3D and Planar Constitutive Relations A School on … on mechanics/ppt... · 3D and Planar...
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3D and Planar Constitutive Relations
A School on Mechanics of Fibre Reinforced Polymer
Composites
Knowledge Incubation for TEQIP
Indian Institute of Technology Kanpur
PM Mohite
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
22-25 January 2017
• Generalized Hooke’s law of the proportionality of stress and strain:
Each of the six component of the stress at any point is a linear function
of the six components of strain at that point.
• Concept of initial state
• Loading under two situations:
- Isothermal and Reversible;
- Adiabatic and Reversible.
• Stress components are the partial differential coefficients of a function (W) of
the strain-components.
Generalized Hooke’s Law:
1
2ij
ij ji
W Wσ
ε ε
∂ ∂= +
∂ ∂
• Form of the Strain Energy Density Function (W):
Homogeneous quadratic function of the strain components.
• W is invariant
• and are tensors.
• W is taken to be zero when body is in the initial state in which are zero.
Then, constant is zero.
Generalized Hooke’s Law:
1constant
2ij ij ijkl ij klW C Cε ε ε= + +
ijC ijklC
ijε
• For unstrained and unstressed body, are zero.
This leads to
and
is a fourth order (stiffness) tensor/matrix.
Generalized Hooke’s Law:
ijC
, , , 1, 2,3ij ijkl klC i j k lσ ε= =
1
2ijkl ij klW C ε ε=
ijklC
( )4
3 81 independent constants!=
• Simply, you can view this as if you have a vector of 9 stress components which
is related to a vector of 9 strain components through a matrix of 9x9!
Generalized Hooke’s Law:
{ } [ ] { }9 1 9 19 9
Cσ ε× ××
=
[ ]
1111 1112 1113 1121 1122 1123 1131 1132 1133
1211 1212 1213 1221 1222 1223 1231 1232 1233
1311 1312 1313 1321 1322 1323 1331 1332 1333
2111 2112 2113 2121 2122 2123 2131 2132 2133
2211 2212 2213 22
C C C C C C C C C
C C C C C C C C C
C C C C C C C C C
C C C C C C C C C
C C C C C= 21 2222 2223 2231 2232 2233
2311 2312 2313 2321 2322 2323 2331 2332 2333
3111 3112 3113 3121 3122 3123 3131 3132 3133
3211 3212 3213 3221 3222 3223 3231 3232 3233
3311 3312 3313 3321 3322 3323 3331 33
C C C C C
C C C C C C C C C
C C C C C C C C C
C C C C C C C C C
C C C C C C C C 32 3333C
• Stress symmetry:
• six independent ways to express when i and j are taken together and still 9
ways to express k and l taken together.
Stress Tensor Symmetry:
jij iσ σ=
( )0 0 ijij kl jiklj kli C Cσ εσ −− ⇒ ==
ij ijkl klCσ ε= ji jikl klCσ ε=
ijkl jiklC C=
6 9 54 independent constants!× =
• Simply, you can view this as if you have a vector of 6 stress components which
is related to a vector of 9 strain components through a matrix of 6x9!
Stress Tensor Symmetry:
{ } [ ] { }6 1 9 16 9
Cσ ε× ××
=
[ ]
1111 1112 1113 1121 1122 1123 1131 1132 1133
2211 2212 2213 221 2222 2223 2231 2232 2233
3311 3312 3313 3321 3322 3323 3331 3332 3333
2311 2312 2313 2321 2322 2323 2331 2332 2333
1311 1312 1313 132
C C C C C C C C C
C C C C C C C C C
C C C C C C C C CC
C C C C C C C C C
C C C C
=
1 1322 1323 1331 1332 1333
1211 1212 1213 1221 1222 1223 1231 1232 1233
C C C C C
C C C C C C C C C
• Strain symmetry:
• six independent ways to express when i and j are taken together and 6 ways to
express k and l taken together.
Stress and Strain Tensor Symmetry:
jij iε ε=
( )0 0 ijij kl ijlki klj C Cσ εσ −− ⇒ ==
ij ijkl klCσ ε= ij ijlk lkCσ ε=
ijkl ijlkC C=
6 6 36 independent constants!× =
• Or simply, you can view this as if you have a vector of 6 stress components
which is related to a vector of 6 strain components through a matrix of 6x6!
Stress and Strain Tensor Symmetry:
{ } [ ] { }6 1 6 16 6
Cσ ε× ××
=
[ ]
1111 1122 1133 1123 1113 1112
2211 2222 2233 2223 2213 2212
3311 3322 3333 3323 3313 3312
2311 2322 2333 2323 2312 2312
1311 1322 1333 1323 1313 1312
1211 1222 1233 1223 1212 1212
C C C C C C
C C C C C C
C C C C C CC
C C C C C C
C C C C C C
C C C C C C
=
• In other words,
Stress and Strain Tensor Symmetry:
1111 1122 1133 1123 1113 111211
2211 2222 2233 2223 2213 221222
3311 3322 3333 3323 3313 331233
2311 2322 2333 2323 2312 231223
1311 1322 1333 1323 1313 131213
1211 12212
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C
σ
σ
σ
σ
σ
σ
=
11
22
33
23
13
2 1233 1223 1212 1212 12C C C C
ε
ε
ε
ε
ε
ε
• Using Voigt notation - a way to represent a symmetric tensor by reducing its
order
• For stress components:
• Strain Components:
Voigt Notation:
11 12 13
22 23
33
σ σ σ
σ σ
σ
1
2
3
4
6 5
{ } { }11 22 33 23 13 12 1 2 3 4 5 6σ σ σ σ σ σ σ σ σ σ σ σ=
{ } { }11 22 33 23 13 12 1 2 3 4 5 62 2 2ε ε ε ε ε ε ε ε ε ε ε ε=
• Instead of writing C as a fourth order tensor, written as a second order tensor
and stress and strains tensors are written as vectors !
Stress and Strain Tensor Symmetries:
1 11 12 13 14 15 16 1
2 21 22 23 24 25 26 2
3 31 32 33 34 35 36 3
4 41 42 43 44 45 46 4
5 51 51 53 54 55 56 5
6 61 62 63 64 65 66 6
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
σ ε
σ ε
σ ε
σ ε
σ ε
σ ε
=
• Existence of W :
Hyperelastic materials
Invariant
Positive Definite
Existence of W:
( ) 0ij ji i jC C ε ε− =
1
2ij i jW C ε ε=
1
2ij j iW C ε ε=
ij jiC C=
•
Existence of W:
1 11 12 13 14 15 16 1
2 22 23 24 25 26 2
3 33 34 35 36 3
4 44 45 46 4
5 55 56 5
6 66 6
C C C C C C
C C C C C
C C C C
C C C
C C
C
σ ε
σ ε
σ ε
σ ε
σ ε
σ ε
=
21 independent constants!
• Stress symmetry
Minor Symmetries
• Strain symmetry
• Existence of W:
Major Symmetry
Symmetries:
ijkl jiklC C=
ijkl jiklC C=
ijkl klijC C=
• Transformations
• Prime denotes the transformed coordinates.
• aij denotes the components of a transformation matrix
Transformations:
• Further reduction in constants obtained by material symmetry
• Symmetry Definition: Any geometrical figure which can be brought to
coincidence with itself, by an operation which changes the position of any of its
points, is said to possess “symmetry”.
• Rotation and Reflection
Material Symmetry:
• Transformation of axes:
• Transformation matrix:
• Transformation of strains:
Material Symmetry: One Plane of Material Symmetry
• Transformation of stiffness:
• Comparison of stress components:
Material Symmetry: One Plane of Material Symmetry
'ij ijC C=
14 15 0C C⇒ = =
Second Approach: Invariance of W
W for
Hyper-
elastic
material
Material Symmetry: One Plane of Material Symmetry
Second Approach: Invariance of W
• For W to be invariant the product terms
must vanish, that is,
Material Symmetry: One Plane of Material Symmetry
1 4 1 5 2 4 2 5 3 4 3 5 4 6 5 6, , , , , , ,ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε
14 15 24 25 34 35 46 56, , , , , , , are zeroC C C C C C C C
• Two Orthogonal Planes of Material Symmetry: Orthotropic Materials
• Transformation of axes:
Transformation matrix:
Transformation of strains:
Material Symmetry: Two Orthogonal Planes of Material Symmetry
• Transformation of stresses:
• Transformation of stiffness:
Material Symmetry: Two Orthogonal Planes of Material Symmetry
'ij ijC C=
• Comparison of stresses:
• Similarly,
Material Symmetry: Two Orthogonal Planes of Material Symmetry
16 0C⇒ =
• Stiffness Tensor:
9 independent constants
Material Symmetry: Two Orthogonal Planes of Material Symmetry
• Second approach: Invariance of w
• W for monoclinic material:
• or form of W:
Material Symmetry: Two Orthogonal Planes of Material Symmetry
2 2 2 211 1 22 2 33 3 44 4
2 255 5 66 6 12 1 2 13 1 3
16 1 6 23 2 3 26 2 6 36 3 6
45 4 5
1
2
C C C C
C C C CW
C C C C
C
ε ε ε ε
ε ε ε ε ε ε
ε ε ε ε ε ε ε ε
ε ε
+ + + + + + + +
= + + + +
2 2 2 2 2 21 2 3 4 5 6
1 2 1 3 1 6
2 3 2 6 3 6 4 5
, , , , , ,
, , ,
, , ,
W W
ε ε ε ε ε ε
ε ε ε ε ε ε
ε ε ε ε ε ε ε ε
=
• For W to be invariant under the strain transformations
• the product terms
must vanish, which is possible when
W for orthotropic material:
Material Symmetry: Two Orthogonal Planes of Material Symmetry
1 6 2 6 3 6 4 5, , ,ε ε ε ε ε ε ε ε
16 26 36 45, , , are zeroC C C C
2 2 2 2 2 211 1 22 2 33 3 44 4 55 5 66 6
12 1 2 13 1 3 23 2 3
1
2
C C C C C CW
C C C
ε ε ε ε ε ε
ε ε ε ε ε ε
+ + + + + +=
+ +
• When material has two orthogonal planes of symmetry then it also symmetric
about a plane which is mutually orthogonal earlier two planes!
•Such materials are called Orthotropic Materials.
• Select now the remaining plane x1-x3 as the third orthogonal plane of material
symmetry in addition to earlier two planes
• Follow the same procedure, either comparing the stresses or invariance of W
• There will be no change in the final stiffness tensor. Number of independent
constants will be still 9!
Material Symmetry: Two Orthogonal Planes of Material Symmetry
• When material has two orthogonal planes of symmetry then it also symmetric
about a plane which is mutually orthogonal earlier two planes!
• Alternately:
• Now select the plane x1-x3 as the second orthogonal plane of material
symmetry in addition to x1-x2 plane
Material Symmetry: Two Orthogonal Planes of Material Symmetry
• Isotropic behaviour of UD lamina in the cross-sectional plane (perpendicular
to fibre s’ length)
• Transformation matrix:
Isotropy in a Plane:
Material Symmetry: Two Orthogonal Planes of Material Symmetry
2 2 2 2 2 21 2 3 4 5 6
1 2 1 3 1 6
2 3 2 6 3 6 4 5
, , , , , ,
, , ,
, , ,
W W
ε ε ε ε ε ε
ε ε ε ε ε ε
ε ε ε ε ε ε ε ε
=
W for monoclinic material: For invariance
1 6 2 6 3 6 4 5, , , 0ε ε ε ε ε ε ε ε =
16 26 36 45, , , are zeroC C C C
• Trigonometric identities for the strains:
• Form of W:
Isotropy in a Plane:
( ) ( )
( ) ( ) ( ) ( )
' '22 33 22 33
22 ' ' '22 33 23 22 33 23
2 222 ' '12 13 12 13
,
,
ε ε ε ε
ε ε ε ε ε ε
ε ε ε ε
+ = +
− = + −
+ = +
( ) ( ) ( )( )( ) ( ) ( )
2 22
11 22 33 22 33 23 12 13
2 2 2' ' ' ' ' ' ' '11 22 33 22 33 23 12 13
, , ,
, , ,
W W
W W
ε ε ε ε ε ε ε ε
ε ε ε ε ε ε ε ε
= + − +
= + − +
• Strain energy density function for orthotropic material:
• rearranging
Isotropy in a Plane:
211 11 12 11 22 13 11 33
2 2 222 22 33 33 23 22 33 44 23
2 255 13 66 12
2 2
+ 2 4
+ 4 4
W C C C
C C C C
C C
ε ε ε ε ε
ε ε ε ε ε
ε ε
= + +
+ + +
+
( )211 11 11 12 22 13 33
2 255 13 66 12
2 2 222 22 33 33 23 22 33 44 23
2
+ 4 4 +
+ 2 4
W C C C
C C
C C C C
ε ε ε ε
ε ε
ε ε ε ε ε
= + +
+
+ + +
• In the second bracket, we take
• In the third bracket, we take
• Rearrange the last bracket with and unchanged
Isotropy in a Plane:
( )211 11 11 12 22 13 33
2 255 13 66 12
2 2 222 22 33 33 23 22 33 44 23
2
+ 4 4 +
+ 2 4
W C C C
C C
C C C C
ε ε ε ε
ε ε
ε ε ε ε ε
= + +
+
+ + +
12 13C C=
55 66C C=
( )
2 2 222 22 33 33 23 22 33 44 23
2 222 22 33 22 22 33 23 22 33 44 23
2 4
2 2 4
C C C C
C C C C
ε ε ε ε ε
ε ε ε ε ε ε ε
+ + +
= + − + +
22 33C C= 23C
• Rearrange the last bracket further as
• we need
Isotropy in a Plane:
( ) ( )
2 2 222 22 33 33 23 22 33 44 23
2 222 22 33 22 23 22 33 44 23
2 4
2 2
C C C C
C C C C
ε ε ε ε ε
ε ε ε ε ε
+ + +
= + − − −
22 2344
2
C CC
−=
• Stiffness tensor
• 5 independent constants
• Such materials are called Transversely isotropic materials.
• Define: where
Isotropy in a Plane:
11 12 12
22 23
22
22 23
66
66
0 0 0
0 0 0
0 0 0
0 02
0
ij
C C C
C C
C
C C C
Sym C
C
= −
Transverse Isotropy with an Additional Orthogonal Plane:
• Consider isotropy in x1-x2 plane as well
• strain
transformation
Transverse Isotropy with an Additional Orthogonal Plane:
• Trigonometric identities:
• Form of W:
( ) ( )
( ) ( ) ( ) ( )
' '11 22 11 22
22 ' ' '11 22 12 11 22 12
2 22 2 ' '13 23 13 23
,
,
ε ε ε ε
ε ε ε ε ε ε
ε ε ε ε
+ = +
− = + −
+ = +
( ) ( ) ( )( )( ) ( ) ( )
2 22
11 22 33 11 22 12 13 23
2 2 2' ' ' ' ' ' ' '11 22 33 11 22 12 13 23
, , ,
, , ,
W W
W W
ε ε ε ε ε ε ε ε
ε ε ε ε ε ε ε ε
= + − +
= + − +
• In the second bracket, we take
• In the third bracket, we take
• Rearrange the last bracket with and unchanged
Transverse Isotropy with an Additional Orthogonal Plane:
12 23C C=
11 22C C= 12C
• Two independent constants !
• Define:
where
Isotropy:
11 12 12
11 12
11
11 12
11 12
11 12
0 0 0
0 0 0
0 0 0
0 02
02
2
ij
C C C
C C
C
C CC
C C
C CSym
− = − −
• Generalized Hooke’s Law: 81 independent constants
• Stress tensor symmetry: 54 independent constants
• Strain tensor symmetry: 36 independent constants
• Existence of W (Hyperelastic/Aelotropic): 21 independent constants
• Existence of one plane of material symmetry: 13 independent constants
• Existence of two/three mutually perpendicular planes of symmetry:
(Orthotropic Material) 9 independent constants
• One plane of isotropy: 5 independent constants
• Two/three/infinite planes of isotropy: 2 independent constants
3D Constitutive Relations: Quick Review
• Strain-stress Relations
• Normal stresses and
strains
Constitutive Relations for Orthotropic Materials:
• Shear stresses and strains
• Poisson’s ratio:
(no sum over i, j)
• In general,
Constitutive Relations for Orthotropic Materials:
ij jiν ν≠
• Matrix-vector form:
where,
Constitutive Relations for Orthotropic Materials:
Always work with compliance
tensor.
It is easy to remember.
• Important Relations:
• Reciprocal relation
where,
Constitutive Relations for Orthotropic Materials:
• For strain energy to be positive definite both Compliance and Stiffness
tensors must be positive definite.
• Strain energy to be positive definite the diagonal entries of the Compliance
tensor must be positive.
• Similarly, the diagonal entries of the Stiffness tensor must be positive
• and the determinant must also be positive
Constraints on Engineering Constants:
Constraints on Engineering Constants:
• Constraint on Poisson’s ratio:
From constraint on determinant:
• For transverse isotropic material:
with
we get, and
Finally, leads to the condition
For isotropic materials:
Constraints on Engineering Constants:
Constitutive Relations: Transformations
• 123 – Principal material directions
• xyz – global reference directions
Transformation matrix
for rotation about z-axis:
• Stress transformation:
For example,
that is,
Transformation matrix:
Constitutive Relations: Stress Transformations
'ij ki lj kla aσ σ=
Constitutive Relations: Stress and Strain Transformations
• Transformation matrix:
where, m=cosθ and n=sinθ
• Stress transformation:
For example,
that is,
Now using
Constitutive Relations: Stress and Strain Transformations
'ij ki lj kla aε ε=
Constitutive Relations: Stress and Strain Transformations
• Strain transformation:
Transformation matrix:
• Stress Transformation:
• Strain Transformation:
Constitutive Relations: Stress and Strain Transformations
• From the first principles:
Writing in global coordinates
leads to
and
Constitutive Relations: Stiffness Transformations
• Results in monoclinic
behaviour!
Constitutive relations:
Constitutive Relations: Stiffness Transformations
• From the first principles:
Writing in global coordinates
Or
and
Constitutive Relations: Compliance Transformations
• The transformed Stiffness and Compliance tensors are symmetric!
• From invariance of W one can show
Constitutive Relations: Compliance Transformations
[ ] [ ] [ ] [ ]1 1
1 2 2 1 and T T
T T T T− −
= =
• Coefficient of thermal expansion is different in 3 directions!
Constitutive Relations: Thermal Effects
• Thermal strain in principal material directions
where,
These strains will not produce stresses unless restricted!
Transforming strains into
global coordinates
We get,
Constitutive Relations: Thermal Effects
Total strains:
Mechanical strains:
Thus,
gives the thermo-elastic
constitutive equations as
Constitutive Relations: Thermo-Elastic Equations
Constitutive Relations: Hygro-Thermal Effects
Total strains:
with,
The hygro-thermo-elastic
constitutive equation:
Constitutive relation in 3D:
Transverse stresses are zero:
leads to
Constitutive Relations: Planar Equations
0zz xz yzσ τ τ= = =
44 45
45 55
0
0
yz yz xz
xz xz yz
S S
S S
γ τ τ
γ τ τ
= + =
= + =
Transverse normal strain:
Constitutive Relations: Transverse Strain
13 23 36 0zz xx yy xyS S Sε σ σ τ= + + ≠
Constitutive relation in 3D (Principal Directions):
Transverse stresses are zero:
leads to
Constitutive Relations: Planar Equations
Transverse normal strain:
that is,
Therefore, for planar case
Planar Relations: Principal Directions
33 13 11 23 22 36S S Sε σ σ= + + 12 0τ ≠
33 13 11 23 22 0S Sε σ σ= + ≠
Transverse normal strain from stiffness relations:
Transverse normal stress:
gives
Planar Relations: Principal Directions
Stresses in principal directions:
Putting
Planar Relations: Principal Directions
33 11 22 in terms of ,ε ε ε
Stresses in principal directions:
And can be written in a form as
Qij and Cij are not same
Inverse f orm:
Sij are same as in 3D relations
Planar Relations: Principal Directions
Reduced Stiffness Matrix:
Stiffness
Compliance in terms
of engineering
constants
Planar Relations: Principal Directions
Stresses in principal directions:
Strains in principal directions:
Planar Relations: Transformation of Stresses and Strains
Stresses in principal directions:
Stresses in principal directions
Stresses in global directions:
Planar Relations: Hygro-thermo-elastic Relations