Microstructure-Properties - Materials Science and Engineering

1
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Microstructure-Properties: I
Lecture 4B:
Anisotropic Elasticity
27-301
Fall, 2007
Prof. A. D. Rollett
Processing
Performance
Microstructure Properties

2
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Objective of Lecture 4B
• The objective of this lecture is to provide a
mathematical framework for the description of
properties, especially when they vary with
direction.
• A basic property that occurs in almost
applications is elasticity. Although elastic
response is linear for all practical purposes, it
is often anisotropic (composites, textured
polycrystals etc.).

3
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Linear properties
• Certain properties, such as elasticity in
most cases, are linear which means that
we can simplify even further to obtain
or if R 0 = 0,
R = R 0 + PF
R = PF.
e.g. elasticity: σ = C ε
stiffness
In tension, C ≡ Young’s modulus, Y or E.

4
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Elasticity
• Elasticity: example of a property that requires tensors
to describe it fully.
• Even in cubic metals, a crystal is quite anisotropic.
The [111] in many cubic metals is stiffer than the
[100] direction.
• Even in cubic materials, 3
numbers/coefficients/moduli are required to describe
elastic properties; isotropic materials only require 2.
• Familiarity with Miller indices is assumed.

5
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Elastic Anisotropy: 1
• First we restate the linear elastic relations for
the properties Compliance, written S, and
Stiffness, written C (!), which connect stress, σ
, and strain, ε. We write it first in vector-tensor
notation with “:” signifying inner product (i.e.
add up terms that have a common suffix or
index in them):
σ = C:ε
ε = S:σ
• In component form (with suffixes),
σ ij = C ijkl ε kl
ε ij = S ijkl σ kl

6
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Elastic Anisotropy: 2
The definitions of the stress and strain tensors mean
that they are both symmetric (second rank) tensors.
Therefore we can see that
ε 23 = S 2311 σ 11
ε 32 = S 3211 σ 11 = ε 23
which means that,
S 2311 = S 3211
and in general,
S ijkl = S jikl
We will see later on that this reduces considerably the
number of different coefficients needed.

7
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Stiffness in sample coords.
• Consider how to express the elastic properties of a
single crystal in the sample coordinates. In this case
we need to rotate the (4 th rank) tensor from crystal
coordinates to sample coordinates using the
orientation (matrix), a (see lecture A):
c ijkl ' = a im a jn a ko a lp c mnop
• Note how the transformation matrix appears four
times because we are transforming a 4th rank tensor!
• The axis transformation matrix, a, is also written as λ.

8
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Young’s modulus from compliance
• Young's modulus as a function of direction
can be obtained from the compliance tensor
as E=1/s' 1111 . Using compliances and a stress
boundary condition (only σ 11 ≠0) is most
straightforward. To obtain s' 1111 , we simply
apply the same transformation rule,
s' ijkl = a im a jn a ko a lp s mnop

9
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
“matrix notation”
• It is useful to re-express the three quantities
involved in a simpler format. The stress and
strain tensors are vectorized, i.e. converted
into a 1x6 notation and the elastic tensors are
reduced to 6x6 matrices.
"
$
#
! 1 1 ! 1 2 ! 1 3
! 2 1 ! 2 2 ! 2 3
! 3 1 ! 3 2 ! 3 3
%
' ( )
'
&
*
"
$
#
( * ) ( ! 1,! 2,! 3,! 4,! 5,! )
6
! 1 ! 6 ! 5
! 6 ! 2 ! 4
! 5 ! 4 ! 3
%
'
&

10
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
“matrix notation”, contd.
• Similarly for strain:
"
$
#
! 1 1 ! 1 2 ! 1 3
! 2 1 ! 2 2 ! 2 3
! 3 1 ! 3 2 ! 3 3
The particular definition of shear strain used in the
reduced notation happens to correspond to that used
in mechanical engineering such that ε 4 is the change
in angle between direction 2 and direction 3 due to
deformation.
%
'
&
( ) *
"
$
$
#
( * ) ( ! 1,! 2,! 3,! 4,! 5,! )
6
! 1
1
2 ! 6
1
2 ! 5
1
2 ! 6
! 2
1
2 ! 4
1
2 ! 5
1
2 ! 4
! 3
%
'
'
&

11
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Work conjugacy, matrix inversion
• The more important consideration is that the
reason for the factors of two is so that work
conjugacy is maintained.
dW = σ:dε = σ ij : dε ij = σ k • dε k
Also we can combine the expressions
σ = Cε and ε = Sσ to give:
σ = CSσ, which shows:
I = CS, or, C = S -1

12
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Tensor conversions: stiffness
• Lastly we need a way to convert the tensor
coefficients of stiffness and compliance to the
matrix coefficients. For stiffness, it is very
simple because one substitutes values
according to the following table, such that
matrix C11 = tensor C 1111 for example.
Tensor 11 22 33 23 32 13 31 12 21
Matrix 1 2 3 4 4 5 5 6 6

13
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
C =
!
#
#
#
#
#
#
#
"
Stiffness Matrix
C 1 1 C 1 2 C 1 3 C 1 4 C 1 5 C 1 6
C 2 1 C 2 2 C 2 3 C 2 4 C 2 5 C 2 6
C 3 1 C 3 2 C 3 3 C 3 4 C 3 5 C 3 6
C 4 1 C 4 2 C 4 3 C 4 4 C 4 5 C 4 6
C 5 1 C 5 2 C 5 3 C 5 4 C 5 5 C 5 6
C 6 1 C 6 2 C 6 3 C 6 4 C 6 5 C 6 6
$
&
&
&
&
&
&
&
%

14
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Tensor conversions: compliance
• For compliance some factors of two are
required and so the rule becomes:
pS = S ijkl mn
p =1 m AND n " [ 1,2,3 ]
p = 2 m XOR n " [ 1,2,3 ]
p = 4 m AND n " [ 4,5,6]

15
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Relationships between coefficients:
C in terms of S
Some additional useful relations between coefficients
for cubic materials are as follows. Symmetrical
relationships exist for compliances in terms of
stiffnesses (next slide).
C 11 = (S 11 +S 12 )/{(S 11 -S 12 )(S 11 +2S 12 )}
C 12 = -S 12 /{(S 11 -S 12 )(S 11 +2S 12 )}
C 44 = 1/S 44 .

16
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
S in terms of C
The relationships for S in terms of C are symmetrical to
those for stiffnesses in terms of compliances (a
simple exercise in algebra!).
S 11 = (C 11 +C 12 )/{(C 11 -C 12 )(C 11 +2C 12 )}
S 12 = -C 12 /{(C 11 -C 12 )(C 11 +2C 12 )}
S 44 = 1/C 44 .

17
Effect of symmetry on stiffness matrix
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
• Why do we need to look at the effect of symmetry?
For a cubic material, only 3 independent coefficients
are needed as opposed to the 81 coefficients in a 4th
rank tensor. The reason for this is the symmetry of
the material.
• What does symmetry mean? Fundamentally, if you
pick up a crystal, rotate [mirror] it and put it back
down, then a symmetry operation [rotation, mirror] is
such that you cannot tell that anything happened.
• From a mathematical point of view, this means that
the property (its coefficients) does not change. For
example, if the symmetry operator changes the sign
of a coefficient, then it must be equal to zero.

18
Effect of symmetry on stiffness matrix
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
• Following Reid, p.66 et seq.:
Apply a 90° rotation about the crystal-z axis
(axis 3),
C’ ijkl = O im O jn O ko O lp C mnop :
C’ = C
C ! =
#
%
%
%
%
%
%
%
%
$ %
O 4 z =
C 22 C 21 C 23 C 25 "C 24 "C 26
C 21 C 11 C 13 C 15 "C 14 "C 16
C 23 C 13 C 33 C 35 "C 34 "C 36
C 25 C 15 C 35 C 55 "C 54 "C 56
"C 24 "C 14 "C 34 "C 54 C 44 C 46
"C 26 "C 16 "C 36 "C 56 C 46 C 66
" 0 !1 0%
$ '
$ 1 0 0'
$ '
# 0 0 1&
&
(
(
(
(
(
(
(
(
' (

19
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Effect of symmetry, 2
• Using P’ = P, we can equate coefficients and
find that:
C 11 =C 22 , C 13 =C 23, C 44 =C 35 , C 16 =-C 26,
C 14 =C 15 = C 24 = C 25 = C 34 = C 35 = C 36 = C 45 = C 46
= C 56 = 0.
C ! =
# C1 1 C1 2 C1 3 0 0 C1 6 &
% C1 2 C1 1 C1 3 0 0 "C1 6(
% C1 3
% 0
%
0
%
$ C1 6
C1 3
0
0
"C1 6
C3 3
0
0
0
0
C4 4
0
0
0
0
C4 4
C4 6
0
0
C4 6
C6 6
(
(
(
(
'

20
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Effect of symmetry, 3
• Thus by repeated applications of the
symmetry operators, one can demonstrate
(for cubic crystal symmetry) that one can
reduce the 81 coefficients down to only 3
independent quantities. These become two in
the case of isotropy.
! C11 C12 C12 0 0 0 $
#
&
# C12 C11 C12 0 0 0 &
#
&
# C12 C12 C11 0 0 0 &
#
0 0 0 C44 0 0
&
#
&
# 0 0 0 0 C44 0 &
#
&
" # 0 0 0 0 0 C44% &

21
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Cubic crystals: anisotropy factor
• If one applies the symmetry elements of
the cubic system, it turns out that only
three independent coefficients remain: C 11 ,
C 12 and C 44 , (similar set for compliance).
From these three, a useful combination of
the first two is
C' = (C 11 - C 12 )/2
• See Nye, Physical Properties of Crystals

22
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Zener’s anisotropy factor
• C' = (C 11 - C 12 )/2 turns out to be the stiffness
associated with a shear in a direction
on a plane. In certain martensitic
transformations, this modulus can approach
zero which corresponds to a structural
instability. Zener proposed a measure of
elastic anisotropy based on the ratio C 44 /C'.
This turns out to be a useful criterion for
identifying materials that are elastically
anisotropic.

23
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Rotated compliance (matrix)
• Given an orientation a ij , we transform the
compliance tensor, using cubic point group
symmetry, and find that:
S 1! 1 = S1 1 a1 1
( 4 4 4
+ a1
2 + ) a1
3
+ 2S 1 2 a 1 2
( 2 2 2 2 2 2
a1 3 + a1 1a1
2 + ) a1 1a1
3
( )
2 2 2 2 2 2
+ S4 4 a1 2a1
3 + a1 1a1
2 + a1 1a1
3

24
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Rotated compliance (matrix)
• This can be further simplified with the aid of the
standard relations between the direction cosines,
a ik a jk = 1 for i=j; a ik a jk = 0 for i≠j, (a ik a jk = δ ij ) to read as
follows.
s 1! 1 = s1 1 "
2( s1 1 " s1 2 " 1
2
s ) 4 4 # 1
{ 2 2 2 2 2 2
#2 + # 2#3
+ }
#3#1
• By definition, the Young’s modulus in any direction is
given by the reciprocal of the compliance, E = 1/S’ 11.

25
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Anisotropy in cubic materials
• Thus the second term on the RHS is zero for
directions and, for C 44 /C'>1, a maximum in
directions (conversely
a minimum for C 44 /C'

26
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Stiffness coefficients, cubics
[Courtney]

27
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Anisotropy in terms of moduli
• Another way to write the above equation is to
insert the values for the Young's modulus in
the soft and hard directions, assuming that
the are the most compliant
direction(s). (Courtney uses α, β, and γ in
place of my α 1 , α 2 , and α 3 .) The advantage of
this formula is that moduli in specific
directions can be used directly.
1
E uvw
= 1
E 100
"
! 3#
$
1
E 100
&
E111' ( 2 2 2 2 2 2
1(
2 + (2(
3
+(3 (1
! 1
%
( )

28
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Example Problem
Should be E = 18.89

29
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Summary
• We have covered the following topics:
– Linear properties
– Non-linear properties
– Examples of properties
– Tensors, vectors, scalars.
– Magnetism, example of linear (permeability), nonlinear
(magnetization curve) with strong
microstructural influence.
– Elasticity, as example as of higher order property,
also as example as how to apply (crystal)
symmetry.

30
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Supplemental Slides
• The following slides contain some useful
material for those who are not familiar with all
the detailed mathematical methods of
matrices, transformation of axes etc.

31
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Notation
F Stimulus (field)
R Response
P Property
j electric current
E electric field
D electric polarization
ε Strain
σ Stress (or conductivity)
ρ Resistivity
d piezoelectric tensor
C elastic stiffness
S elastic compliance
a rotation matrix
W work done (energy)
I identity matrix
O
Y Young’s Young s modulus
δ Kronecker delta
e axis (unit) vector
T tensor
αα direction cosine
symmetry operator (matrix)

32
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Bibliography
• R.E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford University
Press, 2004, 620.112 N55P.
• De Graef, M., lecture notes for 27-201.
• Nye, J. F. (1957). Physical Properties of Crystals. Oxford, Clarendon Press.
• T. Courtney, Mechanical Behavior of Materials, McGraw-Hill, 0-07-013265-8, 620.11292
C86M.
• Chen, C.-W. (1977). Magnetism and metallurgy of soft magnetic materials. New York,
Dover.
• Chikazumi, S. (1996). Physics of Ferromagnetism. Oxford, Oxford University Press.
• Attwood, S. S. (1956). Electric and Magnetic Fields. New York, Dover.
• Newey, C. and G. Weaver (1991). Materials Principles and Practice. Oxford, England,
Butterworth-Heinemann.
• Kocks, U. F., C. Tomé, et al., Eds. (1998). Texture and Anisotropy, Cambridge University
Press, Cambridge, UK.
• Reid, C. N. (1973). Deformation Geometry for Materials Scientists. Oxford, UK, Pergamon.
• Braithwaite, N. and G. Weaver (1991). Electronic Materials. The Open University, England,
Butterworth-Heinemann.

33
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Mathematical Descriptions
• Mathematical descriptions of properties are available.
• Mathematics, or a type of mathematics provides a
quantitative framework. It is always necessary,
however, to make a correspondence between
mathematical variables and physical quantities.
• In group theory one might say that there is a set of
mathematical operations & parameters, and a set of
physical quantities and processes: if the mathematics
is a good description, then the two sets are
isomorphous.

34
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Non-Linear properties, example
• Another important example of non-linear properties is plasticity,
i.e. the irreversible deformation of solids.
• A typical description of the response at plastic yield
(what happens when you load a material to its yield stress)
is elastic-perfectly plastic. In other
words, the material responds
elastically until the yield stress is
reached, at which point the stress
remains constant (strain rate
unlimited).
• A more realistic description is a power-law with
a large exponent, n~50. The stress is scaled by
the crss, and be expressed as either shear stressshear
strain rate [graph], or tensile stress-tensile
strain [equation].
˙
! =
# " &
% (
$ " yield '
n

35
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Einstein Convention
• The Einstein Convention, or summation rule
for suffixes looks like this:
A i = B ij C j
where “i” and “j” both are integer indexes
whose range is {1,2,3}. So, to find each “i th ”
component of A on the LHS, we sum up over
the repeated index, “j”, on the RHS:
A 1 = B 11 C 1 + B 12 C 2 + B 13 C 3
A 2 = B 21 C 1 + B 22 C 2 + B 23 C 3
A 3 = B 31 C 1 + B 32 C 2 + B 33 C 3

36
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Matrix Multiplication
• Take each row of the LH matrix in turn and
multiply it into each column of the RH matrix.
• In suffix notation, a ij = b ikc kj
( a! + b" + c# a$ + b% + cµ a# + b& + c' +
*
-
* d! + e" + f# d$ + e% + fµ d# + e& + f' -
*
-
) l! + m" + n# l$ + m% + nµ l# + m& + n',
=
( a b c +
* -
* d e f -
* -
) l m n,
.
( ! $ # +
* -
* " % & -
* -
) / µ ' ,

37
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Properties of Rotation Matrix
• The rotation matrix is an orthogonal matrix, meaning
that any row is orthogonal to any other row (the dot
products are zero). In physical terms, each row
represents a unit vector that is the position of the
corresponding (new) old axis in terms of the (old)
new axes.
• The same applies to columns: in suffix notation -
a ija kj = δ ik , a jia jk = δ ik
! a b c$
# &
# d e f &
# &
" l m n%
bc+ef+mn = 0
ad+be+cf = 0

38
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Matrix
representation
of the rotation
point groups
Kocks: Ch. 1 Table II

39
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Homogeneity
• Stimuli and responses of interest are, in general, not scalar quantities
but tensors. Furthermore, some of the properties of interest, such as
the plastic properties of a material, are far from linear at the scale of a
polycrystal. Nonetheless, they can be treated as linear at a suitably
local scale and then an averaging technique can be used to obtain the
response of the polycrystal. The local or microscopic response is
generally well understood but the validity of the averaging techniques is
still controversial in many cases. Also, we will only discuss cases
where a homogeneous response can be reasonably expected.
• There are many problems in which a non-homogeneous response to a
homogeneous stimulus is of critical importance. Stress-corrosion
cracking, for example, is a wildly non-linear, non-homogeneous
response to an approximately uniform stimulus which depends on the
mechanical and electro-chemical properties of the material.

40
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
The “RVE”
• In order to describe the properties of a
material, it is useful to define a representative
volume element (RVE) that is large enough to
be statistically representative of that region
(but small enough that one can subdivide a
body).
• For example, consider a polycrystal: how
many grains must be included in order for the
element to be representative of that point in
the material?

41
Objective
Linear
Ferromagnets
Non-linear
properties
Electric.
Conduct.
Tensors
Elasticity
Symmetry
Transformations of Stress & Strain Vectors
!
• It is useful to be able to transform the axes
of stress tensors when written in vector
form (equation on the left). The table
(right) is taken from Newnham’s book. In
vector-matrix form, the transformations
are:
$ # 1"
' +
& ) -
&
# " 2)
-
& # " ) - 3
& ) = -
&
# " 4)
-
& # " 5)
-
& )
% # "
-
6(
,
* 11 * 12 * 13 * 14 * 15 * 16
* 21 * 22 * 23 * 24 * 25 * 26
* 31 * 32 * 33 * 34 * 35 * 36
* 41 * 42 * 43 * 44 * 45 * 46
* 51 * 52 * 53 * 54 * 55 * 56
* 61 * 62 * 63 * 64 * 65 * 66
. $ # ' 1
0 & )
0 &
# 2)
0 & # ) 3
0 & )
0 &
# 4)
0 & # 5)
0 & )
/ % # 6(
!
"
# i = $ ij# j
%1
# i = $ ij
# " j
%1T
& " i = $ ij & j
T
&i = $ ij&
" j