Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
• X
⊗
aaaaaaaaa
bbbbbbbbb
CCCCCCCCC
⊗⊗⊗⊗⊗⊗⊗⊗⊗
AAAAAAAAA
bbbbbbbbb
CCCCCCCCC
tensor analysis
in cartesian coordinates
• literature
• motivation
• definition of tensors
basis vectors order of tensors (zero to four) identity tensors
• products of tensors
product with scalar dyadic product contraction of tensors
− single contraction− double contraction
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors
tensor analysis in cartesian coordinatesÜ 1
• X
⊗
aaaaaaaaa
bbbbbbbbb
CCCCCCCCC
⊗⊗⊗⊗⊗⊗⊗⊗⊗
AAAAAAAAA
bbbbbbbbb
CCCCCCCCC
tensor analysis
in cartesian coordinates
• tensor characteristics
symmetric and skew symmetric parts deviatoric and volumetric parts invariantes eigenvalues and eigenvectors coordinate transformation
• differentiation of tensors
derivations gradient & divergence total differential, variation & increment
• GAUSS integral theorem
• calculation rules
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors
tensor analysis in cartesian coordinatesÜ 2
• X
rotation e′′′i = Q3 · ei
-
-
-
-
6
rbe1
e2
e3 = e′′′3
e′′′1
e′′′2
γ
γ
e′′i = Q2 · e′′′i = Q2 ·Q3 · ei
-
-
--
-
-
-
6
rbe1
e2
e3
e′′1
e′′2e′′3
β
β
e′i= Q1 ·e′′i = Q1 ·Q2 ·Q3 · ei
-
---
-
---
-
-
6
rbe1
e2
e3
e′1
e′2
e′3α
α
• background: rotation of basis vectore′3 = Q1 ·Q2 · e3 about e1 and e2. coloredrepresentation of third component of e′3 asfunction of rotation angles α ∈ [0, 360o] andβ ∈ [0, 360o]. top: 3d rotation of basis vectorswith γ = 30o, β = −30o and α = 30o
• foreground: illustration of dyadic product oftensors
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors
information about title picture ’tensor analysis’Ü 3
• X
⊗
aaaaaaaaa
bbbbbbbbb
CCCCCCCCC
⊗⊗⊗⊗⊗⊗⊗⊗⊗
AAAAAAAAA
bbbbbbbbb
CCCCCCCCC
• general tenor analysis
BASAR&KRÄTZIG (1985)
BASAR&WEICHERT (2000)
BETTEN (1987)
DE BOER (1982), DE BOER (2000) (Appendix B)
FLÜGGE (1972)
IBEN (1995)
KLINGBEIL (1989)
LIPPMANN (1996)
SCHADE (1997)
TROSTEL (1993)
WRIGGERS (2001)
• tensor analysis in cartesian coordinates
HOLZAPFEL (2000): Nonlinear Solid Mechanics. AContinuum Approach for Engineering. John Wiley & Sons
BONET&WOOD (1997): Nonlinear Continuum Mechanics forFinite Element Analysis. Cambridge University Press
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors
tensor analysis - sourcesÜ 4
• X
components matrices tensors
linear strain measureε11 = u1,1
ε22 = u2,2
ε33 = u3,3
ε12 =1
2[u1,2+u2,1]
ε23 =1
2[u2,3+u3,2]
ε13 =1
2[u1,3+u3,1]
ε = Dεuwith
DTε =
∂
∂X1
0 0∂
∂X2
0∂
∂X3
0∂
∂X2
0∂
∂X1
∂
∂X3
0
0 0∂
∂X3
0∂
∂X2
∂
∂X1
ε=∇sym
u
=1
2[ui,j + uj,i] ei⊗ej
balance of momentum in linear continuum mechanics
ρ u1 = σ11,1 + σ12,2 + σ13,3 + ρ b1
ρ u2 = σ21,1 + σ22,2 + σ23,3 + ρ b2
ρ u3 = σ31,1 + σ32,2 + σ33,3 + ρ b3
ρ u = Dσσ + ρ b
with
Dσ=
∂
∂X1
0 0∂
∂X2
0∂
∂X3
0∂
∂X2
0∂
∂X1
∂
∂X3
0
0 0∂
∂X3
0∂
∂X2
∂
∂X1
ρ u= divσ + ρ b
= [σij,j + ρ bi] ei
NEUMANN boundary conditions in linear continuum mechanics
σ11 n1 + σ12 n2 + σ13 n3 = t?1
σ12 n1 + σ22 n2 + σ23 n3 = t?2
σ13 n1 + σ23 n2 + σ33 n3 = t?3
Dt σ = t?
with
Dt=
n1 0 0 n2 0 n3
0 n2 0 n1 n3 0
0 0 n3 0 n2 n1
σ · n = t?
σij nj ei = t?i ei
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.1 motivation
motivation - tensor format continuum mechanicsÜ 5
• X
3a
first order tensor
-6qa
e1
e2
-6qa e
′1
e′2
-
6qae ′′1
e ′′2
3a
a1
a2
-6qa e1
e2
3a a′1
a′2 -
6qa e′1
e′2
3a
-6qa
e1
e2
-6qa e
′1
e′2
• tensor is independent on coordinate system
a = ai ei = a′i e′i = a′′i e
′′i
• components of tensors depend on coordinate system
ai = a · ei a′i = a · e′i
• tensor components can be transformed betweencoordinate systems
a = a′i e′i = Q · [ai ei]
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
definition - properties of tensorsÜ 6
• X
--
-
e1
e2 e3
right hand rule
-
- ei
ϕi, Mi
right thumb rulewithout interest in mechanics
6--
e3
e2
e1rb6-
e2
e1rb6-
e3
e1rb6-tpde2
e1e3
6-tde3
e1e2
• three dimensional - cartesian basis, right-handed (dextral) and orthonormal system
base vectors e1, e2 and e3 are perpendicular basis vectors have length one ‖ei‖ = 1 for i ∈ [1, 3]
generating a right-handed system positive rotations ϕi or moments Mi using right thumb rule
• two dimensional - cartesian basis, right-handed (dextral) and orthonormal system
using e1-e2 plane or e1-e3 plane third base vector defines positive rotations ϕi or moments Mi
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinate systemsÜ 7
• X
--
-
e1
e2 e3
right hand rule
-
- ei
ϕi, Mi
right thumb rulewithout interest in mechanics
6--
e3
e2
e1rb6-
e2
e1rb6-
e3
e1rb6-tpde2
e1e3
6-tde3
e1e2
• three dimensional - cartesian basis, right-handed (dextral) and orthonormal system
base vectors e1, e2 and e3 are perpendicular basis vectors have length one ‖ei‖ = 1 for i ∈ [1, 3]
generating a right-handed system positive rotations ϕi or moments Mi using right thumb rule
• two dimensional - cartesian basis, right-handed (dextral) and orthonormal system
using e1-e2 plane or e1-e3 plane third base vector defines positive rotations ϕi or moments Mi
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinate systemsÜ 7
• X
e1
e2
e1-e2
plane
e2e3
e2-e3
plane
e1
e3
e1-e3
plane
e1
e2e3
spatialcoordinate
system
• thumb: e1
• forefinger: e2
• middle finger: e3
• spatial coordinate system
base vectors
e1, e2, e3
• planar coordinate systems
e1-e2-plane, base vectors
e1, e2
e2-e3-plane, base vectors
e2, e3
e1-e3-plane, base vectors
e1, e3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
spatial and planar coordinate systemsÜ 8
• X
e1
e2
e1-e2
plane
e2e3
e2-e3
plane
e1
e3
e1-e3
plane
e1
e2e3
spatialcoordinate
system
• thumb: e1
• forefinger: e2
• middle finger: e3
• spatial coordinate system
base vectors
e1, e2, e3
• planar coordinate systems
e1-e2-plane, base vectors
e1, e2
e2-e3-plane, base vectors
e2, e3
e1-e3-plane, base vectors
e1, e3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
spatial and planar coordinate systemsÜ 8
• X
e1
e2
e1-e2
plane
e2e3
e2-e3
plane
e1
e3
e1-e3
plane
e1
e2e3
spatialcoordinate
system
• thumb: e1
• forefinger: e2
• middle finger: e3
• spatial coordinate system
base vectors
e1, e2, e3
• planar coordinate systems
e1-e2-plane, base vectors
e1, e2
e2-e3-plane, base vectors
e2, e3
e1-e3-plane, base vectors
e1, e3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
spatial and planar coordinate systemsÜ 8
• X
e1
e2
e1-e2
plane
e2e3
e2-e3
plane
e1
e3
e1-e3
plane
e1
e2e3
spatialcoordinate
system
• thumb: e1
• forefinger: e2
• middle finger: e3
• spatial coordinate system
base vectors
e1, e2, e3
• planar coordinate systems
e1-e2-plane, base vectors
e1, e2
e2-e3-plane, base vectors
e2, e3
e1-e3-plane, base vectors
e1, e3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
spatial and planar coordinate systemsÜ 8
• X
-
-
-e1
e2
e3
cartesian base vectors andright hand rule
e1
e2e3
cartesian basis
representation of vectors
e1
e2
e3
e1
e2
e3
a1
a2
a3
a
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
basis vectors ei, indices i, j ∈ 1, 2, 3
‖ei‖ = 1, ei · ej =
0 for i 6= j
1 for i = j
representation of tensors (first order tensor = vector)
a =3∑i=1
ai ei = a1e1+a2e2+a3e3 = ai ei = aj ej
• EINSTEIN summation conventionaddition over repeated indices• summation indices (twice available, dummy)
can be replaced, e.g. i→ j
tensor components
ai = a · ei
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinates - basis vectorsÜ 9
• X
-
-
-
e1
e2
e3
cartesian base vectors andright hand rule
e1
e2e3
cartesian basis
representation of vectors
e1
e2
e3
e1
e2
e3
a1
a2
a3
a
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
basis vectors ei, indices i, j ∈ 1, 2, 3
‖ei‖ = 1, ei · ej =
0 for i 6= j
1 for i = j
representation of tensors (first order tensor = vector)
a =3∑i=1
ai ei = a1e1+a2e2+a3e3 = ai ei = aj ej
• EINSTEIN summation conventionaddition over repeated indices• summation indices (twice available, dummy)
can be replaced, e.g. i→ j
tensor components
ai = a · ei
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinates - basis vectorsÜ 9
• X
basis vectors
e1 =
1
0
0
e2 =
0
1
0
e3 =
0
0
1
products aiei
a1e1 =
a1
0
0
a2e2 =
0a2
0
a3e3 =
0
0a3
EINSTEIN summation
+
+
=
aiei=
a1
a2
a3
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
representation of first order tensors
a =3∑i=1
ai ei = a1e1+a2e2+a3e3 = ai ei
using basis vectors and tensor components
a = a1
1
0
0
+ a2
0
1
0
+ a3
0
0
1
=
a1
0
0
+
0
a2
0
+
0
0
a3
=
a1
a2
a3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinates - basis vectorsÜ 10
• X
basis vectors
e1 =
1
0
0
e2 =
0
1
0
e3 =
0
0
1
products aiei
a1e1 =
a1
0
0
a2e2 =
0a2
0
a3e3 =
0
0a3
EINSTEIN summation
+
+
=
aiei=
a1
a2
a3
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
representation of first order tensors
a =3∑i=1
ai ei = a1e1+a2e2+a3e3 = ai ei
using basis vectors and tensor components
a = a1
1
0
0
+ a2
0
1
0
+ a3
0
0
1
=
a1
0
0
+
0
a2
0
+
0
0
a3
=
a1
a2
a3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinates - basis vectorsÜ 10
• X
basis vectors
e1 =
1
0
0
e2 =
0
1
0
e3 =
0
0
1
products aiei
a1e1 =
a1
0
0
a2e2 =
0a2
0
a3e3 =
0
0a3
EINSTEIN summation
+
+
=
aiei=
a1
a2
a3
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
representation of first order tensors
a =3∑i=1
ai ei = a1e1+a2e2+a3e3 = ai ei
using basis vectors and tensor components
a = a1
1
0
0
+ a2
0
1
0
+ a3
0
0
1
=
a1
0
0
+
0
a2
0
+
0
0
a3
=
a1
a2
a3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.2 definition
cartesian coordinates - basis vectorsÜ 10
• X
illustration tensor order
3. order
2. order
1. order
0. order
zero,
first
, second and fourth
order tensors
a = a
a =
a1
a2
a3
= ai ei
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei ⊗ ej
A = = Aijkl ei ⊗ ej ⊗ ek ⊗ el
e.g. displacement-, stress- and constitutive tensors
u =
u1
u2
u3
= ui ei
σ =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
= σij ei ⊗ ej
C = = Cijkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.3 characterization
cartesian coordinates - basis vectorsÜ 11
• X
illustration tensor order
3. order
2. order
1. order
0. order
zero,
first, second
and fourth
order tensors
a = a
a =
a1
a2
a3
= ai ei
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei ⊗ ej
A = = Aijkl ei ⊗ ej ⊗ ek ⊗ el
e.g. displacement-, stress- and constitutive tensors
u =
u1
u2
u3
= ui ei
σ =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
= σij ei ⊗ ej
C = = Cijkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.3 characterization
cartesian coordinates - basis vectorsÜ 11
• X
illustration tensor order
3. order
2. order
1. order
0. order
zero,
first, second and fourth order tensors
a = a
a =
a1
a2
a3
= ai ei
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei ⊗ ej
A = = Aijkl ei ⊗ ej ⊗ ek ⊗ el
e.g. displacement-, stress- and constitutive tensors
u =
u1
u2
u3
= ui ei
σ =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
= σij ei ⊗ ej
C = = Cijkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.3 characterization
cartesian coordinates - basis vectorsÜ 11
• X
illustration tensor order
3. order
2. order
1. order
0. order
zero,first, second and fourth order tensors
a = a
a =
a1
a2
a3
= ai ei
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei ⊗ ej
A = = Aijkl ei ⊗ ej ⊗ ek ⊗ el
e.g. displacement-, stress- and constitutive tensors
u =
u1
u2
u3
= ui ei
σ =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
= σij ei ⊗ ej
C = = Cijkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.3 characterization
cartesian coordinates - basis vectorsÜ 11
• X
illustration tensor order
3. order
2. order
1. order
0. order
zero,first, second and fourth order tensors
a = a
a =
a1
a2
a3
= ai ei
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei ⊗ ej
A = = Aijkl ei ⊗ ej ⊗ ek ⊗ el
e.g. displacement-, stress- and constitutive tensors
u =
u1
u2
u3
= ui ei
σ =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
= σij ei ⊗ ej
C = = Cijkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.3 characterization
cartesian coordinates - basis vectorsÜ 11
• X
rb
rb
rb
a2
a1
a
b2
b1
b
c2c1
c
c = a+ b
6-
e2
e1rb
rbrb
rbrb
a2
a1
a
b2
b1
b
d2
d1
d
c2
c1
c
c = a+ b+ d
6-
e2
e1rb
addition of tensors• only tensors of same order can be added• first order tensors
c = a+ b = ai ei + bi ei = [ai + bi] ei =
a1 + b1
a2 + b2
a3 + b3
• second order tensors
A+B = [Aij +Bij ] ei ⊗ ej
=
A11 +B11 A12 +B12 A13 +B13
A21 +B21 A22 +B22 A23 +B23
A31 +B31 A32 +B32 A33 +B33
• calculation rules associative law
A+ [B +C] = [A+B] +C
commutative law
A+B = B +A
identical and inverse elements
A+ 0 = A, A+ [−A] = 0
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.4 addition
addition of tensorsÜ 12
• X
KRONECKER symbol
0
0
1
0
1
00
0
1δij
1
dyadic product
1⊗10
0
1
0
1
00
0
1
0
0
1
0
1
00
0
1
⊗
• KRONECKER symbol
δij = ei · ej =
0 for i 6= j
1 for i = j
• identity tensors
second order identity tensor 1
fourth order identity tensor I′
dyadic product 1⊗ 1
symmetric fourth order identity tensor I
.
1 = δij ei ⊗ ej
I′ = δikδjl ei ⊗ ej ⊗ ek ⊗ el
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ el
I =1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors 1,1⊗1,I ′ and IÜ 13
• X
KRONECKER symbol
0
0
1
0
1
00
0
1δij
1
dyadic product
1⊗10
0
1
0
1
00
0
1
0
0
1
0
1
00
0
1
⊗
• KRONECKER symbol
δij = ei · ej =
0 for i 6= j
1 for i = j
• identity tensors
second order identity tensor 1
fourth order identity tensor I′
dyadic product 1⊗ 1
symmetric fourth order identity tensor I
.
1 = δij ei ⊗ ej
I′ = δikδjl ei ⊗ ej ⊗ ek ⊗ el
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ el
I =1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors 1,1⊗1,I ′ and IÜ 13
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
repetition: KRONECKER symbol & fourth order identity tensors
δij = ei · ej =
0 for i 6= j
1 for i = j
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ elI =
1
2[δikδjl + δilδjk] ei ⊗ ej ⊗ ek ⊗ el
fourth order constitutive tensor C
C = 2µ I + λ 1⊗ 1 =[µ [δilδjk + δikδjl] + λ δijδkl
]ei ⊗ ej ⊗ ek ⊗ el
= Cijkl ei ⊗ ej ⊗ ek ⊗ el
developement of components C1111, C1122, C1112 and C1212
C1111 = µ [δ11δ11 + δ11δ11] + λ δ11δ11
= µ [1 · 1 + 1 · 1] + λ 1 · 1 = 2 µ+ λ
C1122 = µ [δ12δ12 + δ12δ12] + λ δ11δ22
= µ [0 · 0 + 0 · 0] + λ 1 · 1 = λ
C1112 = µ [δ12δ11 + δ11δ12] + λ δ11δ12
= µ [0 · 1 + 1 · 0] + λ 1 · 0 = 0
C1212 = µ [δ12δ21 + δ11δ22] + λ δ12δ12
= µ [0 · 0 + 1 · 1] + λ 0 · 0 = µ
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.5 fundamental tensors
identity tensors (examples)Ü 14
• X
first order tensors
rb
βa
2
βa1
βa
a2
a1
a
c 2
c1
αa
α < 1 < β
6-
e2
e1rb
characterization of products
• multiplication by a scalar
• dyadic product
• contraction
simple contraction double contraction
multiplication by a scalar
• first order tensors
α a = α ai ei = [α ai] ei =
α a1
α a2
α a3
• second order tensors
α A = α Aij ei ⊗ ej =
αA11 αA12 αA13
αA21 αA22 αA23
αA31 αA32 αA33
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.1 multiplication by scalars
products of tensorsÜ 15
• X
⊗
aaaaaaaaa
bbbbbbbbb
CCCCCCCCC
⊗⊗⊗⊗⊗⊗⊗⊗⊗
AAAAAAAAA
bbbbbbbbb
CCCCCCCCC
geometrical interpretationC = a⊗ b
a⊗ b = aibj ei ⊗ ej
C = A⊗ b
A⊗ b = Aijbk ei ⊗ ej ⊗ ek
dyadic products
• increasing order• symbol ⊗two first order tensors
a⊗ b = [ai ei]⊗ [ej bj ] = ai bj ei ⊗ ej
higher order tensors
C = Cijk ei ⊗ ej ⊗ ek =A ⊗ b
= Aij bk ei ⊗ ej ⊗ ekC = Cijkl ei⊗ej⊗ek⊗el =A ⊗ B
= Aij Bkl ei ⊗ ej ⊗ ek ⊗ el
e.g. consitutive tensor
C = 2µ I+ λ 1⊗ 1
with
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
cartesian coordinates - basis vectorsÜ 16
• X
⊗
aaaaaaaaa
bbbbbbbbb
CCCCCCCCC
⊗⊗⊗⊗⊗⊗⊗⊗⊗
AAAAAAAAA
bbbbbbbbb
CCCCCCCCC
geometrical interpretationC = a⊗ b
a⊗ b = aibj ei ⊗ ej
C = A⊗ b
A⊗ b = Aijbk ei ⊗ ej ⊗ ek
dyadic products
• increasing order• symbol ⊗two first order tensors
a⊗ b = [ai ei]⊗ [ej bj ] = ai bj ei ⊗ ej
higher order tensors
C = Cijk ei ⊗ ej ⊗ ek =A ⊗ b
= Aij bk ei ⊗ ej ⊗ ekC = Cijkl ei⊗ej⊗ek⊗el =A ⊗ B
= Aij Bkl ei ⊗ ej ⊗ ek ⊗ el
e.g. consitutive tensor
C = 2µ I+ λ 1⊗ 1
with
1⊗ 1 = δij δkl ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
cartesian coordinates - basis vectorsÜ 16
• X
⊗1
0
0
10
0
A11A11A11A11A11A11A11A11A11
0
00
0
0
0
0
0
e1
e1
A11e1⊗e1
+
+ · · · =
⊗1
0
0
01
0
0
0
00
0
A12A12A12A12A12A12A12A12A12
0
0
0
e1
e2
A12e1⊗e2
A11A11A11A11A11A11A11A11A11
A21A21A21A21A21A21A21A21A21
A31A31A31A31A31A31A31A31A31A12A12A12A12A12A12A12A12A12
A22A22A22A22A22A22A22A22A22
A32A32A32A32A32A32A32A32A32
A13A13A13A13A13A13A13A13A13
A23A23A23A23A23A23A23A23A23
A33A33A33A33A33A33A33A33A33
Aijei⊗ej
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
representation of second order tensors
A =3∑i=1
3∑j=1
Aij ei⊗ej = Aij ei⊗ej
= A11 e1⊗e1 + A12 e1⊗e2 + A13 e1⊗e3
+ A21 e2⊗e1 + A22 e2⊗e2 + A23 e2⊗e3
+ A31 e3⊗e1 + A32 e3⊗e2 + A33 e3⊗e3
using basis vectors and tensor components
A = A11
1 0 0
0 0 0
0 0 0
+A12
0 1 0
0 0 0
0 0 0
+A13
0 0 1
0 0 0
0 0 0
+ · · ·
=
A11 A12 A13
A21 A22 A23
A31 A32 A33
= Aij ei⊗ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product of base vectorsÜ 17
• X
10
0
⊗1
0
0
10
0
1
0
00
0
0
0
0
0
A111A111A111A111A111A111A111A111A111
e1
e1
e1
A111e1⊗e1⊗e1
10
0
⊗0
1
0
00
1
0
0
00
0
0
0
1
0A231A231A231A231A231A231A231A231A231
e2
e3
e1
A231e2⊗e3⊗e1
Aijkei⊗ej⊗ek
+ · · ·+
+ · · · =
cartesian basis - right-handed (dextral) and orthonormal system
e1 =
1
0
0
, e2 =
0
1
0
, e3 =
0
0
1
representation of third order tensors
A =3∑i=1
3∑j=1
3∑k=1
Aijk ei⊗ej⊗ek
= Aijk ei⊗ej⊗ek= A111 e1⊗e1⊗e1 +A112 e1⊗e1⊗e2 + · · ·
representation of fourth order tensors
A =3∑i=1
3∑j=1
3∑k=1
3∑l=1
Aijkl ei⊗ej⊗ek⊗el
= Aijkl ei⊗ej⊗ek⊗el= A1111 e1⊗e1⊗e1⊗e1
+ A1112 e1⊗e1⊗e1⊗e2 + · · ·
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product of base vectorsÜ 18
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej
= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej
= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
e.g. dyadic product of basis tensors ei
e1⊗e1 =
1
0
0
⊗1
0
0
=
1
0
0
[1 0 0
]=
1 0 0
0 0 0
0 0 0
e2⊗e3 =
0
1
0
⊗0
0
1
=
0
1
0
[0 0 1
]=
0 0 0
0 0 1
0 0 0
detailed calculation of a dyadic product
a⊗ b = [ai ei]⊗ [ej bj ] = aibj ei⊗ej= a1bj e1⊗ej + a2bj e2⊗ej + a3bj e3⊗ej= a1b1 e1⊗e1 + a1b2 e1⊗e2 + a1b3 e1⊗e3 + a2b1 e2⊗e1 + · · ·+ a3b3 e3⊗e3
= a1b1
1 0 0
0 0 0
0 0 0
+ a1b2
0 1 0
0 0 0
0 0 0
+ a1b3
0 0 1
0 0 0
0 0 0
+ a2b1
0 0 0
1 0 0
0 0 0
+ · · ·+ a3b3
0 0 0
0 0 0
0 0 1
=
a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3
=
C11 C12 C13
C21 C22 C23
C31 C32 C33
= Cij ei⊗ej
with
Cij = aibj
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.2 dyadic product
dyadic product ⊗ (example)Ü 19
• X
first order tensors
simple contraction
rba
b
ϑ ·a · b = |a| |b| cosϑ
·aaaaaaaaa
ccccccccc
bbbbbbbbb
·
AAAAAAAAA
ccccccccc
bbbbbbbbb
contractive products→ reduction of order
• simple contraction, symbol ·• double contraction, symbol :
simple contraction
a ·b =aiei ·ejbj =ai δijbj =aibi
A ·b =Aijei⊗ej ·ekbk =Aijei δjkbk=Aijbj ei
A ·B=Aijei⊗ej ·ek⊗elBkl= =AijBjl ei⊗el
double contraction
A :B=Aij [ei⊗ej ] : [ek⊗el]Bkl=Aij δikδjlBkl
=Aij Bij
requirements of present course
c = = a · b = ai bi
c = =A :B = Aij Bij
c = ci ei =A · b = Aij bj ei
C = Cij ei ⊗ ej =A ·B = Aik Bkj ei ⊗ ejC = Cij ei ⊗ ej = A :B = Aijkl Bkl ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.3 contraction
contraction of tensors · and :Ü 20
• X
first order tensors
simple contraction
rba
b
ϑ ·a · b = |a| |b| cosϑ
·aaaaaaaaa
ccccccccc
bbbbbbbbb
·
AAAAAAAAA
ccccccccc
bbbbbbbbb
contractive products→ reduction of order
• simple contraction, symbol ·• double contraction, symbol :
simple contraction
a ·b =aiei ·ejbj =ai δijbj =aibi
A ·b =Aijei⊗ej ·ekbk =Aijei δjkbk=Aijbj ei
A ·B=Aijei⊗ej ·ek⊗elBkl= =AijBjl ei⊗el
double contraction
A :B=Aij [ei⊗ej ] : [ek⊗el]Bkl=Aij δikδjlBkl
=Aij Bij
requirements of present course
c = = a · b = ai bi
c = =A :B = Aij Bij
c = ci ei =A · b = Aij bj ei
C = Cij ei ⊗ ej =A ·B = Aik Bkj ei ⊗ ejC = Cij ei ⊗ ej = A :B = Aijkl Bkl ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.3 contraction
contraction of tensors · and :Ü 20
• X
first order tensors
simple contraction
rba
b
ϑ ·a · b = |a| |b| cosϑ
·aaaaaaaaa
ccccccccc
bbbbbbbbb
·
AAAAAAAAA
ccccccccc
bbbbbbbbb
contractive products→ reduction of order
• simple contraction, symbol ·• double contraction, symbol :
simple contraction
a ·b =aiei ·ejbj =ai δijbj =aibi
A ·b =Aijei⊗ej ·ekbk =Aijei δjkbk=Aijbj ei
A ·B=Aijei⊗ej ·ek⊗elBkl= =AijBjl ei⊗el
double contraction
A :B=Aij [ei⊗ej ] : [ek⊗el]Bkl=Aij δikδjlBkl
=Aij Bij
requirements of present course
c = = a · b = ai bi
c = =A :B = Aij Bij
c = ci ei =A · b = Aij bj ei
C = Cij ei ⊗ ej =A ·B = Aik Bkj ei ⊗ ejC = Cij ei ⊗ ej = A :B = Aijkl Bkl ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.3 contraction
contraction of tensors · and :Ü 20
• X
first order tensors
simple contraction
rba
b
ϑ ·a · b = |a| |b| cosϑ
·aaaaaaaaa
ccccccccc
bbbbbbbbb
·
AAAAAAAAA
ccccccccc
bbbbbbbbb
contractive products→ reduction of order
• simple contraction, symbol ·• double contraction, symbol :
simple contraction
a ·b =aiei ·ejbj =ai δijbj =aibi
A ·b =Aijei⊗ej ·ekbk =Aijei δjkbk=Aijbj ei
A ·B=Aijei⊗ej ·ek⊗elBkl= =AijBjl ei⊗el
double contraction
A :B=Aij [ei⊗ej ] : [ek⊗el]Bkl=Aij δikδjlBkl
=Aij Bij
requirements of present course
c = = a · b = ai bi
c = =A :B = Aij Bij
c = ci ei =A · b = Aij bj ei
C = Cij ei ⊗ ej =A ·B = Aik Bkj ei ⊗ ejC = Cij ei ⊗ ej = A :B = Aijkl Bkl ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.3 contraction
contraction of tensors · and :Ü 20
• X
strain vector ε and strain tensor ε
ε =[ε11 ε22 ε33 2ε12 2ε23 2ε13
]T ε =
ε11 ε12 ε13
ε22 ε23
ε33sym
@@@@@R
@@@R
@@R
stress vector σ and stress tensor σ
σ =[σ11 σ22 σ33 σ12 σ23 σ13
]T σ =
σ11 σ12 σ13
σ22 σ23
σ33sym
@@@@@R
@@@R
@@R
specific strain energy
2W = σ : ε = σij εij
= σ1j ε1j + σ2j ε2j + σ3j ε3j
= σ11 ε11 + σ12 ε12 + σ13 ε13 + σ21 ε21 + σ22 ε22 + σ23 ε23
+ σ31 ε31 + σ32 ε32 + σ33 ε33
= σ11 ε11 + σ22 ε22 + σ33 ε33
+ [σ12 ε12 + σ21 ε21] + [σ23 ε23 + σ32 ε32] + [σ13 ε13 + σ31 ε31]
= σ11 ε11 + σ22 ε22 + σ33 ε33 + 2 σ12 ε12 + 2 σ23 ε23 + 2 σ13 ε13
= σ · ε
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.3 contraction
examples of productsÜ 21
• X
scalar dyadic single doubleproduct product contraction contraction
associative law
α[βA] = [αβ]A αa⊗b = a⊗[αb] αA·b = A·[αb] αA : B = A : [αB]
distributive
[α+β]A = αA+βA
α[A+B] = αA+αB
A⊗[b+c] = A⊗b+A⊗c
[A+B]⊗c = A⊗c+B⊗c
A·[b+c] = A·b+A·c[A+B]·c = A·c+B ·c
A : [B+C] = A : B+A : C
[A+B] : C = A : C+B : C
commutative law
αA = Aα
identical element
1A = A 1·a = a I : A = A
inverse element
1A + [−1]A = 0
zero element
0⊗a = 0 0·a = 0 O : A = 0
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.6 products • e1.6.4 calculation laws
products of tensorsÜ 22
• X
symmetric & skew symmetric
+ −
A11
A21
A31
A12
A22
A32
A13
A23
A33A
A11
A12
A13
A21
A22
A23
A31
A32
A33AT
seco
ndor
dert
enso
rtr a
nspo
sted
tens
or
• transposition of a tensor
A = Aij ei ⊗ ej
A = Aijkl ei⊗ej⊗ek⊗el
AT = Aji ei⊗ej
AT = Ailjk ei⊗ej⊗ek⊗el
• symmetric and skew symmetric tensors
A =Asym +Askw =1
2
[A+AT
]+
1
2
[A−AT
]A = Asym + Askw =
1
2
[A + AT
]+
1
2
[A−AT
]
• transposition of a dyadic product
[a⊗ b]T = b⊗ a
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.1 symmetric tensors
transposition of a tensorÜ 23
• X
• additive decomposition of a second order tensor in volumentric and deviatoric parts
A = Avol +Adev
• volumetric tensor (three dimensional and two dimensional)
Avol =1
3[A : 1] 1 Avol =
1
2[A : 1] 1
• deviatoric tensor (three dimensional and two dimensional)
Adev = A−1
3[A : 1] 1 Adev = A−
1
2[A : 1] 1
• calculations rules
tr[Avol] = tr[A] tr[Adev] = 0
• general formulation of deviatoric tensor in ND spatial dimensions
Avol =1
ND[A : 1] 1
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.2 deviator
volumetric deviatoric decompositionÜ 24
• X
invariants of second order tensors
• invariants are scalar valued quantities of a tensor which are independent on the coordinatesystem• invariants of three and two dimensional second order tensors
IA = tr[A]
IIA =1
2
[tr2[A]− tr[A ·A]
]IIIA = det[A] = |A|
IA = tr[A]
IIA = det[A] = |A|
• trace of a tensor
tr[A] = A : 1 = Aii = A11 +A22 +A33
• determinant of a tensor
three dimensional
det[A] = |A| = A11A22A33 +A21A32A13 +A31A12A23
− A11A23A32 −A22A31A13 −A33A12A21
two dimensional
det[A] = |A| = A11A22 −A21A12
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.3 invariants
invariants of tensorsÜ 25
• X
calculation rules for invariants of tensors
• trace of a tensor
tr[1] = 3
tr[AT ] = tr[A]
tr[A ·B] = tr[B ·A]
tr[αA+ βB] = αtr[A] + βtr[B]
tr[A ·BT ] = tr[A ·B]
tr[A] = tr[A · 1] = A : 1
• determinant of a tensor
det[1] = 1
det[AT ] = det[A]
det[A ·B] = det[A] det[B]
det[αA] = α3 det[A]
det[a⊗ b] = 0
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.3 invariants
invariants of tensorsÜ 26
• X
• second order tensors with i, j ∈ [1, 3]
A = Aij ei ⊗ ej
• eigenvalue problem of arbitrary second order tensors
[A− λ 1] ·Φ = 0
• eigenvalues (non-trivial solutions)
det [A− λ 1] = 0
• characteristic polynomial, three dimensional i, j ∈ [1, 3], invariants of tensors A
λ3 − IA λ2 + IIA λ− IIIA = 0 λ1|2|3 = . . .
• characteristic polynomial, two dimensional i, j ∈ [1, 2], invariants of tensors A
λ2 − IA λ+ IIA = 0 λ1|2 =IA
2±
√I2A
4− IIA
• yields ND (spatial dimension) eigenvalues λi• related eigenvectors Φi are solutions of linear systems of equations
[A− λi 1] ·Φi = 0
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.4 eigenvalues
eigenvalue problem of tensorsÜ 27
• X
• general quadratic characteristic polynomial
λ2 + 2p λ+ q = 0
• two solutions
λ1|2 = −p±√D
• discriminant
D = p2 − q
• characterization of solution properties
D > 0: two real solutions D = 0: one double real solution D < 0: two conjugate complex solutions
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.4 eigenvalues
zero values of quadratic polynomialsÜ 28
• X
• general cubic characteristic polynomial
λ3 + aλ2 + bλ+ c = 0
• substitution λ = λ+ a/3, discriminant D
λ3 + 3pλ+ 2q = 0, 2q =2a3
27−ab
3+ c, 3p = b−
a2
3, D = p3 + q2
• characterization of solution properties D > 0: one real and two conjugate complex solutions D = 0, q 6= 0: two real solutions, one is a double solution D = 0, q = 0: tripple real solution D < 0: three different real solutions
• CARDANO solution formula
λ1 = u+ + u−, λ2 = ρ+u+ + ρ−u−, λ3 = ρ−u+ + ρ+u−
with
u± =3√−q ±
√D, ρ± :=
1
2[−1± i
√3]
for one real discriminat D ≥ 0 u+ and u− are uniquely determined in general both complex real third roots u± have to be determined such that u+u− = −p is
fulfilled
• back substitution λi = a/3− λi
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.7 tensor characteristics • e1.7.4 eigenvalues
zero values of cubic polynomialsÜ 29
• X
-
a
-
a
--
--
--
--
--
--
--
--
--
--
-
6
tdq e1
e2
e3
e′1
e′2
ϕ3
ϕ3
-
-
-
6
tdq e1
e2
e3
e′1
e′2
ϕ3
ϕ3 cosϕ3
sinϕ3
cosϕ3
−si
nϕ3
cosϕ3 = e1 · e′1
rotation of tensor basis
• outline
orthogonal tensors
2d transformation of tensor basis
3d transformation of tensor basis
transformation of first order tensor components
transformation of second order tensor components
• schedule
two dimensional explanation and illustration
three dimensional extension
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation
transformation of tensorsÜ 30
• X
e1
e2
ϕ3
e1-e2
plane
e2e3
ϕ1
e2-e3
plane
e1
e3 ϕ2
e1-e3
plane
e1
e2e3
ϕ1
ϕ2ϕ3
• spatial coordinate system
base vectors
e1, e2, e3
rotations
ϕ =
ϕ1
ϕ2
ϕ3
• planar coordinate systems
e1-e2-plane
ϕ =[ϕ3
] e2-e3-plane
ϕ =[ϕ1
] e1-e3-plane
ϕ =[ϕ2
]
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation
rotations in 3d and 2d coordinate systemsÜ 31
• X
e1
e2
ϕ3
e1-e2
plane
e2e3
ϕ1
e2-e3
plane
e1
e3 ϕ2
e1-e3
plane
e1
e2e3
ϕ1
ϕ2ϕ3
• spatial coordinate system
base vectors
e1, e2, e3
rotations
ϕ =
ϕ1
ϕ2
ϕ3
• planar coordinate systems
e1-e2-plane
ϕ =[ϕ3
]
e2-e3-plane
ϕ =[ϕ1
] e1-e3-plane
ϕ =[ϕ2
]
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation
rotations in 3d and 2d coordinate systemsÜ 31
• X
e1
e2
ϕ3
e1-e2
plane
e2e3
ϕ1
e2-e3
plane
e1
e3 ϕ2
e1-e3
plane
e1
e2e3
ϕ1
ϕ2ϕ3
• spatial coordinate system
base vectors
e1, e2, e3
rotations
ϕ =
ϕ1
ϕ2
ϕ3
• planar coordinate systems
e1-e2-plane
ϕ =[ϕ3
] e2-e3-plane
ϕ =[ϕ1
]
e1-e3-plane
ϕ =[ϕ2
]
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation
rotations in 3d and 2d coordinate systemsÜ 31
• X
e1
e2
ϕ3
e1-e2
plane
e2e3
ϕ1
e2-e3
plane
e1
e3 ϕ2
e1-e3
plane
e1
e2e3
ϕ1
ϕ2ϕ3
• spatial coordinate system
base vectors
e1, e2, e3
rotations
ϕ =
ϕ1
ϕ2
ϕ3
• planar coordinate systems
e1-e2-plane
ϕ =[ϕ3
] e2-e3-plane
ϕ =[ϕ1
] e1-e3-plane
ϕ =[ϕ2
]
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation
rotations in 3d and 2d coordinate systemsÜ 31
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation tensor Q1
Q1 =
1 0 0
0 cosϕ1 − sinϕ1
0 sinϕ1 cosϕ1
transformation rule
e′i = Q1 · ei
rotation tensor Q2
Q2 =
cosϕ2 0 sinϕ2
0 1 0
− sinϕ2 0 cosϕ2
transformation rule
e′i = Q2 · ei
rotation tensor Q3
Q3 =
cosϕ3 − sinϕ3 0
sinϕ3 cosϕ3 0
0 0 1
transformation rule
e′i = Q3 · ei
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
elementary rotations about e1, e2, e3Ü 36
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q3 · ei
-
-
-
-
6
rbe1
e2
e3 = e′′′3
e′′′1
e′′′2
ϕ3
ϕ3
ϕ3 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q3 · ei
-
-
--
-
-
-
6
rbe1
e2
e3
e′′1e′′2
e′′3
ϕ2
ϕ2
ϕ2 = −30o
e′i= Q1 ·e′′i = Q1 ·Q2 ·Q3 · ei
-
-
--
-
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3ϕ1
ϕ1
ϕ1 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 38
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q3 · ei
-
-
-
-
6
rbe1
e2
e3 = e′′′3
e′′′1
e′′′2
ϕ3
ϕ3
ϕ3 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q3 · ei
-
-
--
-
-
-
6
rbe1
e2
e3
e′′1e′′2
e′′3
ϕ2
ϕ2
ϕ2 = −30o
e′i= Q1 ·e′′i = Q1 ·Q2 ·Q3 · ei
-
-
--
-
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3ϕ1
ϕ1
ϕ1 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 38
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q3 · ei
-
-
-
-
6
rbe1
e2
e3 = e′′′3
e′′′1
e′′′2
ϕ3
ϕ3
ϕ3 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q3 · ei
--
--
-
-
-
6
rbe1
e2
e3
e′′1e′′2
e′′3
ϕ2
ϕ2
ϕ2 = −30o
e′i= Q1 ·e′′i = Q1 ·Q2 ·Q3 · ei
-
-
--
-
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3ϕ1
ϕ1
ϕ1 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 38
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q3 · ei
-
-
-
-
6
rbe1
e2
e3 = e′′′3
e′′′1
e′′′2
ϕ3
ϕ3
ϕ3 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q3 · ei
--
--
-
-
-
6
rbe1
e2
e3
e′′1e′′2
e′′3
ϕ2
ϕ2
ϕ2 = −30o
e′i= Q1 ·e′′i = Q1 ·Q2 ·Q3 · ei
-
-
--
-
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3ϕ1
ϕ1
ϕ1 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 38
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q1 · ei
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1
ϕ1 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q1 · ei
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′′1
e′′2
e′′3
ϕ2ϕ2
ϕ2 = −30o
e′i= Q3 ·e′′i = Q3 ·Q2 ·Q1 ·ei
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
ϕ3
ϕ3
ϕ3 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 39
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q1 · ei
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1
ϕ1 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q1 · ei
--
-
--
-
-
-
6
rbe1
e2
e3
e′′1
e′′2
e′′3
ϕ2ϕ2
ϕ2 = −30o
e′i= Q3 ·e′′i = Q3 ·Q2 ·Q1 ·ei
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
ϕ3
ϕ3
ϕ3 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 39
• X
-
-
-
-
6
-
--
-
-
6
--
-
-
-
6
rb
elementary rotation about e1
e1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1 rb
elementary rotation about e2
e1
e2
e3
e′1
e′2e′3
ϕ2
ϕ2
rb
elementary rotation about e3
e1
e2
e3 = e′3
e′1
e′2
ϕ3
ϕ3
rotation e′′′i = Q1 · ei
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2e′3
ϕ1 ϕ1
ϕ1 = 30o
e′′i = Q2 · e′′′i = Q2 ·Q1 · ei
--
-
--
-
-
-
6
rbe1
e2
e3
e′′1
e′′2
e′′3
ϕ2ϕ2
ϕ2 = −30o
e′i= Q3 ·e′′i = Q3 ·Q2 ·Q1 ·ei
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
ϕ3
ϕ3
ϕ3 = 30o
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 39
• X
rotation sequence Q3, Q2, Q1
-
---
-
---
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
rotation sequence Q1, Q2, Q3
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
resulting rotation
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q1 ·Q2 ·Q3 · ei
resulting rotation
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence Q3, Q2,Q1
e′i = Q1 ·Q2 ·Q3 · ei
• rotation sequence Q1, Q2,Q3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence can notchanged (rotation is notcommutative)
• free choice of sequences ofelementary rotations aboutbase vectors ei• our decision
elementary rot. about e3
elementary rot. about e2
elementary rot. about e1
• 3d rotation tensor
Q = Q1 ·Q2 ·Q3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 40
• X
rotation sequence Q3, Q2, Q1
-
---
-
---
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
rotation sequence Q1, Q2, Q3
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
resulting rotation
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q1 ·Q2 ·Q3 · ei
resulting rotation
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence Q3, Q2,Q1
e′i = Q1 ·Q2 ·Q3 · ei
• rotation sequence Q1, Q2,Q3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence can notchanged (rotation is notcommutative)
• free choice of sequences ofelementary rotations aboutbase vectors ei• our decision
elementary rot. about e3
elementary rot. about e2
elementary rot. about e1
• 3d rotation tensor
Q = Q1 ·Q2 ·Q3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 40
• X
rotation sequence Q3, Q2, Q1
-
---
-
---
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
rotation sequence Q1, Q2, Q3
-
-
-
-
-
-
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
resulting rotation
-
--
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q1 ·Q2 ·Q3 · ei
resulting rotation
-
-
-
-
-
6
rbe1
e2
e3
e′1
e′2
e′3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence Q3, Q2,Q1
e′i = Q1 ·Q2 ·Q3 · ei
• rotation sequence Q1, Q2,Q3
e′i = Q3 ·Q2 ·Q1 · ei
• rotation sequence can notchanged (rotation is notcommutative)
• free choice of sequences ofelementary rotations aboutbase vectors ei• our decision
elementary rot. about e3
elementary rot. about e2
elementary rot. about e1
• 3d rotation tensor
Q = Q1 ·Q2 ·Q3
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
cumulative rotation about e1, e2, e3Ü 40
• X
6--
e3
e2
e1rb
6--
e3
e2
e1rb
• rotation sequence Q3, Q2, Q1
first rotation Q3 with ϕ3 = 90o
second rotation Q2 with ϕ2 = 90o
third rotation Q1 with ϕ1 = 90o
• rotation sequence Q1, Q2, Q3
first rotation Q1 with ϕ1 = 90o
second rotation Q2 with ϕ2 = 90o
thrid rotation Q3 with ϕ3 = 90o
• result is different to inverse rotation sequenceQ3, Q2, Q1
• rotation is not commutative
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
study of rotation sequenceÜ 41
• X
6--
e3
e2
e1rb
6--
e3
e2
e1rb
• rotation sequence Q3, Q2, Q1
first rotation Q3 with ϕ3 = 90o
second rotation Q2 with ϕ2 = 90o
third rotation Q1 with ϕ1 = 90o
• rotation sequence Q1, Q2, Q3
first rotation Q1 with ϕ1 = 90o
second rotation Q2 with ϕ2 = 90o
thrid rotation Q3 with ϕ3 = 90o
• result is different to inverse rotation sequenceQ3, Q2, Q1
• rotation is not commutative
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
study of rotation sequenceÜ 41
• X
• elementary rotation about basis vector e1, rotated tensor basis, rotated basis vectors
Q1 =
1 0 0
0 cosϕ1 − sinϕ1
0 sinϕ1 cosϕ1
e′1 =
1
0
0
, e′2 =
0
cosϕ1
sinϕ1
, e′3 =
0
− sinϕ1
cosϕ1
• elementary rotation about basis vectors e2 and e3
Q2 =
cosϕ2 0 sinϕ2
0 1 0
− sinϕ2 0 cosϕ2
Q3 =
cosϕ3 − sinϕ3 0
sinϕ3 cosϕ3 0
0 0 1
• rotated basis vectors after elementary rotations about e2 and e3
e′1 =
cosϕ2
0
− sinϕ2
, e′2 =
0
1
0
, e′3 =
sinϕ2
0
cosϕ2
e′1 =
cosϕ3
sinϕ3
0
, e′2 =
− sinϕ3
cosϕ3
0
, e′3 =
0
0
1
• three dimensional rotation tensor Q = Q1 ·Q2 ·Q3
Q=
cosϕ2 cosϕ3 − cosϕ2 sinϕ3 sinϕ2
cosϕ1 sinϕ3 + sinϕ1 sinϕ2 cosϕ3 cosϕ1 cosϕ3 − sinϕ1 sinϕ2 sinϕ3 − sinϕ1 cosϕ2
sinϕ1 sinϕ3 − cosϕ1 sinϕ2 cosϕ3 sinϕ1 cosϕ3 + cosϕ1 sinϕ2 sinϕ3 cosϕ1 cosϕ2
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.3 3d rotation tensor
three-dimensional rotation tensorÜ 42
• X
--
-
6
-tdq e1
e2
e3
e′1
e′2
ϕ3
ϕ3 cosϕ3
sinϕ3
cosϕ3
−si
nϕ3
cosϕ3 = e1 · e′1
a
ϕ3
ϕ3
e1
e2
a1
a2
-
6
e′1
e′2
a′1 a
′2-
6
-
a
tdq
coordinate transformation
• tensors are independent on coordinate system
only tensor components are dependent oncoordinate system
transformation of tensor components is required
• representation of first order tensor in original and rotated basisvectors
a = aj ej = aj Qji e′i a = a′j e
′j = a′j Qij ei
• comparision of both representations
a′i = aj Qji = Qji aj ai = a′j Qij = Qij a′j
• all components of tensor in original and rotated coordinates([a])
[a]′ = QT · [a] [a] = Q · [a]′
• test (example on left hand side)
a′1 = a1 cosϕ3 + a2 sinϕ3 = a1Q11 + a2Q21
a′2 = −a1 sinϕ3 + a2 cosϕ3 = a1Q12 + a2Q22
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.4 first order tensors
transformation of first order tensorsÜ 43
• X
• transformation of basis vectors, tensor basis
e′j = Qij ei ej = Qji e′i
• representation of second order tensors in original and rotated basis vectorstransformation of basis vectors
A = Akl ek ⊗ el = Akl Qki Qlj e′i ⊗ e′j = A′ij e
′i ⊗ e′j
A = A′kl e′k ⊗ e
′l = A′kl Qik Qjl ei ⊗ ej = Aij ei ⊗ ej
• comparision of both representations
A′ij = Qki Akl Qlj Aij = Qik A′kl Qjl
• all components of tensor in original and rotated coordinates ([A])
[A]′ = QT · [A] ·Q [A] = Q · [A]′ ·QT
alternative derivation of second order tensor transformation relations• components of second order tensor A = Akl ek ⊗ el
ei ·A · ej = Akl [ei · ek]⊗ [el · ej ] = Akl δikδlj = Aij
• with respect to rotated coordinate system
Aij = ei ·A · ej = [Qik e′k] ·A · [e′l Qjl] = Qik A
′kl Qjl
A′ij = e′i ·A · e′j = [Qki ek] ·A · [el Qlj ] = Qki Akl Qlj
d.kuhl, wes.online, university of kasselÜÜlinear computational structural mechanics • e1 tensors • e1.8 coordinate transformation • e1.8.5 second order tensors
transformation of second order tensorsÜ 44
• X
functions of one variable functions of two variablesde
rivat
ive
inte
gral
X
f(X
)
X
deriv
ativ
e-s
lope
tang
ent
df(X
)dX
=∂f(X
)∂X
X
f(X
)
a b
area
unde
rfun
ctio
nf
A=b ∫ a
f(X
)dX
X1
X2
f(X
1,X
2)
X1
X2
part
iald
eriv
ativ
es
∂f(X
1,X
2)
∂X
1,∂f(X
1,X
2)
∂X
2
X1
X2
f(X
1,X
2)
a1
b1 a2
b2
volu
me
unde
rfun
ctio
nf
V=b2 ∫ a2
b1 ∫ a1
f(X
1,X
2)dX
1dX
2
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
derivatives of tensorsÜ 45
• X
derivatives of tensors with respect to tensor valued variables→ increase order of a tensor by order of tensor valued variable
∂a∂b
=
∂a
∂b1∂a
∂b2∂a
∂b3
= ∂a∂bi
ei
∂a
∂b=
∂a1
∂b1
∂a1
∂b2
∂a1
∂b3∂a2
∂b1
∂a2
∂b2
∂a2
∂b3∂a3
∂b1
∂a3
∂b2
∂a3
∂b3
= ∂ai∂bj
ei ⊗ ej
∂a
∂B=
∂a
∂B11
∂a
∂B12
∂a
∂B13∂a
∂B21
∂a
∂B22
∂a
∂B23∂a
∂B31
∂a
∂B32
∂a
∂B33
= ∂a∂Bij
ei ⊗ ej
∂A
∂B= =
∂Aij∂Bkl
ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
differentiation of tensorsÜ 46
• X
derivatives of tensors with respect to tensor valued variables→ increase order of a tensor by order of tensor valued variable
∂a∂b
=
∂a
∂b1∂a
∂b2∂a
∂b3
= ∂a∂bi
ei
∂a
∂b=
∂a1
∂b1
∂a1
∂b2
∂a1
∂b3∂a2
∂b1
∂a2
∂b2
∂a2
∂b3∂a3
∂b1
∂a3
∂b2
∂a3
∂b3
= ∂ai∂bj
ei ⊗ ej
∂a
∂B=
∂a
∂B11
∂a
∂B12
∂a
∂B13∂a
∂B21
∂a
∂B22
∂a
∂B23∂a
∂B31
∂a
∂B32
∂a
∂B33
= ∂a∂Bij
ei ⊗ ej
∂A
∂B= =
∂Aij∂Bkl
ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
differentiation of tensorsÜ 46
• X
derivatives of tensors with respect to tensor valued variables→ increase order of a tensor by order of tensor valued variable
∂a∂b
=
∂a
∂b1∂a
∂b2∂a
∂b3
= ∂a∂bi
ei
∂a
∂b=
∂a1
∂b1
∂a1
∂b2
∂a1
∂b3∂a2
∂b1
∂a2
∂b2
∂a2
∂b3∂a3
∂b1
∂a3
∂b2
∂a3
∂b3
= ∂ai∂bj
ei ⊗ ej
∂a
∂B=
∂a
∂B11
∂a
∂B12
∂a
∂B13∂a
∂B21
∂a
∂B22
∂a
∂B23∂a
∂B31
∂a
∂B32
∂a
∂B33
= ∂a∂Bij
ei ⊗ ej
∂A
∂B= =
∂Aij∂Bkl
ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
differentiation of tensorsÜ 46
• X
derivatives of tensors with respect to tensor valued variables→ increase order of a tensor by order of tensor valued variable
∂a∂b
=
∂a
∂b1∂a
∂b2∂a
∂b3
= ∂a∂bi
ei
∂a
∂b=
∂a1
∂b1
∂a1
∂b2
∂a1
∂b3∂a2
∂b1
∂a2
∂b2
∂a2
∂b3∂a3
∂b1
∂a3
∂b2
∂a3
∂b3
= ∂ai∂bj
ei ⊗ ej
∂a
∂B=
∂a
∂B11
∂a
∂B12
∂a
∂B13∂a
∂B21
∂a
∂B22
∂a
∂B23∂a
∂B31
∂a
∂B32
∂a
∂B33
= ∂a∂Bij
ei ⊗ ej
∂A
∂B= =
∂Aij∂Bkl
ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
differentiation of tensorsÜ 46
• X
derivatives of tensors with respect to tensor valued variables→ increase order of a tensor by order of tensor valued variable
∂a∂b
=
∂a
∂b1∂a
∂b2∂a
∂b3
= ∂a∂bi
ei
∂a
∂b=
∂a1
∂b1
∂a1
∂b2
∂a1
∂b3∂a2
∂b1
∂a2
∂b2
∂a2
∂b3∂a3
∂b1
∂a3
∂b2
∂a3
∂b3
= ∂ai∂bj
ei ⊗ ej
∂a
∂B=
∂a
∂B11
∂a
∂B12
∂a
∂B13∂a
∂B21
∂a
∂B22
∂a
∂B23∂a
∂B31
∂a
∂B32
∂a
∂B33
= ∂a∂Bij
ei ⊗ ej
∂A
∂B= =
∂Aij∂Bkl
ei ⊗ ej ⊗ ek ⊗ el
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
differentiation of tensorsÜ 46
• X
gradient (∇) & symmetric part of gradient (∇sym)→ derivative of vector a with repect to position vector X
∇a =∂a
∂X=
∂ai
∂Xjei⊗ej = ai,j ei⊗ej ∇syma =
[∂a
∂X
]sym=
1
2[ai,j+aj,i] ei ⊗ ej
transposition of gradient
∇Ta = [∇a]T = aj,i ei⊗ej
divergence (div) of a tensor→ reduction of order of a tensor
diva = tr[∇a] = ai,i =a1,1+a2,2+a3,3 divA =
A11,1+A12,2+A13,3
A21,1+A22,2+A23,3
A31,1+A32,2+A33,3
= Aij,jei
e.g. displacement gradient and stress tensor
∇u =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
= ui,j ei ⊗ ej divσ =
σ11,1+σ12,2+σ13,3
σ21,1+σ22,2+σ23,3
σ31,1+σ32,2+σ33,3
= σij,j ei
e.g. strain tensor (geometrically linear theory)
ε = ∇symu =1
2
[∇u+∇Tu
]ε =
1
2[ui,j + uj,i] ei ⊗ ej = εij ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
gradient and divergenceÜ 47
• X
gradient (∇) & symmetric part of gradient (∇sym)→ derivative of vector a with repect to position vector X
∇a =∂a
∂X=
∂ai
∂Xjei⊗ej = ai,j ei⊗ej ∇syma =
[∂a
∂X
]sym=
1
2[ai,j+aj,i] ei ⊗ ej
transposition of gradient
∇Ta = [∇a]T = aj,i ei⊗ej
divergence (div) of a tensor→ reduction of order of a tensor
diva = tr[∇a] = ai,i =a1,1+a2,2+a3,3 divA =
A11,1+A12,2+A13,3
A21,1+A22,2+A23,3
A31,1+A32,2+A33,3
= Aij,jei
e.g. displacement gradient and stress tensor
∇u =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
= ui,j ei ⊗ ej divσ =
σ11,1+σ12,2+σ13,3
σ21,1+σ22,2+σ23,3
σ31,1+σ32,2+σ33,3
= σij,j ei
e.g. strain tensor (geometrically linear theory)
ε = ∇symu =1
2
[∇u+∇Tu
]ε =
1
2[ui,j + uj,i] ei ⊗ ej = εij ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
gradient and divergenceÜ 47
• X
gradient (∇) & symmetric part of gradient (∇sym)→ derivative of vector a with repect to position vector X
∇a =∂a
∂X=
∂ai
∂Xjei⊗ej = ai,j ei⊗ej ∇syma =
[∂a
∂X
]sym=
1
2[ai,j+aj,i] ei ⊗ ej
transposition of gradient
∇Ta = [∇a]T = aj,i ei⊗ej
divergence (div) of a tensor→ reduction of order of a tensor
diva = tr[∇a] = ai,i =a1,1+a2,2+a3,3 divA =
A11,1+A12,2+A13,3
A21,1+A22,2+A23,3
A31,1+A32,2+A33,3
= Aij,jei
e.g. displacement gradient and stress tensor
∇u =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
= ui,j ei ⊗ ej divσ =
σ11,1+σ12,2+σ13,3
σ21,1+σ22,2+σ23,3
σ31,1+σ32,2+σ33,3
= σij,j ei
e.g. strain tensor (geometrically linear theory)
ε = ∇symu =1
2
[∇u+∇Tu
]ε =
1
2[ui,j + uj,i] ei ⊗ ej = εij ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
gradient and divergenceÜ 47
• X
gradient (∇) & symmetric part of gradient (∇sym)→ derivative of vector a with repect to position vector X
∇a =∂a
∂X=
∂ai
∂Xjei⊗ej = ai,j ei⊗ej ∇syma =
[∂a
∂X
]sym=
1
2[ai,j+aj,i] ei ⊗ ej
transposition of gradient
∇Ta = [∇a]T = aj,i ei⊗ej
divergence (div) of a tensor→ reduction of order of a tensor
diva = tr[∇a] = ai,i =a1,1+a2,2+a3,3 divA =
A11,1+A12,2+A13,3
A21,1+A22,2+A23,3
A31,1+A32,2+A33,3
= Aij,jei
e.g. displacement gradient and stress tensor
∇u =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
= ui,j ei ⊗ ej divσ =
σ11,1+σ12,2+σ13,3
σ21,1+σ22,2+σ23,3
σ31,1+σ32,2+σ33,3
= σij,j ei
e.g. strain tensor (geometrically linear theory)
ε = ∇symu =1
2
[∇u+∇Tu
]ε =
1
2[ui,j + uj,i] ei ⊗ ej = εij ei ⊗ ej
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
gradient and divergenceÜ 47
• X
X
dX
df
f
f + df
∂f∂Xf
(X)
X
X1
X2
f(X
1,X
2)
X1
X2dX1
dX2
f
f + dfrb rb
functions of one variable
functions of two variable
• illustration of total differential df by means of functions of onevariable
df is the change of the function value f for incrementalchange of the variable dX
mathematical formulation using the derivative ∂f/∂X
df =∂f
∂XdX
• total differential df of a function of two variables
df is the change of the function value f for incrementalchanges of variables dX1 and dX2
mathematical formulation using the partial derivatives∂f/∂X1 and ∂f/∂X2
df =∂f
∂X1dX1 +
∂f
∂X2dX2
• total differential of function df of multiple variables
df is the change of the function value f for incrementalchanges of the variables
• total differential, variation & increment are formally identical,just the interpretation and the symbols (d, δ and ∆) are different
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
total differential, variation & increment of tensorsÜ 48
• X
total differential, variation and increment of a tensor valued function, e.g. A(B, c)
dA(B, c) =∂A(B, c)
∂B: dB +
∂A(B, c)
∂c· dc
δA(B, c) =∂A(B, c)
∂B: δB +
∂A(B, c)
∂c· δc
∆A(B, c) =∂A(B, c)
∂B: ∆B +
∂A(B, c)
∂c·∆c
chain rule of total differential, variation and increment of a tensor valued function, e.g.
δA(B(C(d))) =∂A(B(C(d)))
∂B:∂B(C(d))
∂C:∂C(d)
∂d· δd
=∂A(B(C(d)))
∂B:∂B(C(d))
∂C: δC(d)
=∂A(B(C(d)))
∂B: δB(C(d))
example - increment of second PIOLA-KIRCHHOFF stress tensor (geometrically non-linear)
∆S(E(F (∇u))) =∂S
∂E:∂E
∂F:∂F
∂∇u: ∆∇u =
∂S
∂E:∂E
∂F:∆F =
∂S
∂E:∆E = C :∆E
product rule, example - increment of specific internal virtual work (geometrically non-linear)
δ[A : B] = δA : B +A : δB ∆[δE : S] = ∆δE : S + δE : ∆S
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.9 derivatives
total differential, variation & increment of tensorsÜ 49
• X
GAUSS integral theorem → transformation of volume integral over domain Ω of gradient into anintegral over surface of body Γ∫Ω
∇a dV =
∫Γ
a n dA∫Ω
∇a dV =
∫Γ
a⊗ n dA∫Ω
∇A dV =
∫Γ
A⊗ n dA
∫Ω
a,i dV =
∫Γ
a ni dA∫Ω
ai,j dV =
∫Γ
ai nj dA∫Ω
Aij,k dV =
∫Γ
Aij nk dA
multiplication by KRONECKER symbol δij → form, relevant for course∫Ω
diva dV =
∫Γ
a· n dA∫Ω
divA dV =
∫Γ
A· n dA
∫Ω
ai,i dV =
∫Γ
ai ni dA∫Ω
Aij,j dV =
∫Γ
Aij nj dA
e.g. development of weak form of balance of linear momentum - principle of virtual work∫Ω
div[δu · σ] dV =
∫Γ
δu · σ · n dA =
∫Γσ
δu · σ · n dA =
∫Γσ
δu · t dA
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.10 Gauß integral theorem
tensor analysis - GAUSS integral theorem Ü 50
• X
double contraction of tensor A with a symmetric tensor B = BT
A : B = Asym : B Aij Bij =1
2[Aij +Aji] Bkl
both side double contraction of tensor C with Cijkl = Cjikl = Cjilk = Cijlk
A : C : B = Asym : C : Bsym AijCijklBkl=1
4[Aij+Aji]Cijkl[Bkl+Blk]
gradient and divergence of tensors - formulas
∇[a b] = a ∇b + b⊗∇a∇[a · b] =∇Ta · b +∇Tb · a∇[a B] = a ∇B +B ⊗∇a∇A : 1 = divA
div[a b] = a divb + b · ∇adiv[a⊗ b] = a divb +∇a · bdiv[a ·B] =∇a : B + a · divB
div[a B] = a divB +B · ∇a
chain rule of divergence div[a ·B] = ∇a : B + a · divB and GAUSS integral
theorem consitute divergence theorem∫Ω
a · divB dV =
∫Ω
div[a ·B] dV −∫Ω
∇a : B dV =
∫Γ
a ·B · n dA−∫Ω
∇a : B dV
d.kuhl, wes.online, university of kasselÜ Ülinear computational structural mechanics • e1 tensors • e1.11 tensor reformulations
tensor analysis - more formulas ...Ü 51