ n dV ; pn = ∫ t ψ n dA
∂x
B
∂B
A solution of this system of equations provides an approximate solution of the fields of
displacements, strains and stresses throughout the domain of the body.
WMSReport0003.doc, Brocks, 27.08.02 - - 31
Shape Functions
for Orthogonal Elements
y
u
x
y
b/2 η
3
4
yk
bk
a/2 ξ
2
1
ak
xk
element (k):
x
x k − 12 ak ≤ x ≤ x k + 12 ak
global coordinates
yk − 12 bk ≤ y ≤ yk + 12 bk
local coordinates:
−1 ≤ ξ , η ≤ +1
ξ=
2
2
x
x
−
,
η
=
(
( y − yk )
k)
ak
bk
WMSReport0003.doc, Brocks, 27.08.02 - - 32
4-Node Elements
"linear" plane elements: Nk = 4
shape functions ϕ i (ξ j , η j ) = δ ij
i, j = 1, ..., 4
ϕ1 (ξ , η ) = 14 (1 − ξ )(1 − η)
ϕ 2 (ξ , η ) = 14 (1 + ξ )(1 − η)
ϕ 3 (ξ , η ) = 14 (1 + ξ )(1 + η)
ϕ 4 (ξ , η ) = 14 (1 − ξ )(1 + η)
displacement field: {u} = {Φ}{ˆ}
uk
 u x ( x, y)  ϕ1 0 ϕ 2 0 ϕ 3 0 ϕ 4
 u ( x, y) =  0 ϕ
0 ϕ2 0 ϕ3 0
 y
 
1
 u1( xk ) 
 u(k ) 
 1y 
0   .. 

ϕ 4   .. 
 (k ) 
 u4 x 
 u(k ) 
 4y 
strain matrix
 ε xx 
{ε} =  ε yy  = {D}{u} = {D}{Φ}{ˆ}
u k = {B}{ˆ}
uk
 
 γ xy 
{û}k
nodal displacement matrix of element (k)
WMSReport0003.doc, Brocks, 27.08.02 - - 33
 ∂∂x
differential operator {D} =  0

 ∂∂y
0   a2 ∂ξ∂
 0
∂
∂y =
 
∂
∂
 b2 ∂η
∂x 
k
k
0 
2 ∂ 
b ∂η

2 ∂ 
a ∂ξ 
k
k
 u1 x 
 u1 y 
 
.. 

{uˆ }k = {A}k {uˆ } = {A}k  
..
 
 u Nx 
u 
 Ny 
{û}
{A}k
N =
2N=
global nodal displacement matrix
(8 × 2N) incidence or connectivity matrix
total number of nodes
number of degrees of freedom
{ε} = {B}{A}k {ˆ}
u
stress matrix and HOOKE's law
 σ xx 
{σ} =  σ yy  = {C}k {ε} = {C}k {B}{A}k {ˆ}
u
 
 σ xy 
principle of virtual work
δA − δW = 0
WMSReport0003.doc, Brocks, 27.08.02 - - 34
virtual work of external forces
K
δA = ∫ {t} δ {u} ds = ∑ ∫ {t}T {Φ} ds δ {ˆ}
u k = {fˆ}T δ {ˆ}
u
T
k =1 ∂Sk
∂S
K
{fˆ} = ∑ {A}Tk ∫ {Φ}T {t( s)} ds
nodal forces
k =1
∂Sk
virtual work of stresses
K
δW = ∑
+ 12 ak
+ 12 bk
T
T
{
σ
}
δ
{
ε
}
dx
dy
=
{ˆ}
u
{K}δ {ˆ}
u
∫
∫
k =1 x =− 1 ak y =− 1 bk
2
2
element stiffness matrix
+1
+1
ab
{K}k = k k ∫ ∫ {B}T {C}k {B}dξ dη ={K}Tk
4 ξ =−1 η =−1
global stiffness matrix
K
{K} = ∑ {A}Tk {K}k {A}k = {K}T
k =1
(
)
δA − δW = {fˆ}T − {ˆ}
u T {K} δ {ˆ}
u =0
as δ {ˆ}
u is arbitrary
{K}{ˆ}
u = {fˆ}
linear system of equations for 2N unknowns {û}
WMSReport0003.doc, Brocks, 27.08.02 - - 35
y
3
1
4
3
5
4
1
1
2
3
2
2
2
1
6
4
global nodes
elements
nodal coordinates
(1), ..., (6)
[1], [2]
(1):
0 ,
(2):
0 ,
(3):
L/2 ,
(4):
L/2 ,
(5):
L ,
(6):
L ,
elements
[1]:
(2) (4)
[2]:
(4) (6)
ak = L 2 , bk = H , k = 1, 2
displacement b.c.:
x
H
0
H
0
H
0
(3) (1)
(5) (3)
u1 x = u1 y = u2 x = u2 y = 0
WMSReport0003.doc, Brocks, 27.08.02 - - 36
nodal displacements
 0
 0
 
 0
 0
 
 u3 x 
 u3 y 
{ˆ}
u =   {uˆ }k = {A}k {uˆ }
 u4 x 
u 
 4y 
 u5 x 
u 
 5y 
 u6 x 
u 
 6y 
 u1( xk ) 
 u(k ) 
 1( yk ) 
 u2 x 
 u(k ) 
2y
{ˆ}
u k =  (k )  ,
 u3 x 
 u(k ) 
 3( ky) 
 u4 x 
 u(k ) 
 4y 
0
0

0
0
{ A}2 = 
0
0

0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0

0
1

0
0

0

0
WMSReport0003.doc, Brocks, 27.08.02 - - 37
element stiffness matrix (k = 1, 2)
+1
+1
LH
T
T
{K}k =
{
B
}
{
C
}
{
B
}
ξ
η
{
K
}
d
d
=
k
k
∫ ∫
8 ξ =−
1 η =−1
2
global stiffness matrix
{K} = ∑ {A}Tk {K}k {A}k = {K}T
k =1
{fˆ}k =
element nodal forces
T
Φ
{t( s)}k ds
{
}
∫
∂B k
 0 
{t}k = 

 − p(ξ )
pressure load
+1
L
T
Φ
{fˆ}k =
{
(
ξ
)
}
{t(ξ )}k dξ + {rˆ}k
∫
η =1
4 ξ =−1
or interpolation for arbitrary function {t(ξ )}
{t(ξ )} η =1 = {Φ} η =1{ˆt}k ,
{ˆt}k = (8×1) matrix
 +1

L
T
ˆ
{f}k =  ∫ ({Φ} {Φ}) η =1 dξ {ˆt}k + {rˆ}k
4 ξ =−1

reaction forces (k = 1, only)
{rˆ}1T = (r1 x
r1 y
0 0 0 0 r4 x
r4 y )
2
global nodal forces
{fˆ} = ∑ {A}Tk {fˆ}k
k =1
WMSReport0003.doc, Brocks, 27.08.02 - - 38
system of equations
 K1,1
 K2,1

 K3,1
K
 4,1
 K5,1
K
 6,1
 K7,1
K
 8,1
 0

 0
 0

 0
K1, 2
K2 , 2
K 3, 2
K4, 2
K1, 3
K2 , 3
K 3, 3
K4, 3
K1, 4
K2 , 4
K 3, 4
K1, 5
K2 , 5
K 3, 5
K1, 6
K2 , 6
K 3, 6
K1, 7
K2 , 7
K 3, 7
K1,8
K2 , 8
K 3, 8
0
0
0
0
0
0
0
0
0
K4, 4
K4, 5
K4, 6
K4, 7
K4,8
0
0
0
K 5, 2
K 5, 3
K 5, 4
K 5, 5
K 5, 6
K 5, 7
K 5, 8
K 5, 9
K5,10
K5,11
K6 , 2
K6 , 3
K6 , 4
K6 , 5
K6 , 6
K6 , 7
K6 , 8
K6 , 9
K6,10
K6,11
K 7, 2
K8 , 2
0
K 7, 3
K8 , 3
0
K 7, 4
K8 , 4
0
K7,10
K8,10
K9,10
K7,11
K8,11
K9,11
0
K 7, 7
K8 , 7
K9, 7
K10, 7
K 7, 9
K8 , 9
K9, 9
0
K 7, 6
K8 , 6
K9, 6
K10, 6
K 7, 8
K8 , 8
K9, 8
0
K 7, 5
K8 , 5
K9, 5
K10, 5
K10,8
K10, 9
K10,10
K10,11
0
0
0
0
0
0
K11, 5
K12, 5
K11, 6
K12, 6
K12, 7
K12, 7
K11,8
K12,8
K11, 9
K11,10
K11,11
K12, 9
K12,10
K12,11
  0   0   r1 x 
  0   f1 y   r1 y 
     
  0   0   r2 x 
0   0   0   r2 y 
     
K5,12   u3 x   0   0 
K6,12   u3 y   f3 y   0 
  =   +  
K7,12   u4 x   0   0 
K8,12   u4 y   0   0 
   
K9,12   u5 x   0   0 
     
K10,12   u5 y   f5 y   0 
K11,12   u6 x   0   0 
     
K12,12   u6 y   0   0 
0
0
0
WMSReport0003.doc, Brocks, 27.08.02 - - 39
Lagrangian Interpolation Polynomials
ξj − ξ
g (ξ ) = ∏
j =1, ξ j − ξi
n +1
(n)
i
j ≠i
n = order,
1D:
i = 1, ..., n+1
"truss" element
n = 1 (linear):
1
ξ1 = −1 ; ξ2 = +1
2 nodes
i = 1:
ϕ1 (ξ ) = g1(1) (ξ ) = 12 (1 − ξ )
i = 2:
ϕ 2 (ξ ) = g2(1) (ξ ) = 12 (1 + ξ )
2
ξ
n = 2 (quadratic):
1
2
3
ξ
3 nodes
ξ1 = −1 ; ξ2 = +1 ; ξ3 = 0
1
i=1: ϕ1 (ξ ) = g1( 2 ) (ξ ) = − 12 ξ (1 − ξ ) = 12 (1 − ξ ) − 12 (1 − ξ 2 )
i=2: ϕ 2 (ξ ) = g2( 2 ) (ξ ) = + 12 (1 + ξ )ξ = 12 (1 + ξ ) − 12 (1 − ξ 2 )
i=3: ϕ 3 (ξ ) = g3( 2 ) (ξ ) = (1 + ξ )(1 − ξ ) = 1 − ξ 2
{Φ} = (ϕ1 ϕ 2 ϕ 3 )
WMSReport0003.doc, Brocks, 27.08.02 - - 40
2D:
quadrilateral element
n = 1 (linear)
4 nodes
3
4
η
ξ1 = −1
ξ2 = +1
ξ3 = +1
ξ4 = −1
ξ
η1 = −1
η2 = −1
η3 = +1
η4 = +1
2
1
(1)
(1)
(1)
(1)
 ϕ1 (ξ , η ) ϕ 3 (ξ , η )  g1 (ξ )g1 (η ) g1 (ξ )g2 (η )

 =

 ϕ 2 (ξ , η ) ϕ 4 (ξ , η )  g2(1) (ξ )g1(1) (η ) g2(1) (ξ )g2(1) (η )
 14 (1 − ξ )(1 − η)
= 1
 4 (1 + ξ )(1 − η)
(1 − ξ )(1 + η)

1
1
+
ξ
1
+
η
(
)
(
)

4
1
4
{u} = {Φ}{ˆ}
uk
 ϕ1 0 ϕ 2
{Φ} = 
 0 ϕ1 0
0
ϕ2
ϕ3 0 ϕ4 0 

0 ϕ3 0 ϕ4 
WMSReport0003.doc, Brocks, 27.08.02 - - 41
n = 2 (quadratic):
9 nodes
7
4
3
η
8
ξ5 = 0 η5 = −1
ξ6 = +1 η6 = 0
ξ7 = 0 η7 = +1
ξ8 = −1 η8 = 0
ξ 9 = 0 η9 = 0
9
6
ξ
1
5
2
 ϕ1 ϕ 3 ϕ 8 
ϕ 2 ϕ 4 ϕ 6  =


ϕ5 ϕ 7 ϕ9 
 g1( 2 ) (ξ )g1( 2 ) (η ) g1( 2 ) (ξ )g2( 2 ) (η ) g1( 2 ) (ξ )g3( 2 ) (η )
 g ( 2 ) (ξ )g ( 2 ) (η ) g ( 2 ) (ξ )g ( 2 ) (η ) g ( 2 ) (ξ )g ( 2 ) (η )
1
2
3
2
3
 2( 2 )

(2)
(2)
(2)
(2)
(2)
 g3 (ξ )g1 (η ) g3 (ξ )g2 (η ) g3 (ξ )g3 (η )
 ϕ1 0 ϕ 2
{Φ} = 
 0 ϕ1 0
... ... ... ϕ 9 0 

... ... ... 0 ϕ 9 
WMSReport0003.doc, Brocks, 27.08.02 - - 42
3D:
"brick" element
ϕ1 (ξ , η,ζ ) = g1( n ) (ξ )g1( n ) (η )g1( n ) (ζ )
ϕ 2 (ξ , η,ζ ) = g2( n ) (ξ )g1( n ) (η )g1( n ) (ζ )
ϕ 3 (ξ , η,ζ ) = g2( n ) (ξ )g2( n ) (η )g1( n ) (ζ )
ϕ 4 (ξ , η,ζ ) = g1( n ) (ξ )g2( n ) (η )g1( n ) (ζ )
ϕ 5 (ξ , η,ζ ) = g1( n ) (ξ )g1( n ) (η )g2( n ) (ζ )
etc.
n = 1 (linear)
8 nodes
3
4
7
8
η
ζ
ξ
2
1
6
5
 ϕ1 0 0 ... ... ... ϕ 8 0 0 
{Φ} =  0 ϕ1 0 ... ... ... 0 ϕ 8 0 


 0 0 ϕ1 ... ... ... 0 0 ϕ 8 
n = 2 (quadratic)
27 nodes
WMSReport0003.doc, Brocks, 27.08.02 - - 43
Note:
for quadratic (and higher order) 2D and 3D
elements, internal nodes may be omitted to
reduce the number of degrees of freedom by
means of which some higher polynomial terms
vanish
"boundary node" elements
⇒ 2D quadratic: 8 nodes
term ξ 2η 2 vanishes
1
ξ
ξ2
η
ξη
ξ 2η
η2
ξη 2
ξ 2η 2
⇒ 3D quadratic:
20 nodes
WMSReport0003.doc, Brocks, 27.08.02 - - 44
include only if node i is defined
ϕ1 =
1
4
(1 − ξ )(1 − η)
− 12 ϕ 5
ϕ2 =
1
4
(1 + ξ )(1 − η)
− 12 ϕ 5
ϕ3 =
1
4
(1 + ξ )(1 + η)
ϕ4 =
1
4
(1 − ξ )(1 + η)
ϕ5 =
1
4
(1 − ξ )(1 − η)
ϕ6 =
1
4
(1 + ξ )(1 − η 2 )
ϕ7 =
1
4
(1 − ξ )(1 + η)
ϕ8 =
1
4
(1 − ξ )(1 − η 2 )
− 12 ϕ 8
− 12 ϕ 6
− 12 ϕ 6
− 12 ϕ 7
− 12 ϕ 7
− 12 ϕ 8
2
2
Interpolation functions of four to eight variable-number-nodes for 2D element
WMSReport0003.doc, Brocks, 27.08.02 - - 45
General Isoparametric Elements
curvilinear distorted element
y
3
η
3
4
4
2
ξ
1
1
2
x
x = f x (ξ , η,ζ ) ; y = f y (ξ , η,ζ ) ; z = fz (ξ , η,ζ )
interpolation by shape functions:
{x} = {Ψ}{ˆ}
xk
isoparametric elements: {Ψ} = {Φ}
,
2D:
3D:
 x
{x} =  
 y
 x1( k ) 
 y(k ) 
 1 
x k =  ... 
{ˆ}
 x (k ) 
 (Nk ) 
 yN 
 x
{x} =  y
 
 z
 x1( k ) 
 y(k ) 
 1( k ) 
x k =  z1 
{ˆ}
 ... 


 ... 
ψ i = ϕi
WMSReport0003.doc, Brocks, 27.08.02 - - 46
 ∂∂x
differential operator (2D) {D} =  0

 ∂∂y
0
∂
∂y

∂
∂x 
∂
∂ ∂ξ ∂ ∂η ∂ ∂ζ
=
+
+
∂x ∂ξ ∂x ∂η ∂x ∂ζ ∂x
 ∂   ∂f x
 ∂ξ   ∂ξ
 ∂   ∂f
  = x
 ∂η   ∂η
 ∂   ∂f x
  
 ∂ζ   ∂ζ
∂f y
∂ξ
∂f y
∂η
∂f y
∂ζ
∂
∂
=
{
J
}
 
 
∂ξ
 ∂x 
 
∂fz   ∂ 
∂ξ   ∂x 
∂fz   ∂ 
 
∂η   ∂y 
∂fz   ∂ 

∂ζ   ∂z 
⇒
 ∂  = {J}−1  ∂ 
 
 
 ∂x 
 ∂ξ 
{J} = JACOBI matrix, det{J} = J ≠ 0
 ∂f y
1
{J}−1 =  ∂η
J ∂f x
−
 ∂η
2D:
∂f y 
−
∂ξ 
∂f x 

∂ξ 
∂f x ∂f y ∂f y ∂f x
J=
−
∂ξ ∂η ∂ξ ∂η
∂f x
∂ϕ
= ∑ i xi( k ) , ... etc.
∂ξ i =1 ∂ξ
Nk
WMSReport0003.doc, Brocks, 27.08.02 - - 47
calculation of
strain matrix
{ε} = {D}{Φ}{ˆ}
u k = {B}{ˆ}
uk
stress matrix
{σ} = {C}k {ε} = {C}k {B}{ˆ}
uk
element stiffness matrix
{K}k = ∫∫∫ {B}T {C}k {B} dV
Bk
transformation of the volume differential:
2D:
dV = B dA = B dx dy = B J dξ dη
(B = thickness)
3D:
dV = dx dy dz = J dξ dη dζ
condition: J > 0 ,
{J} positive definite
element stiffness matrix
{K}k =
+1
+1
∫ ∫
+1
T
{
B
}
∫ {C}k {B} J dξ dη dζ
ξ =−1 η =−1 ζ =−1
integration numerically by GAUSS quadrature
global stiffness matrix
K
{K} = ∑ {A}Tk {K}k {A}k
k =1
WMSReport0003.doc, Brocks, 27.08.02 - - 48
GAUSS Quadrature
a function f(ξ) is integrated approximately by a
weighted sum of its values at sampling points ξj
+1
n
w j f (ξ j )
∫ f (ξ ) dξ = ∑
j =1
−1
the formula integrates a polynomial of order (2n-1)
exactly
sampling points (GAUSS points):
number
coordinates
weight
n
ξj
wj
1
0.000 000
2.000 000
2
±0.577 350
1.000 000
3
±0.774 597
0.555 556
0.000 000
0.888 889
±0.861 136
0.347 855
±0.339 981
0.652 145
4
+1
2D:
+1
n
n
∑ w j wk f (ξ j , ηk )
∫ ∫ f (ξ ) dξ dη = ∑
j =1 k =1
η =−1 ξ =−1
WMSReport0003.doc, Brocks, 27.08.02 - - 49
Nonlinear FE Analyses
nonlinear structural behaviour
a) geometrical (large displacements and/or large strain)
b) constitutive nonlinearity (material)
incremental formulation
quasi-static process (equilibrium),
"time" t ≥ 0 monotonously increasing parameter of
load history
principle of virtual work
time t
:
δA(t ) − δW (t ) = 0
time t + ∆t:
δA(t + ∆t ) − δW (t + ∆t ) = 0
f (t + ∆t ) ≈ f (t ) + f˙ t ∆t
t → t + ∆t:
δ∆A − δ∆W = 0
FE formulation:
{K({u})} ∆{u} = ∆{f}
WMSReport0003.doc, Brocks, 27.08.02 - - 50
(a) geometrical non-linearity
t =0
t
t
0
u
∆u
0
r
t
t + ∆t
0
r
O
t + ∆t
configuration
total displacement
t + ∆t
u
r
t + ∆t
r = 0r +
t + ∆t
0
t + ∆t
0
u = t0 u + ∆ u
u = t r + ∆u
total strain (GREEN's quadratic strain)
t + ∆t
0
E(G)
=
T
T
t + ∆t
t + ∆t
t
t

∂
u
∂
u




1
 ∂ 0 u  ∂ 0 u 
0
0
=  0  +  0  +  0  ⋅  0  
 ∂ r  ∂ r 
2  ∂ r   ∂ r 


t + ∆t
0
E + t + ∆0 t G
left subscript:
reference state
left superscript: actual state
reference state 0: Total LAGRANGEan Formulation
reference state t: Updated LAGRANGEan Formulation
WMSReport0003.doc, Brocks, 27.08.02 - - 51
Updated LAGRANGEan Formulation
reference state t
t + ∆t
t
S = tt S + ∆ t S = t T + ∆ t S
S = 2nd PIOLA-KIRCHHOFF stresses
T = CAUCHY stresses
t + ∆t
t
E(G) = ∆ t E(G) ≈ ∆ t E
virtual work during t → t + ∆t (surface tractions only):
δ∆A =
∫
t + ∆t
t
∫
t + ∆t
t
∫
t
t
t ⋅ δ t + ∆t t u dA =
∂ tV
δ∆W =
t
=
∂ tV
∂ tV
S ⋅⋅δ ∆ t + ∆t t E ( G ) dV
V
t
∫ ∆t ⋅ δ
∫ ∆t ⋅ δ ∆u dA
V
t
S ⋅⋅δ ∆ t E dV + ∫ ∆ t S ⋅⋅δ ∆ t E dV
t
V
u dA − ∫ tt S ⋅⋅δ ∆ t E dV = δA(t ) − δW (t ) = 0
t
V
∫ ∆ S ⋅⋅δ ∆ E dV − ∫ ∆t ⋅ δ ( u + ∆u ) dA = 0
t
t
t
t
∂ tV
V
constitutive relation:
∆ t S = t C ⋅⋅∆ t E
(incrementally linear)
{K( ∆{u})} ∆{u} = ∆{f}
t
nonlinear system of equations
WMSReport0003.doc, Brocks, 27.08.02 - - 52
⇒ iterative solution (e.g. NEWTON)
i
i
∆{u} = ∑ ∆{u} = ∑ t{K} {r }
(j)
j =1
−1
(j)
j =1
residual force (out-of balance force)
(j)
{r} =
t + ∆t
(j)
{f} − {f ( {u} + ∆{u})}
t
convergence criterion
(i)
(i)
∆{u} ≤ DTOL
{r } ≤ RTOL and / or
f
t+ ∆t
f
t
(1)
r
∆f
f
(1)
∆u
(2)
∆u
∆u
t
u
t+ ∆t
u
u
WMSReport0003.doc, Brocks, 27.08.02 - - 53
(b) constitutive non-linearity
elastic- plastic material behaviour
(VON MISES, PRANDTL, REUSS)
1. uniaxial tensile test
σ
loading
unloading
RF
εe
εF
εp
ε
σ ≤ R(ε p ) , R(0) = RF
for σ ≤ RF
 Eε
HOOKE's law: σ = 
p
ε
−
ε
E
(
) for σ > RF

σ˙ > 0, ε˙ p > 0 loading
loading condtion:

p
σ˙ < 0, ε˙ = 0 unloading
yield condtion:
WMSReport0003.doc, Brocks, 27.08.02 - - 54
2. general (multiaxial) stress state
yield condtion:
σ ≤ R(ε p )
VON MISES
;
σ=
3
2
σ ij′ σ ij′
equivalent (effective) stress
σ ij′ = σ ij − 13 σ kk deviatoric stresses
HOOKE's law:
 2G ε˙ + ν (ε ) δ 

 ij 1 − 2ν kk ij 
σ˙ ij = 
ν
2G (ε˙ij − ε˙ijp ) +
(ε kk ) δ ij 

1 − 2ν
 
σ ij′ σ˙ ij > 0, ε˙ijp
loading condtion:

p
˙
˙
<
σ
σ
ε
,
0
′
ij
ij
ij

flow rule
for σ ≤ RF
for σ > RF
> 0 loading
= 0 unloading
3 ε˙ p
3 σ˙
ε˙ =
σ ij′ =
σ ij′
p
2σ
2E σ
p
ij
Ep =
σ ij ε˙ijp = σ ε˙ p
Et E
E − Et
⇒ ε˙ p =
, Et =
2
3
dσ
dε
ε˙ijp ε˙ijp
ε̇ ijp is deviatoric, i.e. ε̇ kkp = 0
"true" stress-strain curve required for UL formulation:
F
L
true stresses σ =
vs logarithmic strain ε = ln
A
L0
WMSReport0003.doc, Brocks, 27.08.02 - - 55
p
H
L
A plane panel of dimensions
length L = 200 mm, height H = 100 mm, thickness B = 1mm
is clamped at x = 0 and loaded by a constant pressure p = 100 MPa at y = H
The material is isotropic, elastic with
YOUNG's modulus E = 218 400 MPa and POISSON's ratio ν = 0.3.
Establish the system of equations of the respective finite element model and calculate the
displacement, stress and strain field by applying the FE code ANSYS.
1. analytical solution for a model of two linear elements
y
3
1
4
5
4
3
1
1
2
3
2
2
1
4
2
6
x
WMSReport0003.doc, Brocks, 27.08.02 - - 56
1.1
Calculate the elastic stiffness matrix
 C11
{C} =  C12

 C13
C12
C22
C23
C13 
C23 

C33 
for plane stress conditions.
1.2
Calculate the (symmetric) element stiffness matrix for one element k
{K}k =
1.3
ak bk
4
+1
+1
∫ ∫ {B} {C} {B}dξ dη
T
k
ξ = −1 η = −1
Calculate the global stiffness matrix (see table on page 3)
K
{K} = ∑{A}Tk {K}k {A}k
k =1
1.4
Calculate the column matrix of nodal forces (see table on page 3)
K
{f} = ∑ {A}k ∫ {Φ} {t} ds .
T
k =1
2.
T
∂S k
FE solution by the ANSYS code
Solve the same problem by the FE code ANSYS and compare the results with analytical values
(see lecture notes), especially the displacement u y of node (6) and the stresses at the nodes (1)
and (2) in the table on page 4.
2.1
Use the model of two linear plane stress elements as on page 1.
2.2
Use two quadratic (8-node) elements instead of linear elements.
2.3
Use two other refined meshes and quadratic elements; explain and display the meshes.
WMSReport0003.doc, Brocks, 27.08.02 - - 57
analytical solution (theory of bending)
bending stress
Mz ( x ) 
H
y− 
2
Iz 
pBL2 
x  x2

1− 2
Mz ( x ) = −
+

 L   L  
2 
Mz (0)
max
=
0
=
0
x
x
σ xx = σ xx
= σ xx
=
= 1200 MPa
y= H
y=0
Wz
2 BH 2 10 4
section modulus Wz = I z =
=
mm 3
6
6
H
deflection
1. simple bending (no shear)
pBL4   x  2
x3  x4
b

+
−4
uy ( x ) = −
6
 L   L  
24 EI z   L 
4
pBL
uyb,max = uy ( L) =
= 1.10 mm
8 EI z
2. shear
pL2   x   x  2 
s
−
uy ( x ) = −
2
2GH   L   L  
2
pL
uys,max = uys ( L) =
= 0.24 mm
2GH
total
uymax = uyb,max + uys,max = 1.34 mm
σ xx ( x, y) = −
WMSReport0003.doc, Brocks, 27.08.02 - - 58
uy / L * (E / p)
0
-5
exercise #1:
rectangular panel
-10
bending
shear
total
-15
0.0
0.2
0.4
0.6
0.8
1.0
x/L
WMSReport0003.doc, Brocks, 27.08.02 - - 59
exercise #2:
biaxially loaded panel with circular hole
y
py
px
px
R
2H
x
2H
py
H = 50 mm, R = 10 mm
plane stress, B = 1 mm
py = 100 MPa, px = β py
E = 200000MPa, ν = 0.3
WMSReport0003.doc, Brocks, 27.08.02 - - 60
analytical solution for "infinite" plate: R << H,
uniaxial tension (β = 0)
2
2
4



R
R
R






1
+3
− 1 − 4
cos 2θ  
σ rr (r,θ ) = 2 py 1 −






r
r
r



  R 2 

R 4 

+ 1 + 3
cos
σ θθ (r,θ ) = py 1 +
2
θ






r
r




1
2
2
4


R
R




sin 2θ
−3
σ rθ (r,θ ) = 12 py 1 + 3





r
r 

3
σrr (θ=0)
y
σθθ (θ=0)
r
θ
σij / py
2
σrr (θ=π/2)
x
σθθ (θ=π/2)
1
0
-1
1
2
r/R
3
4
WMSReport0003.doc, Brocks, 27.08.02 - - 61
FE solution: symmetry conditions ⇒1/4 model
y
py
px
x
boundary conditions:
uy
y=0
= 0 ; ux
x =0
=0
WMSReport0003.doc, Brocks, 27.08.02 - - 62
Notation
30.10.99
tensors, vektors, skalars - general
scalar
latin or greek, small or capital (italics)
a , H, α ,
vector
symbolic
indexed
latin or greek, small,
bold or underlined
x , x , σn , σ n
latin or greek, small (italics)
xi , α i
x = xi ei , x = xi ei
2nd order tensor
symbolic
indexed
latin or greek, capital,
bold or underlined
T,T
latin or greek, capital or small (italics)
Tij , σij
T = Tij ei ej , T = Tij ei ej
4th order tensor
symbolic
latin or greek, capital,
< 4>
bold with overhead <4>
indexed
C
or double underlined
C
latin or greek, capital
Cijkl
< 4>
C = Cijkl ei ej ek el , C = Cijkl ei ej ek el
WMSReport0003.doc, Brocks, 27.08.02 - - 63
vector and tensor algebra and analysis
scalar product
α = u⋅v = ui vi ,
t = n⋅S = niσij ej,
A = B⋅C = Bij Cjk ei ek = Aikei ek
double scalar product
α = B⋅⋅C = Bij Cji
< 4>
T = C ⋅⋅E = Cijkl Ekl ei ej
or
α=B:C
...
or
S= C :E
...
tensorial product
T = a b = ai bj ei ej
< 4>
C = A B = Aij Bkl ei ej ek el
a b⋅c = (a b)⋅c = a (b⋅c)
spatial derivatives
NABLA operator
∇ = ei
∂
∂xi
v = grad ϕ = ∇ ϕ = ∂ϕ ei = ϕ,i ei
∂xi
F = grad v = ∇ v =
∂vi
ei ej = vi,j ei ej
∂x j
a = div v = ∇⋅v = = vi,i =
w = div T = ∇⋅T =
∂vi
∂xi
∂Tij
ej = Tij,i ej
∂xi
convention of summation is used in all cases
3
Aij Bjk = ∑ Aij Bjk
j =1
3
,
Ckk = ∑ Ckk
k =1
3
,
3
Aij Bji = ∑ ∑ Aij Bji
i =1 j =1
WMSReport0003.doc, Brocks, 27.08.02 - - 64
special tensors and invariants
I = δij ei ej = ei ei
2nd order unit tensor
T⋅I = I⋅T = T
deviator of a tensor T:
T' = Tij′ ei ej = (Tij - Tkk δij ) ei ej = T - T⋅⋅I
transposed tensor
TT = Tji ei ej
inverse tensor
T-1 , T-1
T⋅T-1 = T-1⋅T = I
1st invariant (trace) of a tensor
J1(T) = tr (T)
2nd invariant of a tensor
J 2 ( T) =
3rd invariant (determinant) of a tensor
J3(T) = det (T)
1
2
(tr T
2
− tr 2 T)
WMSReport0003.doc, Brocks, 27.08.02 - - 65
tensors used in continuum mechanics
deformation gradient
F = I + grad u = (δij + ui,j ) ei ej
GREEN's strain tensor
E( G ) =
linear strain tensor
E=
strain rate tensor
D = dij ei e j =
1
2
1
2
(F
T
⋅ F − I) =
(u
1
2
)
+ u j , i + ui , k uk , j ei e j
i, j
(grad u − grad u) = (u
T
1
2
1
2
i, j
)
+ u j , i ei e j = ε ij ei e j = E T
(grad v − grad v)
T
for small strains
˙ = ε̇ e e
D=E
ij i j
elastic and plastic part
˙ =E
˙e +E
˙ p = ε˙ e + ε˙ p e e
E
ij
ij
i j
(
)
accumulated effective ("equivalent") plastic strain
t
ε =ε
p
e
p
∫
0
CAUCHY ("true") stress tensor
hydrostatic stress
t
2
3
ε˙ijpε˙ijp dτ = ∫
4
3
˙ p ) dτ
J2 ( E
0
S = σij ei ej = ST = σji ei ej
σ h = 13 σ kk = 13 tr S = 13 J1 (S)
VON MISES effective ("equivalent") stress
σ e = σ = 3 J 2 (S′ ) =
2nd PIOLA KIRCHHOFF stress tensor
for small strains
3
2
σ ij′ σ ij′
T = det(F) F −1 ⋅ S ⋅ F − T = T T
T=S
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plastic yielding
R(εp)
yield curve (uniaxial tensile test)
ε p = ε ep = ε −
with
yield strength (at initial plastic flow)
σ
E
R0 = R(0)
especially
lower yield strength
0.2% proof stress
ReL
R p0.2
plastic potential, flow potential (VON MISES)
isotropic hardening
Φ = σ e2 − R2 (ε p ) = 23 σ ij′ σ ij′ − R2 (ε p ) = S′ ⋅⋅S′ − R2 (ε p )
kinematic hardening
Φ = σ e2 − R02 =
3
2
(S′ − X′) ⋅⋅(S′ − X′) − R02
with
X = "back stress" tensor
,
˙ p = λ˙ ∂Φ
E
∂S′
associated flow rule
∂Φ
ε˙ijp = λ˙
∂σ ij′
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matrix notation
general: (n×m) matrix:
{A}
or
A
n = number of rows
m = number of columns
 A11
 .
{A} = 
 .

 An1
 A11
 .

{A}T =  .
 .

 A1m
. . . A1m 
. . . . 

. . . . 

. . . Anm 
special: (n×1) matrix
{u}
or
. .
.
.
.
.
.
.
.
.
An1 
. 

. 
. 

Anm 
u
 u1 
{u} =  . 
 
 un 
{u}T = (u1 . un )
the elements of a matrix do not form the components of a tensor or vector, in general
products
n
{u}T {v} = α = ∑ ui vi
{u} and {v}: (n×1) ; α scalar
1
 u1v1
 .
{u}{v}T = {A} = 
 .

 un v1
. . . u1vm 
. . .
. 

. . .
. 

. . . un vm 
{u}: (n×1) , {v}: (m×1) ; {A}: (n×m)

 m
A1i vi

∑
i =1

{v} = {A}{u} =
.... 

 m
 ∑ Ani vi 

 i =1
{A}: (n×m) ; {u}: (m×1) , {v}: (n×1)
{v}T = {u}T {A}T
{A}T : (m×n) ; {u}T : (1×m) , {v}T : (1×n)
 m
A1i Bi1 . .
∑
i =1
{C} = {A}{B} =  ....
. .
 m
 ∑ Ani Bi1 . .
 i =1


i =1
.... 
m

A
B
∑
ni im 

i =1
m
∑A B
1i
im
{A}: (n×m) ; {B}: (m×p) ; {C}: (n×p)
WMSReport0003.doc, Brocks, 27.08.02 - - 68
Physical Units (SI-Units)
ISO 1000 (1973)
SI = Système International d'Unités
SI units are
the seven basic units and coherently derived units, i.e.
by a factor of 1
Basic Quantities and Units
basic quantity
SI basic unit
name
symbol
length
meter
m
mass
kilogramm
kg
time
second
s
thermodyn. temperature
Kelvin
K
amperage, intensity of
electric current
Ampere
A
amount of substance
Mol
mol
luminous intensity
Candela
cd
WMSReport0003.doc, Brocks, 27.08.02 - - 69
Decimal Parts and Multiples of SI Units
Parts and multiples of SI units which are generated by
multiplication with the factors
10±1, 10±2, 10±3n (n = 1, 2, ...)
have special names and symbols.
They are composed by prefixes.
faktor
10-15
10-12
10-9
10-6
10-3
10-2
10-1
101
102
103
106
109
1012
prefix
femto
pico
nano
micro
milli
centi
deci
deca
hecto
kilo
mega
giga
tera
symbol
f
p
n
µ
m
c
d
da
h
k
M
G
T
Multiples by the factors
10±3n (n = 1, 2, ...)
are to be preferred!
WMSReport0003.doc, Brocks, 27.08.02 - - 70
Derived Quantities and Units
Derived units are formed by products or ratios of basic units.
The same holds for the unit symbols.
quantity
SI unit
name
relation
symbol
frequency
Hertz
Hz
1 Hz = 1 s-1
force
Newton
N
1 N = 1 kg m s-2
pressure, stress
Pascal
Pa
1 Pa = 1 N / m2
energy, work, amount of heat
Joule
J
1J=1Nm
power, heat flow
Watt
W
1W=1J/s
electric charge, quantity of electricity
Coulomb
C
1C=1As
electric potential, voltage
Volt
V
1V=1J/C
electric capacity
Farad
F
1F=1C/V
electric resistance
Ohm
Ω
1Ω=1V/A
magnetic flux
Weber
Wb
1 Wb = 1 V s
magnetic flux density
Tesla
T
1 T = 1 Wb / m2
inductivity
Henry
H
1 H = 1 Wb / A
WMSReport0003.doc, Brocks, 27.08.02 - - 71
Examples for FE meshes
3D FE mesh at the notch of a side-notched flat tensile panel
3D FE mesh of a centre-notched tensile panel
WMSReport0003.doc, Brocks, 27.08.02 - - 72
2D FE mesh of a C(T) specimen, half model accounting for symmetry
Detail of the FE mesh of the C(T) specimen at the crack tip with collapsed elements
2
3
1
WMSReport0003.doc, Brocks, 27.08.02 - - 73
2D FE mesh of a biaxially loaded cruciform specimen with two-fold
symmetry
Detail of the FE mesh of the cruciform specimen at the crack tip with a
regular arrangement of elements
WMSReport0003.doc, Brocks, 27.08.02 - - 74
3D FE mesh of a tubular joint under 3-point bending, having one
symmetry plane
32896
HEXAEDER
ELEMENTE
Detail of the FE mesh of the semi-elliptical surface flaw in the weldment of
the tubular joint
ANSICHT
SYMMETRIEEBENE
Riss
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