Materials & Properties: Mechanical Behaviour...Vacuum, Surfaces & Coatings Group Technology...
Transcript of Materials & Properties: Mechanical Behaviour...Vacuum, Surfaces & Coatings Group Technology...
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 2
Materials & Properties: Mechanical Behaviour C. Garion, CERN TE/VSC
CERN Microcosm exhibition
A dream of engineerโฆ
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Materials & Properties: Mechanical Behaviour C. Garion, CERN TE/VSC
โฆ the hard reality!!!
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1. Material modelling a. Basic notions of material behaviours and strength b. Stress/strain in continuum mechanics c. Linear elasticity d. Plasticity
i. Yield surface ii. Hardening
e. Continuum damage mechanics f. Failure mechanics
2. Structural mechanical analysis a. Vacuum chamber
i. Loads on a chamber ii. Equilibrium equations iii. Stress on tube iv. Instability (buckling)
b. Vacuum system as mechanical system i. Support โ unbalanced force ii. Stability (column buckling)
3. Selection criteria of materials a. Figures of merit of materials b. Some material properties c. Some comparisons based on FoM
4. Conclusion
Plan
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 5
1. Material modelling a. Basic notions of material behaviours and strength b. Stress/strain in continuum mechanics c. Linear elasticity d. Plasticity
i. Yield surface ii. Hardening
e. Continuum damage mechanics f. Failure mechanics
2. Structural mechanical analysis a. Vacuum chamber
i. Loads on a chamber ii. Equilibrium equations iii. Stress on tube iv. Instability (buckling)
b. Vacuum system as mechanical system i. Support โ unbalanced force ii. Stability (column buckling)
3. Selection criteria of materials a. Figures of merit of materials b. Some material properties c. Some comparisons based on FoM
4. Conclusion
Plan
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Basic Notions of Material Strength
ฯeng
ฮตeng
E
ฯy
ฯm
Nominal or engineering stress:
๐๐๐๐๐๐๐๐ =๐น๐น๐๐0
Nominal strain: ๐๐๐๐๐๐๐๐ = ฮ๐ฟ๐ฟ๐ฟ๐ฟ0
= ๐ฟ๐ฟโ๐ฟ๐ฟ0๐ฟ๐ฟ0
Initial section: S0
Force F Initial length: L0
L = L0 + โ
Stress = Force/Section
ฯ[MPa] = F[N]/S[mm2]
Strain = Displacement/length
ฮต = โ[mm]/L[mm]
Basic mechanical properties from tensile tests
E: Young modulus (linear elastic behavior)
ฯy (Rp): Yield strength
ฯm (Rm): Tensile strength
Poisson coefficient: ๐๐ = โ๐๐๐ก๐ก๐ก๐ก๐ก๐ก๐๐๐ก๐ก๐๐๐ก๐ก๐๐๐๐๐ก๐ก๐๐
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๐๐
Elastic brittle behaviour
Basic Material Behaviours
Elastic plastic behaviour
Elastic plastic damageable behaviour Elastic viscoplastic behaviour
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๐๐ = ๐๐๐๐ 1 + ๐๐๐๐๐๐๐๐
๐๐ =๐น๐น๐๐โ ๐๐๐๐๐๐๐๐ โ 1 + ๐๐๐๐๐๐๐๐
True Strain/Stress vs Engineering Strain/Stress
Engineering stress-strain curve
๐๐๐๐ =๐๐๐๐๐๐
True stress-strain curve
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The stress tensor is symmetric: ๐๐ = ๐๐๐๐
dS=dSยทn dF
๐๐(๐๐,๐๐) = lim๐๐๐๐โ0
๐๐๐น๐น๐๐๐๐
๐๐(๐๐,๐๐) = ๐๐ ๐๐ โ ๐๐
Normal stress: ๐๐๐๐ = ๐๐ โ ๐๐ ๐๐ โ ๐๐
Shear stress: ๐๐ = ๐๐ ๐๐ โ ๐๐ โ ๐๐๐๐ โ ๐๐
๐๐๐๐>0: tensile; ๐๐๐๐<0: compression
โ ๐๐1,๐๐2,๐๐3 ,๐๐ =๐๐๐ผ๐ผ 0 00 ๐๐๐ผ๐ผ๐ผ๐ผ 00 0 ๐๐๐ผ๐ผ๐ผ๐ผ๐ผ๐ผ ๐๐1,๐๐2,๐๐3
๐๐ = ๐๐โ๐ผ๐ผ + ๐๐๐ท๐ท with ๐๐โ = 13๐ก๐ก๐ก๐ก ๐๐ the hydrostatic stress, and ๐๐๐ท๐ท the deviatoric stress tensor.
Stress in Continuum Mechanics
Stress vector Stress tensor
M
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ฮฉ0
ฮฉ1 M0
๐๐๐๐0 M
๐๐๐๐
O
ฮฆ ๐๐ = ฮฆ ๐๐0
๐๐๐๐ =๐๐ฮฆ ๐๐0
๐๐๐๐0โ ๐๐๐๐0 = ๐น๐น ๐๐0 โ ๐๐๐๐0
Green Lagrange deformation: ๐ธ๐ธ ๐๐0 =12โ ๐น๐น๐๐ ๐๐0 โ ๐น๐น ๐๐0 โ ๐ผ๐ผ
For small perturbations ๐ข๐ข โช ๐๐0 :
Considering the displacements: (๐๐ = ๐๐0 + ๐ข๐ข ๐๐0 )
๐ธ๐ธ ๐๐0 =12โ๐๐๐ข๐ข ๐๐0
๐๐๐๐0+
๐๐๐ข๐ข ๐๐0
๐๐๐๐0
๐๐
+๐๐๐ข๐ข ๐๐0
๐๐๐๐0
๐๐
โ๐๐๐ข๐ข ๐๐0
๐๐๐๐0
๐๐ ๐ข๐ข =12โ๐๐๐ข๐ข ๐๐0
๐๐๐๐0+
๐๐๐ข๐ข ๐๐0
๐๐๐๐0
๐๐
Strain in Continuum Mechanics
Energy variation: ๐๐๐ผ๐ผ = โซ ๐๐:๐๐๐๐ฮฉ
๐น๐น is the deformation gradient.
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How atoms interact? How material behaves?
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Linear Elasticity
Repulsion
Attraction
U0
The interatomic potential is the sum of 2 contributions:
It presents a minimum, corresponding to the free equilibrium of the atoms. U0 is the bonding energy.
The interactomic force is defined by ๐น๐น = ๐๐๐๐
๐๐๐ก๐ก and can be written: ๐น๐น = ๐๐ โ ๐ก๐ก
This can be generalized in the form (Hookeโs law): ๐๐ = ๐ถ๐ถ: ๐๐๐๐ with ๐ถ๐ถ the stiffness tensor.
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๐๐ = ๐ถ๐ถ: ๐๐๐๐ ๐๐๐๐ = ๐๐:๐๐ with ๐๐ the compliance tensor
๐๐ = ๐๐ โ ๐ก๐ก๐ก๐ก ๐๐๐๐ ๐ผ๐ผ + 2๐๐๐๐๐๐ ๐๐๐๐ =1 + ๐๐๐ธ๐ธ
๐๐ โ๐๐๐ธ๐ธ๐ก๐ก๐ก๐ก ๐๐ ๐ผ๐ผ
๐๐ = ๐๐๐ธ๐ธ
1 + ๐๐ 1 โ 2๐๐ ๐๐ =๐ธ๐ธ
2 1 + ๐๐
๐ธ๐ธ = ๐๐3๐๐ + 2๐๐๐๐ + ๐๐ ๐๐ =
๐๐2 ๐๐ + ๐๐
๐๐ = ๐๐ +2๐๐3 =
๐ธ๐ธ3 1 โ 2๐๐
Linear Isotropic Elasticity
In the worst case, ๐ถ๐ถ has 21 independent parameters!
For homogeneous isotropic material, they can be reduced to two independent parameters !
๐๐ and ๐๐ are the Lamรฉ coefficients. ๐๐, sometimes denoted G, is the shear modulus.
Aluminium Copper Stainless steel Titanium
E [GPa] 70 130 195 110
๐๐ 0.34 0.34 0.3 0.32
Elastic parameters at room temperature of materials commonly used for UHV applications
Bond type Typical range Young modulus [GPa]
Covalent 1000
Ionic 50
Metallic 30-400
Van der Waals 2
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Thermal Linear Isotropic Elasticity
๐๐ = ๐ถ๐ถ: ๐๐๐๐ = ๐ถ๐ถ: ๐๐ โ ๐๐๐ก๐กโ
๐๐๐ก๐กโ = ๐ผ๐ผ โ ๐๐ โ ๐๐๐ก๐ก๐๐๐๐ โ ๐ผ๐ผ
๐๐๐๐๐ก๐กโ = ๐ผ๐ผ ๐๐ โ ๐๐๐๐ โ ๐ผ๐ผ
Aluminium Copper Stainless steel Titanium
๐ผ๐ผ [10-6 ยทK-1] 22 17 16 8.9
In simplified 1D, ๐๐ = ๐ธ๐ธ โ ๐๐ โ ๐๐๐ก๐กโ = ๐ธ๐ธ โ ๐๐ โ ๐ผ๐ผ โ โ๐๐
๐๐๐ก๐กโ is the thermal strain tensor.
ฮฑ is the coefficient of thermal expansion (CTE).
C. Garion 15
Material in free state
Elasticity Brittle failure
ฯ
E ฮต
ฯ
E ฮต
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Strength of Brittle Material
Example on glassy carbon:
Material is very sensitive to stress concentration and therefore the material strength strongly dependent of the initial defects. The strength of the material is represented by the Weibullโs law, defining the survival probability:
๐๐๐๐ฮฉ = ๐๐๐๐๐๐ โ1๐๐0
๐๐๐๐๐๐๐๐0
๐๐
๐๐ฮฉฮฉ
Average strength
[MPa]
Standard deviation
[MPa]
Weibull shape parameter
Weibull scale parameter
[MPa] Flexure 206 37 5.6-6.3 375-416
Compression 1012 73 13.5-14.6 1587-1644
4 points bending test on rods
Damage mechanism: Cleavage, intergranular fracture
Fracture surface of heavy tungsten alloy at 77K
๐๐๐๐๐๐ = max ๐๐๐ผ๐ผ ,๐๐๐ผ๐ผ๐ผ๐ผ ,๐๐๐ผ๐ผ๐ผ๐ผ๐ผ๐ผ , 0
m: shape parameter
Surv
ival
pro
babi
lity
๐๐ ๐๐0
C. Garion 17
Material in free state
Elasticity Plasticity
ฯ
E ฮต
ฯ
E ฮต
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Material with plasticity: irreversible plastic deformation before rupture
Plasticity
Mechanism:
The plastic deformation is associated to the dislocation motions (slip under shear stress).
ฯ
ฯ
ฯ
ฯ
ฯ
ฯ
ฯ
ฯ
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๐๐ = ๐๐๐๐ + ๐๐๐๐
Plasticity
Rp0.2: 0,2 % proof strength
Rp1: 1 % proof strength
๐๐ < ๐๐๐ฆ๐ฆ (๐๐ โ ๐๐๐ฆ๐ฆ < 0) : Elasticity
ฯ
E
Rp0.2
ฮต
๐๐๐๐ ๐๐๐๐ 0.2 %
1 %
Rp1
ฯy
In practice, we consider usually that ๐๐๐ฆ๐ฆ= Rp0.2
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Plasticity โ Yield Surface
Yield surfaces are defined by:
Isotropic criteria:
โข Tresca the maximum shear stress, as a function of the principal stresses, is given by: ๐๐๐๐๐ก๐ก๐๐ = ๐๐๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ Trescaโs yield surface: ๐๐ ๐๐,๐๐๐ฆ๐ฆ = ๐๐๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐ฆ๐ฆ
Initial Trescaโs criteria: ๐๐๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐ ๐ ๐๐ = 0
โข Von Mises
Based on the distortional elastic energy โ ๐ก๐ก๐ก๐ก ๐๐๐ท๐ท โ ๐๐๐ท๐ท = ๐๐๐ท๐ท:๐๐๐ท๐ท
๐๐ ๐๐,๐๐๐ฆ๐ฆ =32 ๐๐๐ท๐ท:๐๐๐ท๐ท โ ๐๐๐ฆ๐ฆ
In principal stress space: 1
2๐๐1 โ ๐๐2 2 + ๐๐1 โ ๐๐3 2 + ๐๐2 โ ๐๐3 2 โ ๐๐๐ฆ๐ฆ
In stress space: 12
๐๐11 โ ๐๐22 2 + ๐๐11 โ ๐๐33 2 + ๐๐22 โ ๐๐33 2 + 6 ๐๐122 + ๐๐132 + ๐๐232 โ ๐๐๐ฆ๐ฆ
โข ๐๐ ๐๐,๐๐๐ฆ๐ฆ, โฆ < 0 : elasticity
โข ๐๐ ๐๐,๐๐๐ฆ๐ฆ, โฆ = 0 : plasticity
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Plasticity โ Yield Surface
Comparison of the Trescaโs and Von Misesโs yield surface for plane stress state:
๐๐๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐๐ผ๐ผ๐ผ๐ผ ๐๐๐ฆ๐ฆ Tresca Von Mises
๐๐๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐ ๐๐๐ฆ๐ฆ Tresca Von Mises
Experimental validity on aluminium alloys
๐๐ =๐๐๐ผ๐ผ 0 00 ๐๐๐ผ๐ผ๐ผ๐ผ 00 0 0
๐๐ =๐๐๐ผ๐ผ ๐๐ 0๐๐ 0 00 0 0
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Plasticity - Hardening
Plastic strain tensor: ๐๐๐๐ = ๐๐ โ ๐๐๐๐
Accumulated plastic strain, p : ๐๐๐๐ =23๐๐๐๐
๐๐:๐๐๐๐๐๐ > 0
๐๐๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐๐ผ๐ผ๐ผ๐ผ ๐๐๐ฆ๐ฆ ฯฮ
ฮต
Elastic Plastic perfect Kinematic hardening Isotropic hardening
Plastic strain are induced by dislocation motions no volumic change tr ๐๐๐๐ = 0
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Plasticity โ Kinematic Hardening
๐๐ ๐๐,๐๐๐ฆ๐ฆ,๐๐ =32 ๐๐๐ท๐ท โ ๐๐ : ๐๐๐ท๐ท โ ๐๐ โ ๐๐๐ฆ๐ฆ
๐๐๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐๐ผ๐ผ๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐๐๐ = ๐๐๐๐
32
๐๐๐ท๐ท โ ๐๐
32 ๐๐๐ท๐ท โ ๐๐ : ๐๐๐ท๐ท โ ๐๐
Normality rule:
๐๐๐๐ =23๐ป๐ป๐๐๐๐๐๐
As a first estimation, the hardening modulus H is estimated by: ๐ป๐ป~๐๐๐ข๐ข โ ๐ ๐ ๐๐0.2
๐๐๐ข๐ข
Consistency equation: ๐๐๐๐ = 0
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๐๐ ๐๐,๐๐๐ฆ๐ฆ ,๐ ๐ =32๐๐๐ท๐ท:๐๐๐ท๐ท โ ๐๐๐ฆ๐ฆ โ ๐ ๐
๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐๐๐ = ๐๐๐๐
32
๐๐๐ท๐ท
32 ๐๐๐ท๐ท:๐๐๐ท๐ท
Normality rule:
๐๐๐ ๐ = ๐๐ โ ๐๐๐๐
The isotropic hardening is non linear and reaches a saturation level. Isotropic and kinematic hardening can be mixed.
๐๐๐ผ๐ผ ๐๐๐ฆ๐ฆ
๐๐๐ผ๐ผ๐ผ๐ผ ๐๐๐ฆ๐ฆ
Consistency equation: ๐๐๐๐ = 0
Plasticity โ Isotropic Hardening
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Continuum Damage Mechanics Material representative element
Damaged state: microvoids and/or microcracks (dislocation stack, decohesion or cleavage of precipitates or inclusions,โฆ)
D
Microdefect size << Element size << Structure size
Continuum damaged element
The degradation is characterized by a damage variable D (a scalar for isotropic damage)
Effective stress and damage variable
n dS
๐๐๐น๐น = ๐๐ ๐๐๐๐๐๐
Micro << Meso << Macro scale
๐๐๐น๐น n
dS
๐๐๐น๐น
dSd: Surface of defects
๐๐๐น๐น = ๐๐โฒ๐๐ ๐๐๐๐ โ ๐๐๐๐๐๐
The damage parameter is defined by ๐ท๐ท = ๐๐๐๐๐๐๐๐๐๐
๐๐โฒ: effective stress tensor ๐๐โฒ =๐๐
1 โ ๐ท๐ท
D=0 : Virgin structure; D=1: crack initiation
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 26
Continuum Damage Mechanics
๐๐ = ๐ถ๐ถ 1 โ ๐ท๐ท : ๐๐๐๐ Strain equivalence principle:
๐ธ๐ธโฒ = ๐ธ๐ธ โ 1 โ ๐ท๐ท
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Continuum Damage Mechanics
Damage evolution
For ductile material: ๐๐๐ท๐ท = ๐๐๐๐๐๐๐๐
Y: Strain energy density release rate ๐๐ =๐๐๐๐๐๐๐ท๐ท =
12 ๐๐
๐๐:๐ถ๐ถ: ๐๐๐๐
๐๐ = 12๐ธ๐ธ
๐๐๐๐๐๐2
1 โ ๐ท๐ท 223
1 + ๐๐ + 3 1 โ 2๐๐๐๐โ๐๐๐๐๐๐
2
Effective Von Mises stress Triaxiality ratio
This relation can be generalized to:
ฮ๐๐ = ฮ๐๐๐๐ + ฮ๐๐๐๐ =๐ถ๐ถ2๐ธ๐ธ ๐๐๐๐
โ1 ๐พ๐พ2โ + ๐ถ๐ถ1๐๐๐๐โ1 ๐พ๐พ1โ
Leading to the Manson-Coffin law used in Low cycle fatigue:
๐๐๐๐ ฮ๐๐๐๐ ๐ฝ๐ฝ = ๐ถ๐ถ๐ถ๐ถ๐ก๐ก
Universal slope equation:
ฮ๐๐ = 3.5๐๐๐๐๐ธ๐ธ ๐๐๐๐โ0.12 + ๐ท๐ท๐ข๐ข0.6๐๐๐๐โ0.6
Typical curve for stainless steel
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Fracture Mechanics ๐๐โ
Crack
For elastic material, the stress singularity reads:
a
K: Stress intensity factor
r: Distance from the crack tip
Mode I: Opening Mode II: In plane shear Mode III: Out-of-plane shear
Solicitation modes:
๐๐ โ๐พ๐พ ๐๐โ,๐๐
2๐๐๐ก๐ก
(after J. Lemaitre, J.L. Chaboche, Mรฉcanique des matรฉriaux solides, Dunod)
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Fracture Mechanics
For elastic material, the stress singularity reads:
K is in the form ๐พ๐พ = ๐๐ โ ๐๐โ ๐๐๐๐ .
In a more global approach [Griffith], the energy release rate is defined as: ๐บ๐บ = โ๐๐๐๐๐๐ก๐ก
Criterion: ๐บ๐บ โฅ ๐บ๐บ๐๐ โถ crack propagation
For elastic material, ๐บ๐บ = ๐พ๐พ2
๐ธ๐ธ
๐๐โ ๐๐โ
๐ผ๐ผ
๐พ๐พ๐ผ๐ผ = ๐๐โ ๐๐๐๐ cos2 ๐ผ๐ผ ๐พ๐พ๐ผ๐ผ๐ผ๐ผ = ๐๐โ ๐๐๐๐ cos๐ผ๐ผ sin๐ผ๐ผ ๐พ๐พ๐ผ๐ผ = 1.122 โ ๐๐โ ๐๐๐๐
๐๐ โ๐พ๐พ ๐๐โ,๐๐
2๐๐๐ก๐ก
a
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Fracture Mechanics Fatigue crack propagation:
https://metallurgyandmaterials.files.wordpress.com/2015/12/liberty-ship-failure.jpg?w=640
Material may exhibit ductile to brittle transition
Toug
hnes
s
Temperature
FCC, HCP materials
BCC materials
Mc Kelvey, 1999
Parisโ law, used for stable crack propagation:
More general law:
๐พ๐พ๐ผ๐ผ๐๐, ๐พ๐พ๐ก๐กโ: fracture parameters
๐ ๐ =๐พ๐พ๐๐๐๐๐๐๐พ๐พ๐๐๐ก๐ก๐๐
๐๐: average load parameter; usually 0.5
๐๐๐๐๐๐๐๐
= ๐ถ๐ถ๐พ๐พ๐๐๐ก๐ก๐๐
1 โ ๐ ๐ 1 โ๐๐๐ ๐ โ ๐พ๐พ๐ก๐กโ
๐พ๐พ๐ผ๐ผ๐๐ โ ๐พ๐พ๐๐๐ก๐ก๐๐
๐๐
๐๐๐๐๐๐๐๐
= ๐ถ๐ถ1โ๐พ๐พ๐๐1
๐พ๐พ๐ผ๐ผ๐๐ = 110 ๐๐๐๐๐๐ โ ๐๐ ๐พ๐พ๐ก๐กโ = 5๐๐๐๐๐๐ โ ๐๐ ๐๐ = 2.8 ๐ถ๐ถ = 2. 10โ6 ๐๐/๐๐๐๐๐๐๐๐๐๐
๐๐1 = 4.6 ๐ถ๐ถ1 = 3. 10โ14 ๐๐/๐๐๐๐๐๐๐๐๐๐
Extrapolated curves for 316
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 31
1. Material modelling a. Basic notions of material behaviours and strength b. Stress/strain in continuum mechanics c. Linear elasticity d. Plasticity
i. Yield surface ii. Hardening
e. Continuum damage mechanics f. Failure mechanics
2. Structural mechanical analysis a. Vacuum chamber
i. Loads on a chamber ii. Equilibrium equations iii. Stress on tube iv. Instability (buckling)
b. Vacuum system as mechanical system i. Support โ unbalanced force ii. Stability (column buckling)
3. Selection criteria of materials a. Figures of merit of materials b. Some material properties c. Some comparisons based on FoM
4. Conclusion
Plan
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 32
Loads on a Vacuum Chamber
1. Vacuum/pressure
At the interface
0. Gravity: specific force: ๐๐๐ฃ๐ฃ = ฯ๐๐
n dS
๐๐๐น๐น = ๐๐ ๐๐๐๐๐๐ = โ๐๐๐๐๐๐๐๐
Rk #1: if p is uniform: โฏ๐๐๐๐๐๐๐๐ = 0
โฌ ๐๐๐๐๐๐๐๐๐ฎ๐ฎ๐๐= โโฌ ๐๐๐๐๐๐๐๐ = โ๐๐โ ๐ฎ๐ฎ๐๐๐๐๐๐๐๐๐๐๐ฎ๐ฎ๐๐
p ฯฮธ
ฯz Rk #2 : For a simple tube (radius R, thickness t):
๐ฎ๐ฎ๐๐ ๐ฎ๐ฎ๐๐1 ๐ฎ๐ฎ๐๐2
๐๐๐๐๐๐ =๐๐๐ ๐ ๐ก๐ก
๐๐๐ง๐ง๐ง๐ง =๐๐๐ ๐ 2๐ก๐ก
Hoop stress:
Longitudinal stress (for a close tube):
๐๐ ๐๐ = โ๐๐๐๐
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 33
Loads on a Vacuum Chamber 2. Electro magnetic forces
Foucaultโs currents are governed by Maxwellโs equation:
B(x,y,t)
rot E=-โฮ/โt
Ez(x,y,t) ~ Bโ
Ohmโs law
Maxwellโs equation
B: magnetic field
E: electric field
j: current density
ฯ: electrical resistivity
B(x,y,t)
jz Laplaceโs law
fv=jโงB B(x,y,t)
jz
f fv ~ BโB
In a long structure, subjected to the magnetic field B:
j = E/ฯ
jz = Ez/ฯ
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 34
For a given magnetic configuration, force intensity ~ t/ฯ
(t/ฯ)st.st. / (t/ฯ)Cu ~ (1/5E-7)/(0.1/1E-9) ~ 0.02
Lorentzโ force are driven by copper
For a (colaminated) copper/stainless steel beam screen at cryogenic temperature:
In a copper tube, 0.1 mm thick, radius of 25 mm, subjected to a magnetic field of 10 T with a decay of 100 T/s:
F ~ 125 N/mm (~ 13 t/m)
F I ~ 12.5 kA
B
Orders of magnitude for cryogenic applications (beam screen)
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 35
Equilibrium equations in static conditions:
Mechanical Problem Formulation:
๐๐๐๐๐๐ ๐๐ + ๐๐๐ฃ๐ฃ = 0 in ฮฉ ฮฉ
๐๐ ๐๐ = ๐น๐น๐ก๐ก on ๐ฎ๐ฎ๐๐
๐ฎ๐ฎ๐๐
Kinematic:
Constitutive model: โข Constitutive law: ๐๐ = ๐ถ๐ถ: ๐๐๐๐ with ๐๐๐๐ = ๐๐ โ ๐๐๐ก๐กโ โ ๐๐๐๐ โ โฆ
โข Plasticity or other
๐ฎ๐ฎ๐๐
boundary conditions: ๐ข๐ข = ๐ข๐ข๐๐๐๐๐๐๐๐๐ก๐ก๐๐๐๐ on ๐ฎ๐ฎ๐๐
๐๐ ๐ข๐ข = 12โ ๐๐๐ก๐ก๐๐๐๐ ๐ข๐ข + ๐๐๐ก๐ก๐๐๐๐๐๐ ๐ข๐ข in ฮฉ
Mechanical solution: ๐ข๐ข , ๐๐
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 36
Design Criteria
Material criterion: โข Maximum stress, โข Elastic regime, โข fatigue, โฆ
Structural criterion: โข Maximum deformation, โข Stability
โ
P
Pth
With imperfections
Tube under external pressure
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 37
I. For an infinite elastic tube subjected to external pressure: Safety factor of 3 is usually applied.
Design rule for stainless steel:
Structural Stability
II. For beam subjected to axial force:
F
Boundary conditions Lr
๐ฟ๐ฟ๐ก๐ก = ๐ฟ๐ฟ
๐ฟ๐ฟ๐ก๐ก = 2 โ ๐ฟ๐ฟ
๐ฟ๐ฟ๐ก๐ก = 0.5 โ ๐ฟ๐ฟ
๐ฟ๐ฟ๐ก๐ก = 2 โ ๐ฟ๐ฟ
L
L
L
L
๐๐๐๐๐ก๐ก =๐ธ๐ธ
4 โ 1 โ ๐๐2๐ก๐ก๐ ๐
3
Eulerโs critical load:
I: Area moment of inertia
~ฯd3t/8 for a tube
wh3/12 for a rectangular cross section
y(x)
Equilibrium equation: ๐๐ = โ๐น๐น๐๐
Bending equation: ๐๐ = ๐ธ๐ธ๐ผ๐ผ๐๐โฒโฒ
Differential equation: ๐ธ๐ธ๐ผ๐ผ๐๐โฒโฒ + ๐น๐น๐๐ = 0
Solution depends on boundary conditions.
๐น๐น๐๐๐ก๐ก =๐๐๐ฟ๐ฟ๐ก๐ก
2
๐ธ๐ธ๐ผ๐ผ
๐ก๐ก โฅ๐ท๐ท
100
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 38
Mechanical System - Supports
Kinematic (longitudinally):
1 fixed support Sliding supports
Only 1 longitudinally-fixed support has to be used.
Static: Distributed forces (tube weight, bake out jacket): q (N/mm)
ฮด0
L
Beam pipe simply supported at its extremities
Deflection: ๐ฟ๐ฟ0 =5๐๐๐ฟ๐ฟ4
384๐ธ๐ธ๐ผ๐ผ
Deflection:
Beam pipe deflection can be minimized by supporting at Gaussian points:
ฮด
L
~0.22L ~0.22L
๐ฟ๐ฟ =๐ฟ๐ฟ050
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 39
Static : Pressure thrust force
So1
So2
๐น๐นโ๐ก๐ก๐ข๐ข๐๐๐๐๐๐๐ก๐ก๐ก๐ก๐ก๐ก = ๐๐๐ฎ๐ฎ๐๐๐๐๐๐๐๐๐๐
P = 0.1 MPa
Be careful to the moment and support anchoring
Mechanical System - Supports
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 40
This phenomenon can occur for a line under internal pressure (not necessary with a bellows).
Deformed tube with internal pressure
The resultant of the pressure forces tend to increase the initial defect instability
Global stability: the bellows and adjacent lines are unstable
Mechanical System โ Global stability
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 41
Deformed tube with internal pressure
y(x)
dx
p
Equilibrium equation (for a tube slice): ๐๐โฒโฒ โ ๐๐ = 0
Bending equation: ๐๐ = ๐ธ๐ธ๐ผ๐ผ๐๐โฒโฒ Differential equation: ๐ธ๐ธ๐ผ๐ผ๐๐(4) + ๐๐๐๐๐๐โฒโฒ = 0
Solutions depends on boundary conditions
1
Mode I
Mode II
0cr
cr
PP
crL 1L
โ
Buckling pressure can read, in a general formulation (similar to Eulerโs formula):
Bending stiffness
Reduced length (depends on boundary conditions)
Radius
Mechanical System โ Global stability
๐๐๐๐๐ก๐ก =๐๐2๐ถ๐ถ๐๐๐ฟ๐ฟ๐ก๐ก2๐ ๐ ๐๐2
๐๐๐น๐น = โ๐๐๐๐๐๐โฒโฒ๐๐๐๐
๐๐ =๐๐๐น๐น๐๐๐๐
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 42
1. Material modelling a. Basic notions of material behaviours and strength b. Stress/strain in continuum mechanics c. Linear elasticity d. Plasticity
i. Yield surface ii. Hardening
e. Continuum damage mechanics f. Failure mechanics
2. Structural mechanical analysis a. Vacuum chamber
i. Loads on a chamber ii. Equilibrium equations iii. Stress on tube iv. Instability (buckling)
b. Vacuum system as mechanical system i. Support โ unbalanced force ii. Stability (column buckling)
3. Selection criteria of materials a. Figures of merit of materials b. Some material properties c. Some comparisons based on FoM
4. Conclusion
Plan
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 43
A few Selection Criteria
Figures of merit:
๐๐0. ๐๐.๐ถ๐ถ.๐๐๐๐
๐๐0.๐๐.๐ถ๐ถ.๐๐๐ฆ๐ฆ๐ธ๐ธ.๐ผ๐ผ
๐๐0. ฮป.๐๐๐๐
๐๐0๐ธ๐ธ13
โข Temperature rise in transient regime:
โข Thermal fatigue:
โข Temperature rise in steady state:
โข Mechanical Stability for transparent vacuum chamber:
Several figures of merit, characterizing the material, can be used depending on the final application.
๐๐๐ธ๐ธ1 3 โข Mechanical Stability for vacuum chamber subjected to fast magnetic field variation:
For beam-material interaction induced heating:
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 44
Beryllium Aluminium Titanium G5 316L Copper Inconel
Density gโcmโ3 1.85 2.8 4.4 8 9 8.2 Heat capacity JโKโ1โKgโ1 1830 870 560 500 385 435 Thermal conductivity WโKโ1โmโ1 200 217 16.7 26 400 11.4
Coefficient of thermal expansion 10โ6โKโ1 12 22 8.9 16 17 13
Radiation length cm 35 9 3.7 1.8 1.47 1.7
Melting temperature K 1560 930 1820 1650 1360 1530
Yield strength MPa 345 275 830 300 200 1100 Young modulus GPa 230 73 115 195 115 208
Electrical resistivity 10โ9โฮฉโm 36 28 1700 750 17 1250
Some Material Properties
Indicative values at room temperature
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 45
Some Material Properties
Some properties depend strongly on temperature or grade or delivery state (annealed, hard,โฆ).
Properties of copper and copper alloys at cryogenic temperatures, Simon et al.
Yield strength of aluminium alloys as a function of temperature
Just two examples:
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 46
Few Figures of Merit
For each application, the correct material.
Vacuum, Surfaces & Coatings Group Technology Department C. Garion, CAS Vacuum, Lund, 7th June 2017 47
Conclusion
Most of the time, the mechanical design of a vacuum system is not really complex. The choice and the knowledge the materials are important to get a robust and reliable mechanical system at the right price.