Download - FMDF cours 181011 - LEM3
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 1
In case there is no infiltration under the dam, the angle α is given by
So that
γb =2 γe
From where
In case with infiltration under the dam, the angle a is given by
So that
From where
A B
y
x
Hα
Sol
O
γ e γ b
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 2
The Airy stress function
associated with this
loading is :
( , ) sinA r cPrθ θ θ=
Show that A(r,θ) is biharmonic.
Determine the components of the stress tensor.
Determine the constant c as a function of angle α.
Calculate the stresses when α=π/2.
Tutorial 1 (continued) : Semi infinite plane under point load
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 3
2 2 2 2
2 2 2 2 2 2
1 1 1 1( )
A A AA
r r r r r r r rθ θ ∂ ∂ ∂ ∂ ∂ ∂∆ ∆ = + + + + ∂ ∂ ∂ ∂ ∂ ∂
( , ) sinA r cPrθ θ θ=
The stress function A(r,θ)
is well biharmonic
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 4
The stresses are defined by the following expressions
( , ) sinA r cPrθ θ θ=
The balance of forces is written to determine the constant c :
If α=π/2, the constant c is -1/πand the stress σr is the given by :
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 5
Tutorial 1 (continued) : Stress field in a plate loaded in tension
and pierced with a tiny hole
σ ∞
σ ∞
x
y
r θM
2 2
2( , ) ln cos 2
fA r br c r d er
rθ θ = + + + +
The Airy stress function
associated with this
loading is :
Show that ( , ) is biharmonic
Determine the stress , ,
( is the radius of the hole)
r r
A r
a
θ θ
θσ σ τ
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 6
2 2
2( , ) ln cos 2
fA r br c r d er
rθ θ = + + + +
σ ∞
σ ∞
x
y
r θM
infinite plate loaded in tension and pierced
by a circular hole of radius a
2 2
2 2 2
1 1A A AA
r r r r θ∂ ∂ ∂∆ = + +∂ ∂ ∂
3
22 2 cos 2
A c fbr er
r r rθ∂ = + + − ∂
2
2 2 4
62 2 cos 2
A c fb e
r r rθ∂ = − + + ∂
22
2 24 cos 2
A fd er
rθ
θ∂ = − + + ∂
2
44 cos 2
dA b
rθ∆ = −
3
8cos 2
A d
r rθ∂∆ =
∂2
2 4
24cos 2
A d
r rθ∂ ∆ = −
∂2
2 2
16cos 2
A d
rθ
θ∂ ∆ =∂
( ) 0A∆ ∆ =
( , ) is well bi-harmonicA r θ
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 7
, , , ,2
,
1 1 1r r rr r
r
A A A Ar r r
θθ θ θ θσ σ τ = + = = −
2 2
2( , ) ln cos 2
fA r br c r d er
rθ θ = + + + +
σ ∞
σ ∞
x
y
rθ
M
3
22 2 cos 2
A c fbr er
r r rθ∂ = + + − ∂
2
2 2 4
62 2 cos 2
A c fb e
r r rθ∂ = − + + ∂
22
2 24 cos 2
A fd er
rθ
θ∂ = − + + ∂
2 2 42 2 4 6 cos 2r
c d fb e
r r rσ θ = + − + +
2 4
62 2 cos 2
c fb e
r rθσ θ = − + +
2 4
62 2 sin 2r
d fe
r rθτ θ = − −
2
22 sin 2
A fd er
rθ
θ∂ − = + + ∂
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 8
Boundary conditions at ∞ , ie far from the hole
0 0 0
( , ) 0 0
0 0 0
x yσ σ ∞
=
( , ) ( , ) tr P x y Pσ θ σ= ⋅ ⋅cos sin 0
sin cos 0
0 0 1
P
θ θθ θ
= −
2
2
sin (1 cos 2 )2
cos (1 cos 2 )2
sin cos sin 22
r
r
θ
θ
σσ σ θ θ
σσ σ θ θ
στ σ θ θ θ
∞∞
∞∞
∞∞
= = −
= = +
= =
at ∞ r >> a2
2
22
b
e
σ
σ
∞
∞
=
=
σ ∞
σ ∞
x
y
rθ
M
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 9
Boundary conditions at r=a
( , ) ( , ) 0r a aσ θ τ θ θ= = ∀
2 2 44 6 cos 2
2 2r
c d f
a a a
σ σσ θ∞ ∞
= + − + +
2 4
62 sin 2
2r
d f
a aθ
στ θ∞
= − −
2
2 42
4
2 4
4 62 2
et2 6 3
2 62 2
d fd a
a ac a
d ff a
a a
σ σσ
σ σ
∞ ∞
∞
∞ ∞
+ = − = −
= − + = =
σ ∞
σ ∞
x
y
rθ
M
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 10
2 2 4
2 2 4
2 4
2 4
2 4
2 4
1 1 4 3 cos 22 2
1 1 3 cos 22 2
1 2 3 sin 22
r
r
a a a
r r r
a a
r r
a a
r r
θ
θ
σ σσ θ
σ σσ θ
στ θ
∞ ∞
∞ ∞
∞
= − − − +
= + + +
= + −
2 2 42 2 4 6 cos 2r
c d fb e
r r rσ θ = + − + +
2 4
62 2 cos 2
c fb e
r rθθσ θ = − + +
2 4
62 2 sin 2r
d fe
r rθτ θ = − −
22
22
b
e
σ
σ
∞
∞
=
=
( ,0) 3aθσ σ ∞=
3σ ∞
σ ∞
2
2c a
σ ∞
= −
2
4
2
36
2
d a
f a
σ
σ
∞
∞
= −
=
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 11
Airy stress function for few loadings (1)
, , ,x yy y xx xy xyA A Aσ σ σ= = = −
2( , ) Beam in tractionA x y ay=
( , ) Beam in shearA x y axy=
3Beam subjected to
( , )bending moment
A x y ay
=
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 12
Airy stress function for few loadings (2)
3( , )A x y axy bxy
= +
2 2 3 2 3 5( , )A x y ax bx y cy dx y ey
= + + + +
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 13
Airy stress function for few loadings (3)
Axisymmetric loading , ,
10r r rr rA A
rθ θσ σ τ= = =
2( , ) ( )A r A r Crθ = =
2( , ) ( ) lnA r A r a r crθ = = +
2 2( , ) ( ) ln lnA r A r a r br r crθ = = + +
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 14
Airy stress function for few loadings (4)
, , , ,2
,
1 1 1r r rr r
r
A A A Ar r r
θθ θ θ θσ σ τ = + = = −
( , ) sinA r crθ θ θ=
( , ) sin 2A r a bθ θ θ= +
( , ) cosA r crθ θ θ=
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 15
Complex formulation of the Airy stress function
- Holomorphic functions (or analytical functions)
z x iy
z x iy
= += −RST
M(x,y)
�x
�y
= +
= −
RS||
T||
xz z
yz z
i
2
2
( , ) ( , )x y Plan g x yg∈ → ( , ) ( , ) ( , )x y z z g z z
g → →
( )
( ), , ,
, , ,
1
2The derivation rules are
1
2
z x y
z x y
g g ig
g g ig
= − = +
g g g
g i g g
x z z
y z z
, , ,
, , ,( )
= += −
RST
P P x y
Q Q x y
==RST
( , )
( , )g P iQ= +
is holomorphic if 0g
gz
∂ =∂
g zg
xi
g
y' ( ) = ∂
∂= − ∂
∂
E� ���� ����
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 16
* Properties oy analytic functions
g P iQdg
dz
g
xi
g
y= + = ∂
∂= − ∂
∂ avec
Cauchy conditions
P Q
x yP Q P Qi i
P Qx x y y
y x
∂ ∂∂ ∂∂ ∂ ∂ ∂∂ ∂∂ ∂ ∂ ∂∂ ∂
=+ = − + = −
∆ ∆P Q= =E
0� ��� ���The real or imaginary parts of an
analytic function, are harmonic
⇐
Conversely, if ( , ) and ( , ) is an analytic function
verify the Cauchy conditions
P x y Q x yg P iQ = +
- If g is an analytical function, its derivative and its integral are also
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 17
Examples of analytic functions
einz, zn and ln z are analytic functions. Their real and imaginary
parts that are harmonic, can be determined.
( )
( ) ( )
( ) (cos sin )
The associated harmonic fonctions are cos and sin
inz in x iy ny
inz in x iy in x iy
ny ny
f z e e e nx i nx
df f df f dfine ine ne i
dz x dz y dz
e nx e nx
+ −
+ +
− −
= = = +∂ ∂= = = = − =∂ ∂
i
Exchanging n by –n, it is seen that eny cosnx and eny sinnx are also
harmonics. It follows that sinhny sinnx, coshny sinnx, sinhny cosnx
and coshny cosnx, obtained by linear combination of the preceding
harmonic functions, are also harmonic.
The hyperbolic sine and hyperbolic cosine functions are defined by
sinh cosh2 2
ny ny ny nye e e eny ny
− −− += =
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 18
1 1 1
( ) ( ) ( ) (cos sin )
( ) ( )
The associated harmonic fonctions are cos and sin
n n ei n n
n n n
n n
f z z x iy r r n i n
df f df f dfnz n x iy in x iy i
dz x dz y dz
r n r n
θ θ θ
θ θ
− − −
= = + = = +∂ ∂= = + = = + =∂ ∂
i
( ) ln ln( ) ln( ) ln
1 1
The associated harmonic fonctions are ln and
if z z x iy re r i
df f df f i dfi
dz z x x iy dz y x iy dz
r
θ θ
θ
= = + = = +∂ ∂= = = = =∂ + ∂ +
i
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 19
Expressions of the Airy stress function
∆ ∆ Αb g = 0 If then 0 is harmonicP A P P= ∆ ∆ =
( ) is analytic with
P Q
x yf z P iQ
P Q
y x
∂ ∂∂ ∂∂ ∂∂ ∂
== + = −
Calculating the harmonic function ( , )Q x y
Q QdQ dx dy
x y
P PQ dQ dx dy
y x
∂ ∂∂ ∂
∂ ∂∂ ∂
= +
= = − +
( ) ( )1 is also analytic function 4 4
4
p qz f z dz p iq P
x y
∂ ∂ϕ∂ ∂
= = + = =
1 1 1 1If then 0 ( ) is analytic functionp px qy p z p iqχ= Α − − ∆ = = +
A px qy p
z z z
z z z z z z= + +
= +
= + + +
RS|T|
1 1
2
Α
Α
Re ϕ χ
ϕ χ ϕ χ
b g b gb g b g b g b g
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 20
Stresses expressions
σ σx xy yy xy x y y z y zz zzi i i i i+ = − = − + = − = −Α Α Α Α Α Α Α, , , , , , , ,d i c h c h2 2
,
,
,
x yy
y xx
xy xy
σσσ
= Α
= Α
= −Α
( ) ( ) ( ) ( )' ' '' ''x xyi z z z z zσ σ ϕ ϕ ϕ χ + = + − −
g g ig
g g ig
z x y
z x y
, , ,
, , ,
= −
= +
RS|
T|
1
21
2
d id i
g g g
g i g g
x z z
y z z
, , ,
, , ,( )
= += −
RST( ) ( )
Expression of the stress function
from the potential complex ( ) and ( )
1( ) ( )
2
z z
z z z z z z
ϕ χ
ϕ χ ϕ χ Α = + + +
σ σy xy xx xy x y x z x zz zzi i i− = + = + = = +Α Α Α Α Α Α Α, , , , , , , , ,d i c h c h2 2
( ) ( ) ( ) ( )' ' '' ''y xyi z z z z zσ σ ϕ ϕ ϕ χ − = + + +
σ σ ϕ ϕ ϕy x z z z+ = + =2 4' ' Re 'b g d ie j b gc h
( ) ( )( )2 2 '' ''y x xyi z z zσ σ σ ϕ χ− + = +
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 21
Displacements expressions
22
2
2
22
2
2
µ ε σ λλ µ
σ σ λ µλ µ
σ σ σ
µ ε σ λλ µ
σ σ λ µλ µ
σ σ σ
x x x y x y y
y y x y x y x
= −+
+ =++
+ −
= −+
+ =++
+ −
RS||
T||
b g d i b g d i
b g d i b g d i
= ++
−
= ++
−
RS||
T||
22
2
22
2
µ ε λ µλ µ
µ ε λ µλ µ
x xx
y yy
b g
b g
∆ Α Α
∆ Α Α
,
,
∆A Pp
x
q
y= = =4 4
∂∂
∂∂
( ) ( )
( ) ( )
,
,
22 2
22 2
x x
y y
u p y
u q x
λ µµ α
λ µλ µ
µ βλ µ
+= − Α + +
+ = − Α + +
avec αβ
y cy d
x cx d
b gb g
= += − +
RST1
2
( ) ( ) ( ), ,
22 2x y x yu iu p iq i
λ µµλ µ++ = + − Α + Α+
g g ig
g g ig
z x y
z x y
, , ,
, , ,
= −
= +
RS|
T|
1
21
2
d id i
+ = ∂∂
Α Α, ,x yiA
zd i 2
( ) ( ) ( ) ( ) ( ) ( )2 22 2 2 2 ' 'x yu iu z z z z z z
z
λ µ ∂ λ µµ ϕ ϕ ϕ ϕ χλ µ λ µ∂+ Α ++ = − = − − −+ +
( ) ( ) ( ) ( )2 ' 'x y
U iU z z z zµ κ ϕ ϕ χ+ = − −
33 4 for plane strain
with 3
for stress plane1
vλ µκλ µ
νκν
+= = −+
−=+
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 22
�eθ
�x
�er
θ
θ
�y
� � �
� � �e x y
e x y
r = += − +
RSTcos . sin .
sin . cos .
θ θθ θθ
�
�
�
�e
e
x
y
r
θ
θ θθ θ
FHGIKJ =FHGIKJ −
FHG
IKJP avec P :
cos sin
sin cos
u
u
u
u
u u
u u
r x
y
x y
x yθ
θ θθ θ
FHGIKJ =FHGIKJ =
+− +FHG
IKJP
cos sin
sin cos
u iu e u iur
i
x y+ = +−θ
θ ( ) ( ) ( ) ( ) ( )( )2 ' 'i
ru iu e z z z zθθµ κ ϕ ϕ χ−+ = − −
σ σθh h� � �
e e x y
t
r
P P, ,
= ( )22 2
r x y
i
r r y x xyi e i
θθ
θ θ
σ σ σ σσ σ σ σ σ σ
+ = + − + = − +
( ) ( )( )22 2 '' ''i
r ri e z z zθθ θσ σ σ ϕ χ− + = +
* System coordinates change
Because we will use of polar coordinates in the solution of many
problems in elasticity, the previous governing equations will now be
developed in this curvilinear system.
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 23
3 3Let ( ; , , ) denote the Cartesian coordinates system and ( ; , , )
a coordinate system associated with curvilinear coordinates , .
O x y x M xα βα β
��� � � �
The complex number is associated with the , coordinates and
the complex is associated with the curvilinear coordinates , .
z x iy x y
iζ α β α β= +
= +
As ( , ) and ( , ), then we have :
( ) and '( )
x x y y
dzz f f
d
α β α β
ζ ζζ
= =
= =
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 24
We easily show that '( ) | '( ) | , in other words that the argument
of the complex number is equal to , the angle between the two coordinate
systems respectively associated with , and , .
if f e
x y
θζ ζθ
α β
=
� arg '( ) arg arg arg , ,dz
f dz d u x ud
ζ ζ α θζ
= = − = − =�� � �
So we have '( ) | '( ) | and '( ) | '( ) | so thati if f e f f eθ θζ ζ ζ ζ −= =
2'( )=
'( )
ife
f
θζζ
( ) and '( )dz
z f fd
ζ ζζ
= =
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 25
Summary of key findings
The resolution of a plane elasticity problem comes down to the
search for a stress function, called the Airy function A, which is bi-
harmonic, that is to say ∆(∆A)=0.
The expression of this stress function, from the complex potentials ϕand χ which are analytical functions of the complex variable z, is
given by :
The search for the Airy stress function is therefore to find these
complex potentials. The components of the stress tensor and the
displacement vector are then determined by the following
relationships :
( ) ( ) ( ) ( ) ( ) ( )1Re
2z z z z z z z z zϕ χ ϕ χ ϕ χ Α = + = + + +
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 26
In a Cartesian coordinates system (x,y)
In a curvilinear coordinates system associated to varaibles (α,β)
3 4 for plane strain
with 3 for stress plane
1
vκνκν
= −−=+