/ DSM / IRAMIS / LLB)

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23/10/2012

ANF Métallurgie Fondamentale |Vincent Klosek (CEA / DSM / IRAMIS / LLB)

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Residual Stresses ?

Static multiaxial stresses within an isolated solid in mechanical equilibrium,

neither subjected to external force nor to external moment

Result from the thermo-mechanical history

of the considered material

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8 NOVEMBRE 2012 CEA | 23 OCTOBRE 2012| PAGE 3

n

T

(

M

,

)

=

σ

⋅

M da n

A

B

(4)

1D

M x

dx

du

x

u













=













∆

=

→ ∆

lim

0

ε

))

(

)

(

2

1 )

(

X

=

∇

ξ

X

+

t

∇

ξ

X

ε

(5)

Constitutive Laws

Local relations between σσσσ, εεεε and T Thermoelasticity

Thermoelasticity in the framework of infinitesimal strain theory ( linearized relation):

)

(

:

−

T

−

T

₀

=

ε

k

σ

A

Isotropic elastic material, isothermal transformation:

ij ij ij

λ

ε

δ

µε

σ

=

⋅

(

₁₁

+

₂₂

+

₃₃

)

⋅

+

2

Young modulus: Poisson ratio:

µ

λ

µ

λ

µ

+

=

(

3

2 )

E

)

(

2 λ

µ

λ

ν

+

=

)

(

:

−

T

−

T

₀

=

ε

k

σ

A

(6)

0 )

rot

(

rot

g d

ε

=

ion accommodat elastic free stress total

ε

=

+

(7)

Strain Incompatibility

Application of a compatible strain field to a heterogeneous material:

B

A+B

σ

RESIDUAL STRESSES

A

B

A

A+B

ε

0

A res

σ

B res

σ

(8)

Main sources

various scales

Atomic scale: point defects (very local influence) Single crystal scale: dislocations, precipitates, etc…

Polycrystal scale:

Plastic strain incompatibilities

phase transformations

(9)

Various scales…

1st order stresses: in equilibrium at the whole sample scale ( scale considered by engineers for structures calculations)

)

(

)

(

)

(

)

(

X

I

X

II

X

III

X

res

σ

=

+

∫

=

res I

V

(

)

1 )

(

X

σ

X

σ

2nd _{order stresses: in equilibrium at the scale of a group of crystallites}

3rd _{order stresses: in equilibrium at the scale of a crystallite}

Defects of crystallographic lattice (dislocations, precipitates, vacancies, grain boundaries, etc…)

∫

=

V

(

)

(

X

σ

X

σ

∫

=

v res II

v

(

)

1 )

(

X

σ

X

σ

))

(

)

(

)

(

)

(

X

res

X

I

X

II

X

III

σ

=

−

+

(10)

Why evaluating them is so important?...

Strongly influence the mechanical behaviour of a component - harmful: premature fracture (fatigue, cracking, etc…)

- favourable: example of prestress treatments (shot peening, etc…)

Significant effects on performance, safety, reliabillity of a component, a structure. Improvement of the processes (thermal or mechanical treatment required ?...)

At a more fundamental point of view:

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Examples of significant manifestations

Silver Bridge (WV, USA), 1967 Drying of wood

(12)

Experimental determination of elastic strains

Destructive or semi-destructive methods

(13)

(semi-) destructive methods

Principle: some matter is removed mechanical equilibrium is affected

system tends to reach a new equilibrium it changes its shape

Hole drilling method (incremental or not) or ring core method

plane stress approximation, isotropic elasticity law, empirical coefs to determine… Contour method

(14)

Non destructive methods

Ultrasonic technique

dependence of the propagation velocity of ultrasonic waves on stress state

(non linear elasticity, use of acoustoelastic coefficients)

Barkhausen noise (ferromagnetic materials) Discontinuous motion of Bloch walls

stress inverse magnetostrictive effect

technique very sensitive to microstructural defects (…tricky to isolate contributions)

Raman spectrometry

dependence of optic phonon frequencies on stress state Diffraction

(15)

Theory of diffraction by distorted crystals

(Krivoglaz, 1969)

s

g

K

=

+

Diffraction vector

Bragg:

I

≠

0 ⇔

s

=

0 Bragg:

I

≠

0 ⇔

s

=

0

(non distorted perfect crystal)

Scattered intensity (normalised):

∫∫

⋅

_⋅

∆ ⋅

=

x

n

K

e

n s

e

u K

d

I

(

)

2πi 2πi

)

(

)

(

)

,

(

x

n

u

x

u

x

n

u

=

−

∆

initial atom positions

(16)

-Intensity distribution I(K) directly related to the projection of the relative

displacement between atoms

∆

u along K

Directional measurement

!

nK

K

n KK

K

u

K

(x)

ε

K

x

=

⋅

=

∆

⋅

→0

lim

²

1 )

(

ε

- Homogeneous deformation

diffraction peak shift

(back to Bragg law!)

(17)

Selectivity of diffraction methods:

Strain is measured for grains in diffraction condition only !!!

K

(18)

K

Selectivity of diffraction methods:

(19)

What do we measure ?...

1st order moment

∫

+∞ ∞ −

⋅

=

s

I

(

s

)

ds

) 1 (

µ

2nd _{order (centered) moment}

∫

+∞ ∞ −

⋅

=

s

²

I

(

s

)

ds

) 2 (

µ

(pos. of the « gravity centre » of the peak

K

KK ) 1 (

µ

ε

_Ω

=

−

Strain along K averaged over the diffracting volume 0 0 tan

θ

ε

= ∆ =− ∆ hkl hkl KK d d

If homogeneous strain over Ω!

(« peak width »)

²

) 2 ( 2

K

KK

µ

ε

=

Ω

Variance of elastic strains along K over the diffracting volume

Ideal

Ideal for

for coupling

coupling with

with homogenization

homogenization techniques!

techniques!

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Interpretation of peak shifts

Stress at a given point x:

)

(

:

)

(

)

(

_x

_B

_x

σ

x

σ

res

+

=

σ

=

0

res with

σ

=

0

Strain:

ε

(

x

)

=

S

(

x

)

:

σ

(

x

)

σ

=

0 +

General Formulation :

Ω Ω Ω

=

⊗

+

⊗

>

<

₂

:

₂

:

S

:

σ

res

K

σ

B

S

K

KK

ε

(21)

The « sin²

ψ

law »…

1st approximation

σ

_res

=

0 <

>

_Ω

=

K

⊗

₂

K

:

S

:

B

_Ω

:

σ

K

KK

ε

2

nd

_{approximation: isotropic elasticity…}

σ

S

K

:

2

K

KK

⊗

=

>

<

ε

_Ω

ε

_Ω

=

1 +

ν

(

σ

₁₁

cos

²

ϕ

+

σ

₁₂

sin

2 ϕ

+

σ

₂₂

sin

²

ϕ

−

σ

₃₃

)

sin

²

ψ

E

KK

σ

S :

:

2

K

KK

>

=

<

ε

_Ω

ψ

ϕ

σ

ϕ

σ

ν

σ

ν

σ

ν

2 sin

)

sin

cos

(

1 )

(

1 ²

sin

)

²

sin

2 sin

²

cos

(

23 13 33 22 11 33 33 22 12 11

+

−

+

Ω

E

KK

Too often applied out of

hypothesis validity…

33 22

11

,

σ

,

σ

(22)

Example 1: welding

Process involving melting

Temperature gradients

Withdrawal during solidification

Residual

Residual stresses

stresses

Residual

Residual stresses

stresses

If no cohesion

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Determination by means of neutron diffraction

tangential stress tangential stress

Tensile residual stress

distance from centre (mm)

d e p th f ro m i n n e r s u rf a c e ( m m ) (MPa) tangential stress tangential stress axial stress axial stress (coll. CEA/DEN)

(24)

Example 2: Intergranular stresses in Zr

0 =

σ

∆

T = -600°C,

SC estimates – isotropic elastic behaviour

{00.2} reflections

Isotropic texture

a

c

α

≈

2

(25)

Experimental observations (neutron diffraction)

Example 2: Intergranular stresses in Zr

(Coll. O. Castelnau (PIMM))

ΣΣ

= 450

= 450 MPa

MPa

Importance to

Importance to take

take microstructure

microstructure into

into account

account for data

for data analysis

analysis !!!

!!!

(26)

Various scales – various mechanisms : need to consider physical, chemical and mechanical phenomena (i.e. metallurgicalmetallurgical phenomenaphenomena!)

Stress fields are most of the time complex and heterogeneous

Characterization of residual stresses is of fundamental and applied importances (but relevant scales are not the same!)

(but relevant scales are not the same!)

Various techniques to determine residual stresses

Diffraction (RX & neutrons) allows to characterize intra- and intergranular heterogeneities – ideal for coupling with micromechanical modelling

BUT in every case: interpretating measurement results is far from being trivial

RESIDUAL STRESS MEASUREMENT RESIDUAL STRESS MEASUREMENT ! Numerical prediction of residual stresses within components ?

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