ABSTRACT
E v E
v E E
G
=
−
−
1 0
1 0
0 0 1
σ
σ
τ
(11.1)
Problem:Howthisrelationshiptransformswhenitisexpressedinaxes(x, y)distinctfrom(ℓ, t) and forming any angle θ with the (ℓ, t) coordinates? (See Figure 11.1.)‡
First, let us recall the following:
◾ Recall 1: ”e stress σ acting on a side with normal vector n is given by
σ σ{ } = { }ij n
Columnmatrix of components of stress Stress matrix Column mat
σ rix of directional cosines of
n
(11.2)
◾ Recall 2: ”e coordinates of asame vector V
in two distinct coordinate systems (x, y) and (ℓ, t), such that ( )
x , = θ, are
V V V t V x V yt x y
= + = +
with the relation
V
V
c s
s c
V
V
c
s x
−
= =
=
cos
sin
θ θ
(11.3)
In axes (ℓ, t), let us express, through the relationship in Equation 11.2, the stress acting on a facet ofnormalx :
c
s { } = { } =
where {σ/x} is the stress vector [σij] is the stress matrix
And in axes (x, y), following Equation 11.3,
c s
s c
c
s ij{ } = −
In a similar manner, the stress acting on a facet with the normal y is written in the (x, y)
axes as
c s
s c
s
c ij{ } = −
−
”erefore, the stress matrix in (x, y) axes is
σ σ σ σij ijx y tx y
c s
s c
c s
s c =
= −
−
, ,/ /,
By setting
P
c s
s c [ ] =
−
and observing that matrix [P] is orthogonal, that is, t[P] = [P]−1, we have*
σ σij t
P P = [ ] [ ], ,
where t[P] is the transpose of matrix [P]. In developing that expression,
σ τ τ σ
σ τ τ σ
c s
s c
c s
s c
=
−
−
which can be rearranged to give
σ σ τ
σ σ
xc s cs
s c cs
sc sc c s
=
−
− −
( ) y
xyτ
(11.4)
”en
σ σ[ ] = [ ][ ], ,t x yT
with*
T
c s sc
s c sc
sc sc c s
[ ] = −
− −
( )
With consideration of strains allowing a similar calculation procedure, we can write parallel to this
ε ε ε
ε ε
c s cs
s c cs
cs cs c s
= −
− −
( )
tε
or
ε ε γ
ε ε
c s cs
s c cs
cs cs c s
= −
− −
2 22 2 ( )
tγ
”en
ε γ
ε γ
= ′[ ]
with
′[ ] = − − −
= [ ]T
c s cs
s c cs
cs cs c s
2 22 2 ( )
In this way, we can express Equation 11.1 in axes (x, y), since we have written
ε γ
ε γ
ε γ
= ′[ ]
=
−
T
E v E
v E
;
1 0
1 0
0 0 1
E
G
{ } { } = [ ]{ }σ σ σ, , ,;
from which by substituting
ε
ε
γ
T
E v E
v E E
= ′[ ]
−
−
1 0
1 0
0 0
1 G
T
[ ]
σ
σ
τ
Aftercalculation,thefollowingbehaviorrelationshipappears,writtenintechnicalformincoordinates(x, y)thatmakeanangleθwithaxes(ℓ, t).ItrevealstheelasticmoduliandPoisson’sratios relatingtothesedirections.”enonconventionalcouplingcoeªcientsdenotedbyηandμ* show, for example, that a normal stress induces a distortion.†
ε
ε
γ
η
µ
E
v
E G
v
E E G
=
−
−
E E G η µ
σ
σ
τ1
= + + −
with:
E c E
s E
c s G
v E
( )θ 1
1 2
= + + −
E s E
c E
c s G
v E
G
( )θ 1
1 2
c s E E
v E
c s G
v
E
( ) ( )
( )
θ
θ
= + +
+
− 1
4 1 1
= + − + −
= −
v E
c s c s E E G
G cs
c
( )
( )
1 1 1
2 η
θ E
s E
c s v E G
G cs
− + − −
= −
( )
( ) µ
θ s E
c E
c s v E Gt t
− − − −
( )
(11.5)
σ
σ
τ
E v v
v E v v
=
− −( ) ( )1 1 0
v E v v
E v v
G
( ) ( )1 1 0
0 0
− −
ε
ε
γ
whereappearelasticsti¨nesscoeªcientsasopposedtothoseofEquation11.1referredtoas §exibility coeªcients. To ease writing, it will be preferably noted:
σ σ τ
ε ε γ
E v E
v E E
G
=
0 0
(11.6)
Anidenticalproceduretothatfollowedabovetoobtainstrain-stressbehaviorleadstothe stress-strain relation:
σ σ τ
c s cs
s c cs
cs cs c s
T
= −
− −
( )
σ σ τ
ε ε γ
=
−
− −
=
c s cs
s c cs
cs cs c s
tT T
2 22 2
( )
ε ε γ
(11.7)
Recallthataxes(x, y)arederivedfromaxes(ℓ, t)byrotationθaboutthethirdaxisz.Substituting Equations 11.7 into 11.6, we obtain
σ σ τ
T
E v E
v E E
G
T
= [ ]
[ ]1 1
0 0
′ ε ε γ
which can be rewritten as
σ σ τ
ε ε
xE E E
E E E
E E E
=
xyγ
Oncethecalculationisperformed,thefollowingexpressionsofstinesscoeªcientsEij are obtained, in which c = cos θ and s = sin θ:
σ σ τ
ε ε
xE E E
E E E
E E E
=
E c E s
γ
θ
= +
with
:
E c s v E G
E s E c E c s v E
+ +
= + +
2 2
( )
( ) (
θ
+
= + − + −
G
E c s E E v E c s G
)
( ) ( ) ( )
θ
E c s E E G c s v E
E
4( ) ( ) ( )
(
θ = + − + +
θ
θ
) ( )( )
( )
= − − − − +{ } = −
cs c E s E c s v E G
E cs
s E c E c s v E Gt t t 2 2 2 2 2 − + − +{ }( )( )
expressions in which:
; E E v v
E E v vt t
= −( )
= −( )1 1
(11.8)
”evariationofthesestinesscoeªcientsEij as functions of angle θ is pictured in Figure 11.2 for aplycharacterizedbyverydierentvaluesofmoduliEℓ and Et, corresponding, for example, to the case of unidirectionalber/resin layers.*
11.3 Case of Thermomechanical Loading 11.3.1 Flexibility Coefcients When considering the temperature variations,* the behavior relation in Equation 11.1 should be replaced with the amended form in Equation 10.9, namely,
ε
ε
γ
E v E
v E E
G
=
−
−
1 0
1 0
0 0 1
+
σ
σ
τ
α
α
T∆ t
in which αℓ and αt are the thermal expansion coeªcients of the unidirectional layer along the longitudinaldirectionℓandtransversedirectiont,respectively.Followingthesameprocedureas in Section 11.1 with the same notations, we can write
ε γ
ε γ
σ σ
= [ ]
{ } = [ ]{ } x y t
;′
from where, by substituting,
ε
ε
γ
T
E v E
v E E
= [ ]
−
−′
1 0
1 0
0 0
1 G
T
[ ]
σ
σ
τ
+ [ ]
∆T T t′
α
α
Inthisrelationship,wendagainthe©exibilitymatrixontherightside,thetermsofwhichare described in details in Equation 11.5. ”e second term on the right side is written as
∆ ∆T c s cs
s c cs
cs cs c s
T
c
2 22 2 0
−
− −
=
( )
α α
α α α α α α
+ + −
s
s c
cs
t( )
”erefore, the thermomechanical behavior relationship for a unidirectional layer, written in axes (x, y) other than the specic coordinates (ℓ, t) of unidirectional, can be summarized as follows:
ε
ε
γ
η
µ
E
v
E G
v
E E G
=
−
−
xy E E G η µ
σ
σ
τ1
+
∆T
E E G v v
α
α
α
, , , , x xy xy, ,η µ are given by relations (11.5)
α α αx tc s= +2 2
α α α
α α
s c
cs
= +
= −
α
θ θ
( )
= = c scos ; sin
(11.9)
11.3.2 Stiffness Coefcients By inversion of Equation 10.9, we get
σ σ τ
E v v
v E v v
v E v v
=
−( ) −( )
−
1 1 0
( ) −( )
E v v
G
t1 0
0 0
ε
ε γ
− −( )
+ −( )
−( ) +
∆T
E v v
v E v v
v E v v
1 1
1 1
α α
α −( )
α
Following the procedure of Section 11.2, with the same notations, we can write
σ σ
ε γ
ε γ
{ } = [ ]{ }
= ′
T T , ,
from where, by replacing,
σ σ τ
T
E v E
v E E
G
T
= [ ]
′ 1 1
0 0
− [ ]
+ +
ε ε γ
α α α α
E v E
v E E∆ 1 0
Inthersttermontherightside,wendagainthematrixdetailedinEquation11.8.”esecond term can be developed as follows:
− − − −
+ ∆T
c s cs
s c cs
cs cs c s
E v E
v E t t
( )
α α α +
= ⋅ ⋅ ⋅Et tα
⋅ ⋅ ⋅ − +( ) + +( ) +( ) +∆T
c E v s E v
s E v c E t t t t t
α α α α α α t t t
v
cs E v E v
α α
α α α α
+( ) +( ) − +( )
”erefore,thethermomechanicalbehaviorrelationshipwritteninaxes(x, y)otherthanthe specic unidirectional coordinates (ℓ, t) can be summarized as follows:
σ σ τ
ε ε
xE E E
E E E
E E E
=
T
E
E
E
E E E E
γ
α
α
α
−
∆
E c E vt t
are given by relations (11.8)
α α α= +( ) + +( )
= +( ) + +( )
=
s E v
E s E v c E v
E cs E
α α
α α α α α
α
v E v
c s
E E v v
α α α α
θ θ
+( ) − +( ) = =
= −(
cos ; sin
1 )
= −( )
E E v v
(11.10)
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