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11/10/2018 1 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 γ b =2 γ e From where In case with infiltration under the dam, the angle a is given by So that From where A B y x H α 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 : (, ) sin Ar cPr θ 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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### Transcript of FMDF cours 181011 - LEM3

FMDF_cours_181011A. 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
From where
In case with infiltration under the dam, the angle a is given by
So that
From where
A B
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 2
The Airy stress function
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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2
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 3
2 2 2 2
1 1 1 1 ( )
A A A A
( , ) sinA r cPrθ θ θ=
The stress function A(r,θ)
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/π and the stress σr is the given by :
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3
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 θ θ = + + + +
r r
A r
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 6
2 2
r θ θ = + + + +
2 2
3
r r r θ∂ = + + − ∂
r r r θ∂ = − + + ∂
A f d er
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 7
, , , ,2
r
θθ θ θ θσ σ τ = + = = −
2 2
r θ θ = + + + +
r r r θ∂ = + + − ∂
r r r θ∂ = − + + ∂
A f d er
c d f b e
r r r σ θ = + − + +
2 4
c f b e
r r θσ θ = − + +
d f e
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 8
Boundary conditions at ∞ , ie far from the hole
0 0 0
( , ) ( , ) tr P x y Pσ θ σ= ⋅ ⋅ cos sin 0
sin cos 0
0 0 1
r
r
θ
θ
∞ ∞
∞ ∞
∞ ∞
= = −
= = +
= =
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 9
Boundary conditions at r=a
( , ) ( , ) 0r a aσ θ τ θ θ= = ∀
2 2 4 4 6 cos 2
2 2 r
c d f
a a a
σ σσ θ ∞ ∞
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 10
2 2 4
2 2 4
1 1 3 cos 2 2 2
1 2 3 sin 2 2
r
r
c d f b e
r r r σ θ = + − + +

c f b e
r r θθσ θ = − + +
d f e
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 11
, , ,x yy y xx xy xyA A Aσ σ σ= = = −
2( , ) Beam in tractionA x y ay=
( , ) Beam in shearA x y axy=
3 Beam subjected to
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 12
3 ( , )A x y axy bxy
= +
= + + + +
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 13
r θ 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
, , , ,2
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
i
2
2
( , ) ( , )x y Plan g x y g∈ → ( , ) ( , ) ( , )x y z z g z z
g → →
1
2
g z
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 16
* Properties oy analytic functions
g P iQ dg
P Qx x y y
y x
analytic function, are harmonic
verify the Cauchy conditions
P x y Q x y g 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.
( )
( ) ( )
inz in x iy ny
inz in x iy in x iy
ny ny
df f df f df ine 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 cosh 2 2
− −− += =
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 18
1 1 1
n n ei n n
n n n
f z z x iy r r n i n
df f df f df nz n x iy in x iy i
dz x dz y dz
r n r n
if z z x iy re r i
df f df f i df i
dz z x x iy dz y x iy dz
r
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
y x
Q Q dQ dx dy
x y
y x
4
x y
= = + = =
1 1 1 1If then 0 ( ) is analytic functionp px qy p z p iqχ= Α − − = = +
A px qy p
= +
= + + +
χ χ
b g b g b g b g b g b g
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 20
Stresses expressions
,
,
,
= Α
= Α
= −Α
( ) ( ) ( ) ( )' ' '' ''x xyi z z z z zσ σ χ + = + − −
g g ig
g g ig
z x y
z x y
1 ( ) ( )
2
χ
χ 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 χ − = + + +
σ σ y 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 χ− + = +
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A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 21
Displacements expressions
2 2
σ σ σ
σ σ σ
= − +
+ = + +
+ −
= − +
+ = + +
+ −
= + +

= + +

x xx
y yy
b g
b g
Α Α
Α Α
2 2 2x y x yu iu p iq i
λ µµ λ µ ++ = + − Α + Α +
g g ig
g g ig
z x y
z x y
z d i 2
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 ' 'x yu iu z z z z z z
z
λ µ ∂ λ µµ χ λ µ λ µ∂ + Α ++ = − = − − − + +
( ) ( ) ( ) ( )2 ' ' x y
U iU z z z zµ κ χ+ = − −
3 3 4 for plane strain
with 3
νκ ν
A. Zeghloul Fracture mechanics, damage and fatigue – Plane elasticity 22

x
er
θ
θ
y
F HG
+ − + F HG
I KJP
cos sin
sin cos
i
x y+ = +− θ
θ ( ) ( ) ( ) ( ) ( )( )2 ' 'i
ru iu e z z z zθ θµ κ χ−+ = − −
σ σ θ h h
e e x y
θ θ
θ θ
+ = + − + = − +
( ) ( )( )22 2 '' ''i
r ri e z z zθ θ θσ σ σ 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 β α β

the complex is associated with the curvilinear coordinates , .
z x iy x y
iζ α β α β = +
= +
( ) and '( )
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
f dz d u x u d
ζ ζ α θ ζ
= = − = − =
So we have '( ) | '( ) | and '( ) | '( ) | so thati if f e f f eθ θζ ζ ζ ζ −= =
2'( ) =
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 and 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 :
( ) ( ) ( ) ( ) ( ) ( )1 Re
2 z z z z z z z z z χ χ 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