Technical Note GKSS/WMS/00/03interner Bericht
Finite Element Modelling
Lecture NotesFaculty of Engineering Kiel
W. Brocks
Februar 2000
Institut für Werkstofforschung
GKSS-Forschungszentrum Geesthacht
WMSReport0003.doc, Brocks, 27.08.02 - - 2
Inhalte Wiederholung von Grundbegriffen der Festigkeitslehre und Einführung in dieKontinuumsmechanik fester Körper: Spannungs- und Verzerrungstensor,Gleichgewichtsbedingungen, elastisches Stoffgesetz,Formulierung des Festigkeitsproblems als Randwertaufgabe und alsVariationsaufgabe (Prinzip der virtuellen Arbeiten)Einführung in die Methode der finiten Elemente (FEM): Diskretisierung desKontinuums, Verschiebungsansätze in den Elementen, (elastisches) Stoffgesetz,Steifigkeitsmatrix und Lastvektoren, Lösung des GleichungssystemsAufbau eines FE-Programmes: erforderliche Eingaben, Programmablauf,ErgebnisdatenEinführung in die Anwendung des FE-Programmsystems ANSYS mit Übungenam Rechner: Erstellung von FE-Netzen und sonstigen Eingabedaten,Durchführung von FE-Rechnungen, Ergebnisdarstellung undErgebnisauswertung.Selbständige Bearbeitung einfacher Probleme: Stab unter Zugbelastung,Biegebelastung einer eingespannten Scheibe mit Rechteckquerschnitt, gelochteScheibe 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", Springer-Verlag, 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 ofvirtual work)introduction into the finite element method (FEM): discretization of thecontinuum, shape and displacement functions, isoparametric elements, (elastic)constitutive equations, local and global stiffnes matrices and load vectors, solutionof equationsgeneral structure of a FE programme: input data, flow diagram of problemsolution, resultsapplication of the FE programme ANSYS with computer exercises: meshes andother input data, FE simulations, presentation, evaluation and interpretation ofresults,individual working on case and parameter studies: tensile bars, bending of aclamped 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, NewYork, 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 - - 3
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 - - 4
Continuum Mechanics
Continuum mechanics is a
phenomenological field theory.
Basing on observed phenomena, mathematical modelsfor the mechanical behaviour of matter are formulated.
As everybody knows, the behaviour of matter isdetermined by interactions of atoms and molecules.However, an engineering modeling cannot be done onthis 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 ofheterogeneous materials with strong local gradients ofthe microstructure a phenomenological theory is notsufficient, in general. For this purpose,
micromechanical models on a "meso level"between a microstructural and a phenomenologicalmodeling are developed and applied.
WMSReport0003.doc, Brocks, 27.08.02 - - 5
Equations of Continuum Mechanics
1. material independent principles
kinematics• body• configuration• motion• deformation
kinetics• external actions: forces• internal reactions: stresses
balance equations (conservation laws)• matter• momentum• moment of momentum• energy / work / power• entropy (2nd principle of thermodynamics)
2. material dependent equations
constitutive equations: solids, fluids, gasesreversible, irreversible processes
The combination of all principles leads to theformulation of initial boundary value problems.
WMSReport0003.doc, Brocks, 27.08.02 - - 6
KinematicsBody
A body B is a three-dimensional differentiablemanifold, the elements of which are called particlesx ∈ B.
A body B is endowed with a non-negative scalarmeasure, m, which is called the mass of the body.
ConfigurationA configuration χχχχ of a body B is a smooth
homeomorphism of B onto a region, BB ⊂ EE3, of
three-dimensional EUKLID ean space EE3, x = χχχχ(x),
called the region occupied by the body B in theconfiguration χχχχ.
MotionA motion of a body B is a one-parameter family tχχχχof configurations, x = tχχχχ(x) = χχχχ(x,t). The realparameter t is the time.
Deformation
The "local" deformation results from observing aninfinitesimal vicinity of a particle in its presentconfiguration with respect to a referenceconfiguration.
WMSReport0003.doc, Brocks, 27.08.02 - - 7
Deformation
Any general motion of a body includes a deformationwhich can be described as follows. A particle, x, which isidentified by its place
0 0x = ( )χ x in a referenceconfiguration at time t=0, occupies the place
t t t tt tx x x x x u= ( ) = ( ) = ( ) = ( ) = +−χ χ χ φ φx 0 1 0 0 0 0
0( ), ,
at the time t.The mapping tφφφφ is called deformation,
and 0t u is the displacement of the particle.
Observing an infinitesimal vicinity of the particle in thecourse of motion,
t t t tt
d d dx x x x xx
x+ = +( ) ≈ ( ) + ⋅φ φ φ0 0 00
0∂∂
leads to the definition of the deformation gradient,
0 00
0 0t
tt tF
xu I H I= = ∇ + = +∂
∂φ
Various tensor-valued measures of the local deformation,commonly addressed as strains, can be derived from
0t F.
WMSReport0003.doc, Brocks, 27.08.02 - - 8
Strain Tensors polar decomposition
0 0 0 0 0t t t t tF R U V R= ⋅ = ⋅
with 0 0t t TR R I⋅ =
right CAUCHY-GREEN Tensor
0 0 0 02t t T t t T TC F F U I H H H H= ⋅ = = + + + ⋅
• GREEN's quadratic strain tensor
012 0
2
12
00
00
00
00
012
00
00
t t
t t T t t T
t t t T
E U I
u u u u
E u u
( )G = −( )= ∇ + ∇( ) + ∇( ) ⋅ ∇( )[ ]= + ∇( ) ⋅ ∇( )0t E strain tensor for small deformations
• HENCKY's logarithmic ("true") strain tensor
0 0t H tE U( ) ln= ( )
tensor of deformation ratest t t t t TD v v= ∇ + ∇( )[ ]1
2
t t T t tD F E F= ⋅ ⋅− −0 0
1˙ ( )G
WMSReport0003.doc, Brocks, 27.08.02 - - 9
Forces and Stresses
The concept describes the action of the outside worldon the body B in motion and the interaction betweenthe different parts P of the body.
(a) external body force
F b xb t dV( ) ( , ) ,P P B
P
= ⊂∫ ρ
over the part P of B in the configuration χχχχ.
(b) contact force
F t xc dS( ) ( ; )P P
P
= ∫∂
extended over the boundary ∂P of P .
(c) total resultant force F F F( ) ( ) ( )P P P= +b c
(d) stress principle t x t x n( ; ) ( , )P =
where n is the exteriour unit normal vector at thepoint x on the boundary ∂P of PThis implies the existence of a stress-tensor fieldS(x) such that the stress vector t(x,n) may beexpressed by
t x n n S x( , ) ( )= ⋅
WMSReport0003.doc, Brocks, 27.08.02 - - 10
Principal Stresses and Invariants
"eigenvalue" problem S n n⋅ = σcondition det S I−( ) =σ 0
characteristic equation σ σ σ31
22 3 0− − − =J J J
invariants of stress tensor
J
J
J
ii
ij ij ii jj
ij jk ki
1
212
2 2 12
313
( ) tr( )
( ) tr( ) tr ( )
( ) det( )
S S
S S S
S S
= =
= −[ ] = −( )= =
σ
σ σ σ σ
σ σ σ
three real solutions ("eigenvalues") = principal stresses
σ σ σI II III≥ ≥with corresponding principal directions n n nI II III, ,
S n n⋅ = =i i i i I II IIIσ ( , , )
S n n=
σσ
σ
I
II
III
i j
0 0
0 0
0 0
invariants in terms of principal stresses
J
J
J
I II III
I II II III I III
I II III
1
2
3
( )
( )
( )
S
S
S
= + +
= − + +( )=
σ σ σ
σ σ σ σ σ σσ σ σ
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stress deviator ′ = −S S I13 σ ii
hydrostatic stress σ σh ii= 13
invariants of stress deviator
J
J
J
ij ij
ij jk ki ij jk ki
1
212
16 11 22
222 33
233 11
2
122
232
132
313
13
0′( ) =
′( ) = ′ ′
= −( ) + −( ) + −( ) +[+ + + ]
′( ) = ′ ′ ′ =
S
S
S
σ σ
σ σ σ σ σ σ
σ σ σ
σ σ σ σ σ σ
VON M ISES equivalent (effective) stress
σ e J= ′3 2 ( )S
VON M ISES yield condition
σ e R≤ 0
upper limit of purely elastic deformation
beginning plastic deformation
R0 = yield strength
WMSReport0003.doc, Brocks, 27.08.02 - - 12
plane stress
S e e=
σ σσ σ
xx xy
xy yy i j
0
0
0 0 0
principal stresses
σσ
σ σ σ σ σI
IIxx yy xx yy xy
= +( ) ± −( ) +12
214
2 2
direction of first (maximum) principal stress
tan 22
0ϕσ
σ σ=
−xy
yy xx
maximum shear stress
τ σ σ σ σ σmax = −( ) + = −( )14
2 2 12xx yy xy I II
direction of maximum shear stress
ϕ ϕ π1 0 4
= ±
VON M ISES equivalent (effective) stress
σ σ σ σ σ σ
σ σ σ σ
e xx xx xx yy xy
I II I II
= + − +
= + −
2 2 2
2 2
3
WMSReport0003.doc, Brocks, 27.08.02 - - 13
Balance Equations
conservation of mass
global: any finite part P of B
˙ ( )m
ddt
dVPP
= =∫ ρ 0
local (equation of continuity)˙ div ˙ρ ρ+ =x 0
balance of momentum
global
F x( ) ˙P
P
= ∫ddt
dVρ
local (CAUCHY's law of motion)div ˙S b x+ =ρ ρ
balance of moment of momentumfor non-polar media
global
x b x t x x× + × = ×∫∫ ∫ρ ρ
∂
dV dSddt
dVPP P
˙
local (symmetry of stress tensor)S S= T
WMSReport0003.doc, Brocks, 27.08.02 - - 14
General Principlesgoverning the mechanical behaviour of materials
1. material frame-indifference
Constitutive equations must be invariant underchanges of frame reference. If a constitutiveequation is satisfied for a process with a motion anda symmetric stress tensor given by
x S S= χ( , ), = ( , )xt txthen it must be satisfied also for the motion andstress tensor given by
˜ ˜ ˜ ( ) ( )
˜ ˜ ˜ ) )
˜
x c Q
S S Q S Q
= + ⋅
⋅ ⋅= −
χ( , ) = χ( , ) ,
= ( , ) = ( ( , ) (
x x
x x
t t t t
t t t t
t t
T
τ
2. determinism
The stress in a body is determined by the history ofthe motion of that body.
3. local action
In determining the stress at a given particle x, the
motion outside an arbitrary neighbourhood of x maybe disregarded.
WMSReport0003.doc, Brocks, 27.08.02 - - 15
Constitutive Equations
principle of determinism:
S( , ) ( , ),x xyt
t
= =−∞f
ττχ x, y ∈ B
transformation of time: τ = −t s
S( , ) ( , ),x xyt t s
s
= − =
∞f
0
χ
principle of material frame-indifference:
Q Q Q⋅ ⋅ = ⋅ f fT χ
S( , ) ( , ) ( , )x xyt t s t s
s
= − − − =
∞f
0
χ χ
principle of local action:
Ω : ; ( (), )y x x y− ≤ = =δ χ χx y
S F F( , ) ( , ( ,...) )x x xt
s
t s t s t s= ∇( ) =
∞
−∞− −
−∞−f
0
"simple materials" S F( , ) ( )x xt
s
t s= =
∞
−∞−f
0
principle of fading memory:
the memory of a simple material fades in time
WMSReport0003.doc, Brocks, 27.08.02 - - 16
Hooke's Law of Elasticity
general: S C E= ⋅⋅< >4
, σ εij ijkl klC=
E C S=
⋅⋅
< > −4 1
, ε σij ijkl klC= −1
C< >4
stiffness tensor (4th order)
C< > −
4 1
compliance tensor (4th order)
isotropic: C G K Gijkl ik jl ij kl= + −( )2 23δ δ δ δ
or
S E E I= +−
21 2
Gν
ν(tr )
σ ε νν
ε δij ij kk ijG= +−
21 2
( )
E S S I= −+
12 1G
νν
(tr )
ε σ νν
σ δij ij kk ijG= −
+
12 1
( )
G = shear modulusK = bulk modulusν = POISSON's ratio
WMSReport0003.doc, Brocks, 27.08.02 - - 17
Material Parameters for Linear ElasticityElastizitätskonstanten
λ = µ = E = ν = K = G =
λ, µ λ µ µ λ µλ µ3 2+( )
+λ
λ µ2 +( )λ µ+ 2
3µ
G, K K G− 23
G 33
K G
K G
⋅+
3 26 2
K G
K G
−+
K G
E, ν Eνν υ( )( )1 1 2+ −
E
2 1( )+ νE ν E
3 1 2( )− νE
2 1( )+ ν
λ, µ LAMÉ's coefficients LAMÉsche KonstantenG shear modulus SchubmodulK bulk modulus KompressionsmodulE YOUNG's modulus Elastizitätsmodulν POISSON's ratio Querkontraktionszahl
WMSReport0003.doc, Brocks, 27.08.02 - - 18
Variational Principles
variational principles in mechanics
Ø replace the (differential) equations of motion
or equilibrium, as there are
• balance of momentum,
• balance of angular momentum,
• CAUCHY's field equations;
Ø 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 - - 19
Variational Calculusproblem: find a set of functions xi(t), i = 1, ..., n
for which the integral
I F t x x x x dtn n
t
t
= ( )∫ , ,..., , ˙ ,..., ˙1 1
0
1
becomes an extremum under given "boundary" conditionsx t x x t xi i i i( ) ; ( )0
01
1= = (b.c.)
definition of "varied" functions
x t x t t t ti i i i i( ) ( ) ( ) ( ) ( )= + = =εξ ξ ξwith 0 1 0
ξi t( ) are arbitrary, differentiable functionsmeeting the b.c.
⇒ I F t x x dtt
t
( ) , ,..., ˙ ˙ ,...ε εξ εξ= + +( )∫ 1 1 1 1
0
1
the condition for I becoming an extremum
is ∂∂ε ε
I
=
=0
0
δ ξ δ ξx t x ti i i i= =( ) ; ˙ ˙ ( ) are variations of xi ; xi
δ ∂∂ε ε
II=
=0
is the (first) variation of I
δI = 0 is the variational problem
WMSReport0003.doc, Brocks, 27.08.02 - - 20
the variational problem leads to
δ ∂∂ε
∂∂
ξ ∂∂
ξε
II F
xFx
dti
ii
i
t
t
=
= +
==
∫0
0
1
0˙
˙
partial integration yields
˙˙ ˙ ˙
ξ ∂∂
ξ ∂∂
ξ ∂∂i
it
t
ii t
t
iit
tFx
dtFx
ddt
Fx
dt
=
−
∫ ∫
0
1
0
1
0
1
where [..] = 0 due to b.c.
δ ξ ∂∂
∂∂
IFx
ddt
Fx
dtii it
t
= −
=∫ ˙0
1
0
and as ξi t( ) are arbitrary ("test functions")
⇒ EULER's differential equationof the variational problem
∂∂
∂∂
Fx
ddt
Fxi i
− =˙
0
WMSReport0003.doc, Brocks, 27.08.02 - - 21
Variational Problem in Mechanics
u x x= −( ) ( )t t0 displacement vektorw arbitrary "virtual" displacement,
• independent of t• w( )t0 0= in the reference configuration
Z energy functional
δ εε
∂ ε∂εε ε
ZZ Z Z= + −
= +
→ =lim
( ) ( ) ( )0
0
u w u u w
variation of Z at u in w direction(GÂTEAUX derivative)
δ δ ∂ ε∂ε ε
u xu w
w= = + ==
( )
0
virtual displacement
(1) δ α αδZ Z( , ) ( , )u w u w=(2) δ δ δZ Z Z( , ) ( , ) ( , )u w w u w u w1 2 1 2+ = +
example: Z = ⋅ − ⋅12 u u b u
δε
ε ε εε
Z = + ⋅ + − ⋅ + − ⋅ + ⋅[ ]→
lim ( ) ( ) ( )0
12
12
1u w u w b u w u u b u
= ⋅ + ⋅ − ⋅[ ]→
limε
ε0
12u w w w b w
= − ⋅( )u b uδ δZ is linear in δu
WMSReport0003.doc, Brocks, 27.08.02 - - 22
Principle of Virtual WorkCAUCHY's field equations of motion
∂σ∂
ρ ρij
ij jx
b u+ = ˙ , j = 1, 2, 3
multiply by virtual displacement δuj and integrate over
the volume V:
∂σ∂
δ ρ δ ρ δij
ij
V
j j
V
j j
Vx
u dV b u dV u u dV∫ ∫ ∫+ = ˙
∂σ∂
δ ∂∂
σ δ σ∂ δ
∂ij
ij
V iij j
V
ijj
iVx
u dVx
u dVu
xdV∫ ∫ ∫= ( ) −
( )
GAUß' theorem
∂∂
σ δ σ δ δ∂ ∂x
u dV n u dA t u dAi
ij j
V
i ij j
V
j j
V
( ) = =∫ ∫ ∫
σ∂ δ
∂σ δ
∂∂
σ δεijj
iij
j
iVV
ij ij
V
u
xdV
u
xdV dV
( )= =∫∫ ∫
ρ δ ρ δ δ ρ˙ ˙ ˙ ˙u u dVddt
u u dV u u dVj j
V
j j
V
j j
V∫ ∫ ∫= − 1
2
δ δ δ δA W P E− = −
WMSReport0003.doc, Brocks, 27.08.02 - - 23
virtual work of external forces
δ δ ρ δ∂
A t u dA b u dVj j
V
j j
V
= +∫ ∫virtual work of stresses (virtual strain energy)
δ σ δεW dVij ij
V
= ∫virtual power of momentum
δ ρ δPddt
u u dVj j
V
= ∫ ˙
virtual kinetic energy
δ δ ρE u u dVj j
V
= ∫ 12 ˙ ˙
virtual work of mass acceleration
δ δ δ ρ δB P E u u dVj j
V
= − = ∫ ˙
WMSReport0003.doc, Brocks, 27.08.02 - - 24
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 δ ε δεW C dVijkl kl ij
V
= ∫
W C dVijkl ij kl
V
= ∫12 ε ε elastic strain energy
• static problem (equilibrium)
δ δ δB P E= − = 0
WMSReport0003.doc, Brocks, 27.08.02 - - 25
Example: plane panel
LH
p
thickness = 1
( , )
( , ),
tboundary
=
=−
≤ ≤ =
t x y
t x y px
y x L y H
0
0
displacement field ( , )
( , )u =
u x y
u x yx
y
boundary conditions (b.c.) u y u yx y( , ) ( , )0 0 0= =
dimensionless coordinates ξ η= =xL
yH
;
global shape functions, ϕ ξ ηi ( , ), fulfilling b. c.
ϕ ξ ϕ ξ ϕ ξηϕ ξ ϕ ξ η ϕ ξη
1 22
3
43
52
62
= = == = =
, ,
, ,
Φ =
ϕ ϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ ϕ
1 2 3 4 5 6
1 2 3 4 5 6
0 0 0 0 0 0
0 0 0 0 0 0
WMSReport0003.doc, Brocks, 27.08.02 - - 26
series expansion of displacement field
u =
==
+=
∑
∑
α ϕ
α ϕα
i ii
i ii
1
6
61
6 Φ with .
.α
α
α
=
1
12
α is (12×1) matrix of unknowns
strain matrix
εεεγ
α α
∂∂∂∂
∂∂
∂∂
=
=+
= = =
xx
yy
xy
uxuy
uy
ux
x
y
x y
D u D BΦdifferential operator
D =
=
∂∂
∂∂
∂∂
∂∂
∂∂ξ
∂∂η
∂∂η
∂∂ξ
x
y
y x
L
H
H L
0
0
0
0
B =
1 2 3 2
2
2 1 2 3 2
2 2
2
2 2 2
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0
L L L L L L
H H H
H H H L L L L L L
ξ η ξ ξη η
ξ ξ ξη
ξ ξ ξη ξ η ξ ξη η
WMSReport0003.doc, Brocks, 27.08.02 - - 27
stress matrix and HOOKE's law
σσσσ
ε=
=xx
yy
xy
C
C stiffness matrix
for plane stress:σ zz = 0
C C=−
−( )
=E T
1
1 0
1 0
0 0 12
12
ν
νν
ν
for plane strain: ε zz = 0
( )( )
C C=+ −
−−
−( )
=E T
1 1 2
1 0
1 0
0 0 1 212
ν ν
ν νν ν
ν
WMSReport0003.doc, Brocks, 27.08.02 - - 28
virtual work of external forces
δ δ ξ δ αηξ
A LdT T= [ ] ===∫ t u f1
0
1
virtual work of stresses
δ σ δ ε ξ η α δ αηξ
W Ld HdT T= ===∫∫
0
1
0
1
K
"generalized" forces
f t( ) ( ) t ( )T T TLd L d
pL
= =
= − ( )
==
==
∫ ∫ξ ξ ξ ξ ξηξ
ηξ
Φ Φ1
0
1
1
0
1
12
13
12
14
13
120 0 0 0 0 0
global stiffness matrix
K B C B K= ===∫∫LH d dT T
ηξ
ξ η0
1
0
1
principle of virtual work
δ δ α δ αA W T T− = −( ) f K
as δ α is arbitrary
K fα =
linear system of equations for unknowns αi
WMSReport0003.doc, Brocks, 27.08.02 - - 29
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
δ δ ρ δ δ
∂
Π = − ⋅⋅ ( ) + ⋅ + ⋅ =∫ ∫ ∫S u b u t ugradB B B
dV dV dA 0
• 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
B = B
=k
k
K
1U
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 u xn n n N= =( ), ,...1
are the basic unknown parameters of the problem.
• A set of functions, ϕ ik( )( )ξ , i = 1, ...Nk, ξξξξ being local coordinates, is chosen to
WMSReport0003.doc, Brocks, 27.08.02 - - 30
define uniquely the state of displacement, ˜ ( )( )u k ξ , within each element, k, in terms of its nodal
displacements, uik( ) ,
˜ ( ) ( )( ) ( ) ( )u uki
kik
i
Nk
ξ ξ==∑ϕ
1
with u uik k
ik
ik
jk
ij( ) ( ) ( ) ( ) ( )˜ ,= ( ) ( ) =ξ ξϕ δ .
• The displacement functions uniquely define the state of deformation, i.e. some strain tensor,˜ ( )( )E k ξ , within each element in terms of the nodal displacements, e.g., small strain
˜ ( )( )( )
( )( )
( )Ex
ux
uk ik
ik i
k
ik
T
i
Nk
ξξ
ξξ
ξ= ⋅ ⋅ + ⋅ ⋅
=
∑12 1
∂ϕ∂
∂∂
∂ϕ∂
∂∂
,
with ∂∂
ξx
= JACOBIAN matrix.
• 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, tik( ) concentrated at the nodes equilibrating the boundary stresses,
˜ ˜( ) ( ) ( )t n Sk k k= ⋅ , is determined, resulting in a force-displacement or "stiffness" relationship for
each element.
t C uik
ijk
i
N
jk
k( ) ( ) ( )=
=∑
1
.
• Nodal displacements, uik( ) , nodal forces, ti
k( ) , and element stiffnesses, Cijk( ) , are
assembled according to the conditions of connectivity, u uik
ink
nn
N
A( ) ( )==
∑1
,
for all elements to compose the system of equations ensuring the conditions of compatibility
and equilibrium throughout.
˜ ( ) ˜ ( ) ( )( )u x u x x u= == =
∑ ∑k
k
K
n nn
N
1 1
ψ
where
˜ ( )( )( ) ( )
u x u xk ik
ik
i
N
k
k
= ∈
=∑ϕ
1
0
B
else, ψ ϕn i
kin
k
i
N
k
K
Ak
( ) ( ) ( )x ===∑∑
11
• 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 - - 31
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 f p 0n n n n N+ + = =, ,...,1
s S
xf b p tn
nn n n ndV dV dA= − ⋅ = =∫ ∫ ∫∂ψ
∂ρ ψ ψ
∂B 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 - - 32
Shape Functionsfor Orthogonal Elements
x
yu
x
y
a/2 ξ
b/2 η
1 2
34
bk
ak
xk
yk
element (k):
global coordinatesx a x x a
y b y y bk k k k
k k k k
− ≤ ≤ +− ≤ ≤ +
12
12
12
12
local coordinates: − ≤ ≤ +1 1ξ η,
ξ η= −( ) = −( )2 2a
x xb
y yk
kk
k,
WMSReport0003.doc, Brocks, 27.08.02 - - 33
4-Node Elements
"linear" plane elements: Nk = 4
shape functionsϕ ξ η δi j j ij( , ) = i, j = 1, ..., 4
ϕ ξ η ξ η
ϕ ξ η ξ η
ϕ ξ η ξ η
ϕ ξ η ξ η
114
214
314
414
1 1
1 1
1 1
1 1
( , )
( , )
( , )
( , )
= −( ) −( )= +( ) −( )= +( ) +( )= −( ) +( )
displacement field: ˆu u= Φ k
u x y
u x y
u
u
u
u
x
y
xk
yk
xk
yk
( , )
( , )
..
..
( )
( )
( )
( )
=
ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ
1 2 3 4
1 2 3 4
1
1
4
4
0 0 0 0
0 0 0 0
strain matrix
ˆ ˆε Φ=
= = =εεγ
xx
yy
xy
k kD u D u B u
u k nodal displacement matrix of element (k)
WMSReport0003.doc, Brocks, 27.08.02 - - 34
differential operator D =
=
∂∂
∂∂
∂∂
∂∂
∂∂ξ
∂∂η
∂∂η
∂∂ξ
x
y
y x
a
b
b a
k
k
k k
0
0
0
0
2
2
2 2
ˆ ˆ..
..u A u A = =
k k k
x
y
Nx
Ny
u
u
u
u
1
1
u global nodal displacement matrixA k (8 × 2N) incidence or connectivity matrix
N = total number of nodes2 N = number of degrees of freedom
ˆε = B A uk
stress matrix and HOOKE 's law
ˆσ ε=
= = σσσ
xx
yy
xy
k k kC C B A u
principle of virtual work
δ δA W− = 0
WMSReport0003.doc, Brocks, 27.08.02 - - 35
virtual work of external forces
δ δ δ δ∂ ∂
A ds dsT
S
T
S
kk
KT
k
= = =∫ ∫∑=
ˆ ˆ ˆt u t u f uΦ1
nodal forces ˆ f A t( )= ∫∑=
kT T
Sk
K
s dsk
Φ∂1
virtual work of stresses
δ δ δW dx dyT
y b
b
x a
a
k
KT
k
k
k
k
= ==−
+
=−
+
=∫∫∑ ˆ ˆσ ε
12
12
12
12
1
u K u
element stiffness matrix
K B C B Kkk k T
k kTa b
d d= ==−
+
=−
+
∫∫41
1
1
1
ξ ηηξ
global stiffness matrix
K A K A K= ==
∑ kT
kk
K
kT
1
δ δ δA W T T− = −( ) =ˆ ˆ ˆf u K u 0
as δˆu is arbitrary
ˆ ˆK u f=
linear system of equations for 2N unknowns u
WMSReport0003.doc, Brocks, 27.08.02 - - 36
1 2
1 2
34
1 2
34
2
1
4 6
3 5
x
y
global nodes (1), ..., (6)
elements [1], [2]
nodal coordinates (1): 0 , H
(2): 0 , 0
(3): L/2 , H
(4): L/2 , 0
(5): L , H
(6): L , 0
elements [1]: (2) (4) (3) (1)
[2]: (4) (6) (5) (3)
a L b H kk k= = =2 1 2, , ,
displacement b.c.: u u u ux y x y1 1 2 2 0= = = =
WMSReport0003.doc, Brocks, 27.08.02 - - 37
nodal displacements
ˆ
( )
( )
( )
( )
( )
( )
( )
( )
u k
xk
yk
xk
yk
xk
yk
xk
yk
u
u
u
u
u
u
u
u
=
1
1
2
2
3
3
4
4
, ˆu =
0
0
0
0
3
3
4
4
5
5
6
6
u
u
u
u
u
u
u
u
x
y
x
y
x
y
x
y
ˆ ˆu A u = k k
A =
2
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
WMSReport0003.doc, Brocks, 27.08.02 - - 38
element stiffness matrix (k = 1, 2)
K B C B KkT
k kTLH
d d= ==−
+
=−
+
∫∫81
1
1
1
ξ ηηξ
global stiffness matrix K A K A K= ==
∑ kT
k kk
T
1
2
element nodal forces
ˆ f t( )kT
ks dsk
= ∫ Φ∂B
pressure load ( )
t k p=
−
0
ξ
ˆ ˆf ( ) t( ) rkT
k k
Ld= +
==−
+
∫4 11
1
Φ ξ ξ ξη
ξ
or interpolation for arbitrary function )t(ξ
) ˆt(ξ η η= ==1 1Φ t k , ˆt k = (8×1) matrix
ˆ ˆ ˆf rkT
k k
Ld= ( )
+ =
=−
+
∫4 11
1
Φ Φη
ξ
ξ t
reaction forces (k = 1, only)
ˆr 1 1 1 4 40 0 0 0Tx y x yr r r r= ( )
global nodal forces ˆ ˆf A f= =
∑ kT
kk 1
2
WMSReport0003.doc, Brocks, 27.08.02 - - 39
system of equations
K K K K K K K K
K K K K K K K K
K K K K K K K K
K K K
1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8
2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8
3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8
4 1 4 2
0 0 0 0
0 0 0 0
0 0 0 0
, , , , , , , ,
, , , , , , , ,
, , , , , , , ,
, , 44 3 4 4 4 5 4 6 4 7 4 8
5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 5 10 5 11 5 12
6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 10
0 0 0 0, , , , , ,
, , , , , , , , , , , ,
, , , , , , , , , ,
K K K K K
K K K K K K K K K K K K
K K K K K K K K K K K66 11 6 12
7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 8 7 9 7 10 7 11 7 12
8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 10 8 11 8 12
9 5 9 60 0 0 0
, ,
, , , , , , , , , , , ,
, , , , , , , , , , , ,
, ,
K
K K K K K K K K K K K K
K K K K K K K K K K K K
K K K99 7 9 8 9 9 9 10 9 11 9 12
10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12
11 5 11 6 12 7 11 8 11 9 11 10 11 11 11 12
12 5 12 6 12 7 12
0 0 0 0
0 0 0 0
0 0 0 0
, , , , , ,
, , , , , , , ,
, , , , , , , ,
, , , ,
K K K K K
K K K K K K K K
K K K K K K K K
K K K K 88 12 9 12 10 12 11 12 12
3
3
4
4
5
5
6
6
0
0
0
0
K K K K
u
u
u
u
u
u
u
u
x
y
x
y
x
y
x
y, , , ,
=
+
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
5
1
1
2
2
f
f
f
r
r
r
r
y
y
y
x
y
x
y
WMSReport0003.doc, Brocks, 27.08.02 - - 40
Lagrangian Interpolation Polynomials
gin j
j ijj i
n( )
,
( )ξξ ξξ ξ
=−−=
≠
+
∏1
1
n = order, i = 1, ..., n+1
1D: "truss" elementn = 1 (linear):
1 2ξ2 nodes ξ ξ1 21 1= − = +;
i = 1: ϕ ξ ξ ξ1 11 1
2 1( ) ( )( )= = −( )g
i = 2: ϕ ξ ξ ξ2 21 1
2 1( ) ( )( )= = +( )g
n = 2 (quadratic):
1 23
ξ
3 nodes ξ ξ ξ1 2 31 1 0= − = + =; ; 1
i=1: ϕ ξ ξ ξ ξ ξ ξ1 12 1
212
12
21 1 1( ) ( )( )= = − −( ) = −( ) − −( )g
i=2: ϕ ξ ξ ξ ξ ξ ξ2 22 1
212
12
21 1 1( ) ( )( )= = + +( ) = +( ) − −( )g
i=3: ϕ ξ ξ ξ ξ ξ3 32 21 1 1( ) ( )( )= = +( ) −( ) = −g
Φ = ( )ϕ ϕ ϕ1 2 3
WMSReport0003.doc, Brocks, 27.08.02 - - 41
2D: quadrilateral elementn = 1 (linear) 4 nodes
ξ ηξ ηξ ηξ η
1 1
2 2
3 3
4 4
1 1
1 1
1 1
1 1
= − = −= + = −= + = += − = +
ξ
η
1 2
34
ϕ ξ η ϕ ξ ηϕ ξ η ϕ ξ η
ξ η ξ ηξ η ξ η
1 3
2 4
11
11
11
21
21
11
21
21
( , ) ( , )
( , ) ( , )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
=
g g g g
g g g g
=−( ) −( ) −( ) +( )+( ) −( ) +( ) +( )
14
14
14
14
1 1 1 1
1 1 1 1
ξ η ξ ηξ η ξ η
ˆu u= Φ k
Φ =
ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ
1 2 3 4
1 2 3 4
0 0 0 0
0 0 0 0
WMSReport0003.doc, Brocks, 27.08.02 - - 42
n = 2 (quadratic): 9 nodes
ξ ηξ ηξ ηξ ηξ η
5 5
6 6
7 7
8 8
9 9
0 1
1 0
0 1
1 0
0 0
= = −= + == = +
= − == =
ξ
η
1 2
34
5
6
7
8 9
ϕ ϕ ϕϕ ϕ ϕϕ ϕ ϕ
ξ η ξ η ξ ηξ η ξ η ξ
1 3 8
2 4 6
5 7 9
12
12
12
22
12
32
22
12
22
32
22
3
=
g g g g g g
g g g g g g
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) (( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
2
32
12
32
22
32
32
ηξ η ξ η ξ ηg g g g g g
... ... ...
... ... ...Φ =
ϕ ϕ ϕϕ ϕ
1 2 9
1 9
0 0
0 0 0
WMSReport0003.doc, Brocks, 27.08.02 - - 43
3D: "brick" element
ϕ ξ η ζ ξ η ζ
ϕ ξ η ζ ξ η ζ
ϕ ξ η ζ ξ η ζ
ϕ ξ η ζ
1 1 1 1
2 2 1 1
3 2 2 1
4 1
( , , ) ( ) ( ) ( )
( , , ) ( ) ( ) ( )
( , , ) ( ) ( ) ( )
( , , )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
=
=
=
=
g g g
g g g
g g g
g
n n n
n n n
n n n
n (( ) ( ) ( )
( , , ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
ξ η ζ
ϕ ξ η ζ ξ η ζ
g g
g g g
n n
n n n
2 1
5 1 1 2=etc.
n = 1 (linear) 8 nodes
ξ
η
5 6
78
34
1 2
ζ
... ... ...
... ... ...
... ... ...
Φ =
ϕ ϕϕ ϕ
ϕ ϕ
1 8
1 8
1 8
0 0 0 0
0 0 0 0
0 0 0 0
n = 2 (quadratic) 27 nodes
WMSReport0003.doc, Brocks, 27.08.02 - - 44
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 η2 2 vanishes
1 ξ ξ 2
η ξη ξ η2
η2 ξη2 ξ η2 2
⇒ 3D quadratic: 20 nodes
WMSReport0003.doc, Brocks, 27.08.02 - - 45
include only if node i is defined
ϕ1 =14 1 1−( ) −( )ξ η − 1
2 5ϕ − 12 8ϕ
ϕ2 =14 1 1+( ) −( )ξ η − 1
2 5ϕ − 12 6ϕ
ϕ3 =14 1 1+( ) +( )ξ η − 1
2 6ϕ − 12 7ϕ
ϕ4 =14 1 1−( ) +( )ξ η − 1
2 7ϕ − 12 8ϕ
ϕ5 =14
21 1−( ) −( )ξ η
ϕ6 =14
21 1+( ) −( )ξ η
ϕ7 =14
21 1−( ) +( )ξ η
ϕ8 =14
21 1−( ) −( )ξ η
Interpolation functions of four to eight variable-number-nodes for 2D element
WMSReport0003.doc, Brocks, 27.08.02 - - 46
General Isoparametric Elementscurvilinear distorted element
ξ
η
1 2
34
x
y
1
2
3
4
x f y f z fx y z= = =( , , ) ; ( , , ) ; ( , , )ξ η ζ ξ η ζ ξ η ζ
interpolation by shape functions: ˆx x= Ψ k
isoparametric elements: Ψ = Φ , ψ ϕi i=
2D: x =
x
yˆ ...
( )
( )
( )
( )
x k
k
k
Nk
Nk
x
y
x
y
=
1
1
3D: x =
x
y
z
ˆ
...
...
( )
( )
( )x k
k
k
k
x
y
z=
1
1
1
WMSReport0003.doc, Brocks, 27.08.02 - - 47
differential operator (2D) D =
∂∂
∂∂
∂∂
∂∂
x
y
y x
0
0
∂∂
∂∂ξ
∂ξ∂
∂∂η
∂η∂
∂∂ζ
∂ζ∂x x x x
= + +
∂∂ξ∂
∂η∂
∂ζ
∂∂ξ
∂∂ξ
∂∂ξ
∂∂η
∂∂η
∂∂η
∂∂ζ
∂∂ζ
∂∂ζ
∂∂∂∂∂∂
=
f f f
f f f
f f f
x
y
z
x y z
x y z
x y z
∂∂ξ
∂∂
=
Jx
⇒ ∂∂
∂∂ξx
J
=
− 1
J = JACOBI matrix, det J J= ≠ 0
2D: JJ
− =−
−
1 1∂∂η
∂∂ξ
∂∂η
∂∂ξ
f f
f f
y y
x x
J = −∂∂ξ
∂∂η
∂∂ξ
∂∂η
f f f fx y y x
∂∂ξ
∂ϕ∂ξ
fxx i
i
N
ik
k
==∑
1
( ), ... etc.
WMSReport0003.doc, Brocks, 27.08.02 - - 48
calculation of
strain matrix ˆ ˆε Φ= =D u B uk k
stress matrix ˆσ ε= =C C B uk k k
element stiffness matrix
K B C BkT
k dVk
= ∫∫∫B
transformation of the volume differential:
2D: dV BdA Bdx dy B d d= = = J ξ η(B = thickness)
3D: dV dx dy dz d d d= = J ξ η ζ
condition: J > 0 , J positive definite
element stiffness matrix
K B C B JkT
k d d d==−
+
=−
+
=−
+
∫∫∫ ξ η ζζηξ 1
1
1
1
1
1
integration numerically by GAUSS quadrature
global stiffness matrix
K A K A==
∑ kT
kk
K
k1
WMSReport0003.doc, Brocks, 27.08.02 - - 49
GAUSS Quadrature
a function f(ξ) is integrated approximately by a
weighted sum of its values at sampling points ξj
f d w fj jj
n
( ) ( )ξ ξ ξ−
+
=∫ ∑=1
1
1
the formula integrates a polynomial of order (2n-1)
exactly
sampling points (GAUSS points):
number
n
coordinates
ξj
weight
wj
1 0.000 000 2.000 000
2 ±0.577 350 1.000 000
3 ±0.774 597
0.000 000
0.555 556
0.888 889
4 ±0.861 136
±0.339 981
0.347 855
0.652 145
2D: f d d w w fj k j kk
n
j
n
( ) ( , )ξ ξ η ξ ηξη =−
+
=−
+
==∫∫ ∑∑=
1
1
1
1
11
WMSReport0003.doc, Brocks, 27.08.02 - - 50
Nonlinear FE Analyses
nonlinear structural behavioura) 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
time
t A t W t
t t A t t W t t
: ( ) ( )
: ( ) ( )
δ δδ δ
− =+ + − + =
0
0∆ ∆ ∆
f t t f t f tt
( ) ( ) ˙+ ≈ +∆ ∆
t t t→ + ∆ : δ δ∆ ∆A W− = 0
FE formulation:
) K( u u f∆ ∆=
WMSReport0003.doc, Brocks, 27.08.02 - - 51
(a) geometrical non-linearity
0rtr
tu0
t =0 t
O
t+∆t
t+∆tr
∆u
0ut+∆t
configuration t t t t t+ += + = +∆ ∆ ∆r r u r u00
total displacement 0 0t t t+ = +∆ ∆u u u
total strain (GREEN's quadratic strain)
000
00
00
00
0 0
12
t tt t t t t t
t t t t
++ +
+ +
=
+
+
⋅
= +
∆∆ ∆
∆ ∆
Eu
r
u
rur
ur
E G
G
T T
( ) ∂∂
∂∂
∂∂
∂∂
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 - - 52
Updated LAGRANGEan Formulation
reference state t
tt t
tt
tt
t+ = + = +∆ ∆ ∆S S S T S
S = 2nd PIOLA-KIRCHHOFF stresses
T = CAUCHY stresses
tt t
t t+ = ≈∆ ∆ ∆E E EG G( ) ( )
virtual work during t t t→ + ∆ (surface tractions only):
δ δ δ∂ ∂
∆ ∆ ∆∆ ∆A dA dAtt t
tt t
V Vt t
= ⋅ = ⋅+ +∫ ∫t u t u
δ δ
δ δ
∆ ∆
∆ ∆ ∆
∆ ∆W dV
dV dV
tt t
tt t
V
tt
t
V
t t
V
t
t t
= ⋅⋅
= ⋅⋅ + ⋅⋅
+ +∫
∫ ∫
S E
S E S E
G( )
∆ ∆t u S E⋅ − ⋅⋅ = − =∫ ∫δ δ δ δ∂
t
V
tt
t
V
dA dV A t W tt t
( ) ( ) 0
∆ ∆ ∆ ∆t t
V
t
V
dV dAt t
S E t u u⋅⋅ − ⋅ +( ) =∫ ∫δ δ∂
0
constitutive relation: ∆ ∆t t tS C E= ⋅⋅(incrementally linear)
t ) K( u u f∆ ∆ ∆=
nonlinear system of equations
WMSReport0003.doc, Brocks, 27.08.02 - - 53
⇒ iterative solution (e.g. NEWTON)
∆ ∆ u u K r(j) (j)
= ==
−
=∑ ∑j
i
tj
i
1
1
1
residual force (out-of balance force)
( )r f f u u(j) (j)
= − ++t t t∆ ∆convergence criterion
r RTOL and / or u DTOL(i) (i)
≤ ≤∆
f
u
∆u(1)
∆u(2)
∆u
∆ft f
t+ ∆tf
tu t+ ∆tu
r(1)
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(b) constitutive non-linearity
elastic- plastic material behaviour(VON MISES, PRANDTL, REUSS)
1. uniaxial tensile test
RF
σ
εεF
εe
εp
loading
unloading
yield condtion: σ ε≤ =R R Rp( ) , ( )0 F
HOOKE's law: σε σ
ε ε σ=≤
−( ) >
E R
E R
for
forF
pF
loading condtion:˙ , ˙
˙ , ˙
σ εσ ε
> >< =
0 0
0 0
p
p
loading
unloading
WMSReport0003.doc, Brocks, 27.08.02 - - 55
2. general (multiaxial) stress state
yield condtion: σ ε≤ R p( ) ; σ σ σ= ′ ′32 ij ij
VON MISES equivalent (effective) stress
′ = −σ σ σij ij kk13 deviatoric stresses
HOOKE's law:
˙˙ ( )
˙ ˙ ( )σ
ε νν
ε δ σ
ε ε νν
ε δ σij
ij kk ij
ij ijp
kk ij
G R
G R=
+−
≤
−( ) +−
>
21 2
21 2
for
for
F
F
loading condtion:′ > >′ < =
σ σ εσ σ ε
ij ij ijp
ij ij ijp
˙ , ˙
˙ , ˙
0 0
0 0
loading
unloading
flow rule ˙˙ ˙
ε εσ
σ σσ
σijp
p
ij p ijE= ′ = ′3
232
EE E
E EEp =
−=t ,
d
tt d
σε
σ ε σ ε ε ε εij ijp p p
ijp
ijp˙ ˙ ˙ ˙ ˙= ⇒ = 2
3
ε ijp is deviatoric, i.e. εkk
p = 0
"true" stress-strain curve required for UL formulation:
true stresses σ = FA
vs logarithmic strain ε = lnLL0
WMSReport0003.doc, Brocks, 27.08.02 - - 56
LH
p
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
1 2
1 2
34
1 2
34
2
1
4 6
3 5
x
y
WMSReport0003.doc, Brocks, 27.08.02 - - 57
1.1 Calculate the elastic stiffness matrix
C =
C C C
C C C
C C C
11 12 13
12 22 23
13 23 33
for plane stress conditions.
1.2 Calculate the (symmetric) element stiffness matrix for one element k
K B C Bkk k T
k
a bd d=
=−
+
=−
+
∫∫41
1
1
1
ξ ηηξ
1.3 Calculate the global stiffness matrix (see table on page 3)
K A K A==
∑ kT
kk
K
k1
1.4 Calculate the column matrix of nodal forces (see table on page 3)
f A t= ∫∑=
k
T T
Sk
K
dsk
Φ∂1
.
2. 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 uy 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 - - 58
analytical solution (theory of bending)
bending stress
σ xxz
z
x yM x
Iy
H( , )
( )= − −
2
M xpBL x
LxLz ( ) = − −
+
2 2
21 2
σ σ σxx xx xy H xx x
y
z
z
M
Wmax ( )
= = = ===
==
0 00
01200 MPa
section modulus W IH
BHz z= = =2
610
6
2 4
mm3
deflection1. simple bending (no shear)
u xpBL
EIxL
xL
xLy
b
z
( ) = −
−
+
4 2 3 4
246 4
u u LpBL
EIyb
yz
,max ( ) .= = =4
81 10 mm
2. shear
u xpLGH
xL
xLy
s ( ) = −
−
2 2
22
u u LpLGHy
sys,max ( ) .= = =
2
20 24 mm
total u u uy yb
ysmax ,max ,max .= + = 1 34 mm
WMSReport0003.doc, Brocks, 27.08.02 - - 59
0.0 0.2 0.4 0.6 0.8 1.0-15
-10
-5
0
x / L
u y /
L *
(E /
p)
bendingsheartotal
rectangular panel
exercise #1:
WMSReport0003.doc, Brocks, 27.08.02 - - 60
exercise #2:
biaxially loaded panel with circular hole
x
y
px
py
px
py
2 H
2 H
R
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 - - 61
analytical solution for "infinite" plate: R << H,
uniaxial tension (β = 0)
σ θ θ
σ θ θ
σ θ
θθ
θ
rr y
y
r y
r pRr
Rr
Rr
r pRr
Rr
r p
( , ) cos
( , ) cos
( , )
= −
− −
+
= +
+ +
=
12
2 2 4
12
2 4
12
1 1 4 3 2
1 1 3 2
11 3 3 22 4
+
−
Rr
Rr
sin θ
x
y
r
θ
1 2 3 4-1
0
1
2
3
r / R
σ ij /
py
σrr (θ=0)
σθθ (θ=0)
σrr (θ=π/2)
σθθ (θ=π/2)
WMSReport0003.doc, Brocks, 27.08.02 - - 62
FE solution: symmetry conditions ⇒1/4 model
x
y
py
px
boundary conditions:
u uy y x x= == =0 0
0 0;
WMSReport0003.doc, Brocks, 27.08.02 - - 63
Notation30.10.99
tensors, vektors, skalars - general
scalar latin or greek, small or capital (italics) a , H, α ,
vector
symbolic latin or greek, small,
bold or underlined x , x , σσσσn , σ n
indexed latin or greek, small (italics) xi , α i
x = xi ei , x = xi ei
2nd order tensor
symbolic latin or greek, capital,
bold or underlined T , T
indexed 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,
bold with overhead <4> C< >4
or double underlined C
indexed latin or greek, capital Cijkl
C< >4
= Cijkl ei ej ek el , C = Cijkl ei ej ek el
WMSReport0003.doc, Brocks, 27.08.02 - - 64
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 or α = B : C ...
T = C< >4
⋅⋅E = Cijkl Ekl ei ej or S = C : E ...
tensorial product
T = a b = ai bj ei ej
C< >4
= A B = Aij Bkl ei ej ek el
a b⋅c = (a b)⋅c = a (b⋅c)
spatial derivatives
NABLA operator ∇ = eiix
∂∂
v = grad ϕ = ∇∇∇∇ ϕ = ∂ϕ∂xi
ei = ϕ,i ei
F = grad v = ∇∇∇∇ v = ∂∂
v
xi
j
ei ej = vi,j ei ej
a = div v = ∇∇∇∇⋅v = = vi,i = ∂∂
v
xi
i
w = div T = ∇∇∇∇⋅T = ∂∂T
xij
i
ej = Tij,i ej
convention of summation is used in all cases
A B A Bij jk ij jkj
==
∑1
3
, C Ckk kkk
==
∑1
3
, A B A Bij ji ij jiji
===
∑∑1
3
1
3
WMSReport0003.doc, Brocks, 27.08.02 - - 65
special tensors and invariants
2nd order unit tensor I = δij ei ej = ei ei
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 J212
2( ) tr trT T T= −( )2
3rd invariant (determinant) of a tensor J3(T) = det (T)
WMSReport0003.doc, Brocks, 27.08.02 - - 66
tensors used in continuum mechanics
deformation gradient F = I + grad u = (δij + ui,j ) ei ej
GREEN's strain tensor E F F I e e( ) T, , , ,
Gi j j i i k k j i ju u u u= ⋅ −( ) = + +( )1
212
linear strain tensor E u u e e e e E= −( ) = +( ) = =12
12grad grad , ,
T u ui j j i i j ij i jTε
strain rate tensor D e e v v= = −( )dij i j12 grad gradT
for small strains D E e e= =˙ ε ij i j
elastic and plastic part ˙ ˙ ˙ ˙ ˙E E E e e= + = +( )e p e pε εij ij i j
accumulated effective ("equivalent") plastic strain
ε ε ε ε τ τep p p p= =∫ ∫p
ij ij
t t
d J d23
0
43 2
0
˙ ˙ ( ˙ )E
CAUCHY ("true") stress tensor S = σij ei ej = ST = σji ei ej
hydrostatic stress σ σh = = =13
13
13 1kk Jtr ( )S S
VON MISES effective ("equivalent") stress
σ σ σ σe = = ′ = ′ ′3 232J ij ij( )S
2nd PIOLA KIRCHHOFF stress tensor T F F S F T= ⋅ ⋅ =− −det( ) 1 T T
for small strains T S=
WMSReport0003.doc, Brocks, 27.08.02 - - 67
plastic yielding
yield curve (uniaxial tensile test) R(εp)
with ε ε ε σp e
p= = −E
yield strength (at initial plastic flow) R0 = R(0)
especially
lower yield strength ReL
0.2% proof stress Rp0.2
plastic potential, flow potential (VON MISES)
isotropic hardening Φ = − = ′ ′ − = ′ ⋅ ⋅ ′ −σ ε σ σ ε εe p p p2 2 3
22 2R R Rij ij( ) ( ) ( )S S
kinematic hardening Φ = − = ′ − ′( ) ⋅ ⋅ ′ − ′( ) −σ e2
02 3
2 02R RS X S X
with X = "back stress" tensor
associated flow rule
˙ ˙ε λ ∂∂σij
ij
p =′
Φ , ˙ ˙ES
p =′
λ ∂∂Φ
WMSReport0003.doc, Brocks, 27.08.02 - - 68
matrix notation
general: (n×m) matrix: A or A
n = number of rows
m = number of columns
. . .
. . . . .
. . . . .
. . .
A =
A A
A A
m
n nm
11 1
1
. .
. . . .
. . . .
. . . .
. .
A T
n
m nm
A A
A A
=
11 1
1
special: (n×1) matrix u or u
.u =
u
un
1
.u Tnu u= ( )1
the elements of a matrix do not form the components of a tensor or vector, in general
products
u vTi i
n
u v= = ∑α1
u and v: (n×1) ; α scalar
. . .
. . . . .
. . . . .
. . .
u v AT
m
n n m
u v u v
u v u v
= =
1 1 1
1
u: ( n×1) , v: (m×1) ; A: ( n×m)
....v A u= =
=
=
∑
∑
A v
A v
i ii
m
ni ii
m
11
1
A: ( n×m) ; u: ( m×1) , v: (n×1)
v u AT T T= A T : (m×n) ; u T : (1×m) , v T : (1×n)
. .
.... . . ....
. .
C A B= =
= =
= =
∑ ∑
∑ ∑
A B A B
A B A B
i ii
m
i imi
m
ni ii
m
ni imi
m
1 11
11
11 1
A: ( n×m) ; B: ( m×p) ; C: ( n×p)
WMSReport0003.doc, Brocks, 27.08.02 - - 69
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 ofelectric current
Ampere A
amount of substance Mol mol
luminous intensity Candela cd
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Decimal Parts and Multiples of SI Units
Parts and multiples of SI units which are generated bymultiplication with the factors
10±1, 10±2, 10±3n (n = 1, 2, ...)have special names and symbols.They are composed by prefixes.
faktor prefix symbol
10-15 femto f10-12 pico p10-9 nano n10-6 micro µ10-3 milli m
10-2 centi c10-1 deci d101 deca da102 hecto h
103 kilo k106 mega M109 giga G1012 tera T
Multiples by the factors10±3n (n = 1, 2, ...)
are to be preferred!
WMSReport0003.doc, Brocks, 27.08.02 - - 71
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 relation
name 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 1 J = 1 N m
power, heat flow Watt W 1 W = 1 J / s
electric charge, quantity of electricity Coulomb C 1 C = 1 A s
electric potential, voltage Volt V 1 V = 1 J / C
electric capacity Farad F 1 F = 1 C / V
electric resistance Ohm Ω 1 Ω = 1 V / 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
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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 - - 73
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
1
2
3 1
2
3
WMSReport0003.doc, Brocks, 27.08.02 - - 74
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 - - 75
3D FE mesh of a tubular joint under 3-point bending, having one
symmetry plane
32896HEXAEDERELEMENTE
Detail of the FE mesh of the semi-elliptical surface flaw in the weldment of
the tubular joint
ANSICHTSYMMETRIE-EBENE
Riss