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Final

EARTH 135 Lecture 4: Lecture4.1Exam


Department
Earth Science
Course Code
EARTH 135
Professor
Tanimoto
Study Guide
Final

This preview shows pages 1-2. to view the full 6 pages of the document.
Lecture'4'Stress-Strain'Relationship':'Elastic'constants''
4.1'General'linear'case'
'
When%stress%and%strain%are%small,%the%relationship%between%them%is%linear.%
%
(1) Stress%has%six%independent%components%
σ
xx
,%
σ
yy
,%
σ
zz
,%
σ
xy
,%
σ
yz
,%
σ
zx
%with%its%
symmetry%property%
σ
ij =
σ
ji
.%Strain%also%has%six%independent%components:%
ε
xx
,%
ε
yy
,%
ε
zz
,%
ε
xy
,%
ε
yz
,%
ε
zx
.%%Then%it%follows%the%relationship%that%connects%them%
can%be%written%by%a%6x6%matrix.%We%write%it%as%
%
%
σ
xx
σ
yy
σ
zz
σ
xy
σ
yz
σ
zx
=
Cxxxx Cxxyy Cxxzz Cxxxy Cxxyz Cxxzx
Cyyxx Cyyyy Cyyzz Cyyxy Cyyyz Cyyzx
Czzxx Czzyy Czzzz Czzxy Czzyz Czzzx
Cxyxx Cxyyy Cxyzz Cxyxy Cxyyz Cxyzx
Cyzxx Cyzyy Cyzzz Cyzxy Cyzyz Cyzzx
Czxxx Czxyy Czxzz Czxxy Czxyz Czxzx
ε
xx
ε
yy
ε
zz
ε
xy
ε
yz
ε
zx
% % (1)%
Matrix%elements%in%this%6x6%matrix%are%elastic%constants%and%are%written%a%%
,%where%the%indices%i,%j,%k%and%l%vary%x,%y,%and%z.%%
%
%
(2) There%are%36%independent%components.%%But%an%additional%condition,%the%
existence%of%strain%energy,%reduces%this%number%to%21%by%the%following%
reason:%
%
When%deformation%occurs%in%an%elastic%medium,%there%is%always%a%strain%
energy%change.%This%is%given%by%
(1 / 2)
σ
ij
ε
ij
%where%we%sum%over%i%and%j%(over%x,%
y%and%z).%%Since%
σ
ij =Cijkl
ε
kl
,%we%can%write%also%this%formula%as%
(1 / 2)Cijkl
ε
ij
ε
kl
.%
We%sum%over%i,%j,%k%and%l%(they%change%as%x,%y,%z).%%This%is%a%scalar.%
%
You%should%get%the%same%value%if%we%flip%(ij)%and%(kl)%as%in%
(1 / 2)Cklij
ε
ij
ε
kl
.%This%
means%there%is%symmetry%
Cijkl =Cklij
,%meaning%that%the%above%matrix%should%
be%a%symmetric%matrix.%This%reduces%the%independent%components%of%elastic%
constants%to%21.%
%
In%the%most%general%case,%the%number%of%elastic%constants%is%21.%%
This%is%the%most%anisotropic%case%(material).%
%
'
'
'

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

4.2'Isotropic'case'
When%elastic%constants%are%the%same%in%all%directions,%the%medium%is%called%
isotropic.%%In%an%isotropic%medium,%because%of%symmetry,%the%number%of%elastic%
constants%is%reduced%to%2%from%21.%
%
There%are%typically%two%different%choices%of%independent%parameters.%We%call%them%
the%case%of%Lame’s%constants%
(
λ
,
µ
)
%and%the%case%of%Young’s%modulus%and%Poisson’s%
ratio,%
(E,
ν
)
.%For%each%case,%the%relationships%are%
%
%
(
λ
,
µ
)
%
%
σ
xx
σ
yy
σ
zz
σ
xy
σ
yz
σ
zx
!
"
#
#
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
&
&
=
λ
+2
µλ λ
000
λ λ
+2
µλ
000
λ λ λ
+2
µ
000
0 0 0 2
µ
0 0
0 0 0 0 2
µ
0
0 0 0 0 0 2
µ
!
"
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
ε
xx
ε
yy
ε
zz
ε
xy
ε
yz
ε
zx
!
"
#
#
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
&
&
%(2)%
%
%
(E,
ν
)
%
ε
xx
ε
yy
ε
zz
ε
xy
ε
yz
ε
zx
!
"
#
#
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
&
&
=
1 / E
ν
/E
ν
/E000
ν
/E1 / E
ν
/E000
ν
/E
ν
/E1 / E000
0 0 0 (1+
ν
) / E0 0
0 0 0 0 (1+
ν
) / E0
0 0 0 0 0 (1+
ν
) / E
!
"
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
σ
xx
σ
yy
σ
zz
σ
xy
σ
yz
σ
zx
!
"
#
#
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
&
&
%%
%%%%%%(3)%
%
'
'
'
'
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