Modeling of additive manufactured materials magnetic behavior

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Olivier Hubert. Modeling of additive manufactured materials magnetic behavior. Doctoral. GDR -ALMA, Toulouse - visio, France. 2020. �hal-02970489�

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Modeling of additive manufactured materials magnetic behavior

Olivier HUBERT

September 2020, 30th

Universit´e Paris-Saclay, ENS Paris-Saclay - CNRS LMT - Laboratoire de M´ecanique et Technologie 4, avenue des sciences, 91190 Gif-sur-Yvette, France.

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1 _INTRODUCTION

Definitions and scope

Additive manufacturing of soft/medium magnetic materials

2 MAGNETIC BEHAVIOR

Definitions and basic properties Magnetomechanics

Some specificities of AM materials magnetic behavior

3 _{MODELING OF AM MATERIALS MAGNETIC BEHAVIOR}

Typical scales

Gibbs free energy at the magnetic domain family scale Stochastic modeling, localization, homogenization Some results

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1 _INTRODUCTION

2 MAGNETIC BEHAVIOR

Typical scales

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Strong increase of AM materials and structures Many available materials

Most alloys with dominant Fe, Ni, Co: get neutral, desired or su↵ered magnetic state

Relationship between magnetism and additive manufacturing: recent topic [1, 2] Mainly metallic materials even if

ferrimagnetic/antiferromagnetic/magneto-rheologic materials could be considered Many possible materials however

Only ferromagnetic materials considered in this talk Possible extension to other magnetic orders

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Many possible materials remain

Soft magnetic materials: FeSi, FeCo, FeNi, Ni, electrical steels

Medium magnetic materials: all carbon steels from ferritic/ferrito-pearlitic to quenched/aged martensitic alloys

Medium magnetic materials: ferritic, martensitic and (unstable) austenitic stainless steels (Fe-Cr-Mn)

Hard magnetic materials

Magnetic shape memory alloys Magneto-caloric materials

...

Only soft/medium ferromagnetic materials considered in this talk Below Curie temperature of course...

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Soft/medium ferromagnetic materials

Interest of AM for these materials?

Wide range of chemical composition / gradient Prototyping

Lightening of structures (lattices)

Complex designs (where conventional machining fails) Repairing

(a) (b)

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AM Techniques

Powder bed fusion by energy source: laser in the case of selective laser melting (SLM) or an electron beam in electron beam manufacturing (EBM)

Directed energy deposition (DED) and binder jetting Stress-relief, sintering or homogenization heat treatment

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1 _INTRODUCTION

2 MAGNETIC BEHAVIOR

Typical scales

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−8000 0 8000 −1.5 0 1.5 H (A/m) M ( 1 0 A/ m) 6 Ms Hc χi Mr W

Figure 3: Cyclic magnetic behavior of carbon steel and main quantities [6].

~

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Definitions and basic properties

Ferromagnetic domains

100 µm

(a) (b)

Figure 4: (a) Fe-27%Co sheet [7]; (100)-oriented silicon-iron crystal [8] - Kerr microscopy.

~

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Ferromagnetic domains - interaction with defects

Figure 5: Defect / 180 domain wall interaction / silicon-iron crystal [8] - Kerr microscopy.

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Definitions and basic properties Anisotropy 0 1 2 3 4 5 0 2 4 6 8 10 12 14 16 18 H (10 A/m) M (10 A/m) <100> <110> <111> 4 5 W an (kJ/m 3 ) <010> <001> <100> 0 2 4 6 8 10 12 14 (a) (b)

Figure 6: Magnetic crystalline anisotropy of pure iron: (a)[9]; (b)[10].

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−8000−2 −4000 0 4000 8000 −1 0 1 2 H (A/m) M ( 1 0 A/ m) FeCo FeSi Ni 6

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Definitions and basic properties Composition / phases −8000 −4000 0 4000 8000 −1.5 −1 −0.5 0 0.5 1 1.5 H (A/m) M (1 0 A/ m) Fe 42CD4T Z20C13 6

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Grain size 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 200 400 600 800 GS−1 (μ m−1 ) Hc 0 .1 H z − 2 0 kA/ m (A/ m) LC steel Carbon steel (a) (b)

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Magnetomechanics

Magnetostriction

100 µm

(a) (b)

Figure 10: (a) Fe-27%Co sheet [7]; (100)-oriented silicon-iron crystal [8] - Kerr e↵ect

~

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Magnetostriction

100 µm

(a) (b)

Figure 11: (a) Fe-27%Co sheet [7]; (100)-oriented silicon-iron crystal [8] - Kerr e↵ect

~ M↵ = Ms~e↵ = Ms i~ei ✏µ↵ = 3 2 0 @ 100 ( ₁2 1₃) 111 1 2 111 1 3 111 1 2 100( ₂2 1₃) 111 2 3 111 1 3 111 2 3 100( ₃2 1₃) 1 A

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Magnetomechanics Magnetostriction −1.5 −1 −0.5 0 0.5 1 1.5 −12 −9 −6 −3 0 3 6 9 12 H (10 A/m) M (10 A /m ) 5 4 anhysteretic cyclic −12 −9 −6 −3 0 3 6 9 12 −1 −0.5 0 0.5 1 1.5 2 2.5 3 anhysteretic cyclic 5 M (10 A/m) ε (ppm ) μ //

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Constant stress, variable magnetic field 0 5000 10000 15000 0 5 10 15 5 H (A/m) M ( 10 A /m ) 0MPa -180 MPa 180 MPa Villari reversal -200 -100 0 100 200 2 1 0 -1 -2 6 -5 0 5 10 x 10-6 σ (MPa) M (10 A/m)

ε

μ //

Figure 13: Stress influence on magnetic and magnetostrictive behavior of a low carbon steel [14].

~

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Magnetomechanics

Constant magnetic field, variable stress

−200 −100 0 100 200 −2 −1 0 1 x 105 σ(MPa) M(A/m) H =1000 A/m 0 −6 −4 −2 0 2 4 6 8 (10 )−4 −150 −100 −50 0 50 100 150 ε σ (MP a) ε ε 11 22

Figure 14: Piezomagnetic behavior and E e↵ect - low carbon steel and Fe-Co - [15, 16].

~

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Plastic straining 0 1000 2000 0 4 8 12 H (A/m) M (10 5 A /m ) Undeformed ε p = 0.01% 0 5 10 15 0 2 4 6 8 M (105 A/m) ε // Undeformed ε p = 0.01% (ppm )

Figure 15: Anhysteretic magnetic behavior of NO Fe-3%Si w/wt plastic strain [13]

~

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Magnetomechanics Plastic straining 0 5 10 15 20 3 4 5 6 0 5 10 15 20 600 700 800 900 1000 1100

ε (%)

_p

ε (%)

p Mr (1 0 A/ m) H c (A/ m) 5

Figure 16: Remanent magnetization and coercive field evolution with large plastic strain - pipeline steel [17]

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Porosities Cracks

Surface finish

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Metallurgical (process) issues I

Chemical heterogeneities / unexpected phases Residual stresses, cracks

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Crystal texture

Crystal defects, plasticity

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Production of high silicon content Fe-Si alloy

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Main modeling difficulties Crystallographic texture

Residual stresses (order I and II)

Micro/macro segregations and unexpected phases

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1 _INTRODUCTION

2 MAGNETIC BEHAVIOR

Typical scales

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Magnetic domains Grains and phases RVE 30 μm 30 μm 30 μ_m RD TD ND Figure 21: Microstructure of DP600 [20].

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Typical scales

Multiscale modeling: [10, 21, 22]

Di↵erent scales involved depending on the problem

Polycrystal : RVE Single crystal : Grain scale Variant : Variant scale Domain : Domain scale Phase : Phase scale

RVE - representative volume element = polycrystal (ODF) g : grain scale

': phase family scale : variant family scale

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first step: build an energy function at the domain scale where anisotropic crystallographic phenomena are significant and some fields can be simplified. second step: energy conservation at the local scale (energy density)

constant velocity - removal of kinetic energy and associated power density (Body forces, Maxwell forces)

direct relationship between the variation of internal energy density and power sources:

du↵ = dh↵ + T↵ds↵ + ↵ : d✏↵ + ~H↵.d ~B↵

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Gibbs free energy at the magnetic domain family scale

More usual control variables: temperature, stress, and magnetic field free Helmholtz energy density

↵ = u↵ T↵s↵

magnetic free enthalpy

k↵ = ↵ H~↵.~B↵

Gibbs free energy (mechanical free enthalpy)

g↵ = k↵ ↵ : ✏↵

Since chemical bound is constant over a domain inside a phase, this leads to: dg↵ = s↵dT↵ ✏↵ : d ↵ B~↵.d ~H↵

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Consequences:

Entropy density, strain and magnetic induction finally derive from the Gibbs free energy function following

s↵ = @g↵ @T↵ ~ B↵ = @g↵ @ ~H ✏↵ = @g↵ @ ↵

Definition of Gibbs free chemical, mechanical and magnetic energy densities separately

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Gibbs free chemical energy:

g_↵T(T↵) = h↵ T↵s↵ = h↵ T↵s_↵0 + ⇢↵c_↵p T↵ T_↵0 + T↵ln(T 0 ↵

T↵

) Gibbs free mechanical energy

dg ( ↵) = ✏↵ : d ↵

Two sources of deformation: elastic + inelastic of multiphysic origin (thermal expansion, magnetostriction, phase transformation...) - small perturbations hypothesis ✏ = ✏e_↵ + ✏l_↵ g_↵( ↵) = 1 2 ↵ : C 1 ↵ : ↵ Z _↵ O ✏ l ↵ : d

! Integration CANNOT be simplified without assumptions on inelastic strain / stress relationship

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Gibbs free magnetic energy

Usual simplification by physicians: magnetization ~M instead of magnetic induction ~B since ~M = ~0 in vacuum ~ B = µ0(~H + ~M) ~ H↵ = @g↵ @ ~B↵ = @g↵ µ0@ ~M↵ g_↵H(~H↵) = ↵( ~M↵) µ0H~↵. ~M↵

! Expression of Helmholtz magnetic energy ↵ CANNOT be simplified without

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Taylor expansion of the Helmholtz magnetic energy ↵ that must be an even function

of magnetization (magnetic behavior odd function)

↵( ~M↵) = M~ ↵.P↵. ~M↵ + ~M↵ ⌦ ~M↵ : P0_↵ : ~M↵ ⌦ ~M↵+

~

M↵ ⌦ ~M↵ ⌦ ~M↵ ) P”↵ ) ~M↵ ⌦ ~M↵ ⌦ ~M↵

Magneto-elastic coupling ?

! Taylor expansion as well - of 1rst order in stress (keeping sti↵ness independent of stress)

H

↵ ( ~M↵, ↵) = M~ ↵.E↵ : ↵. ~M↵

E↵ 4th order magnetostriction tensor

~

M↵E↵. ~M↵ = ~M↵ ⌦ ~M↵ : E↵ = ✏µ_↵

homogeneous to a deformation _{! MAGNETOSTRICTION tensor}

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At scale ↵ of cubic symmetry:

constant magnetization norm: saturation magnetization

~ M = Ms i~ei ~ M _{⌦ ~}M = M_s2 0 @ 2 1 1 2 1 3 1 2 ₂2 2 3 1 3 2 3 ₃2 1 A

simplification of local magnetostriction tensor using 3 constants (cubic symmetry) minus 1 (incompressibility) ✏µ_↵ = 3 2 0 @ 100 ( ₁2 1₃) 111 1 2 111 1 3 111 1 2 100( ₂2 1₃) 111 2 3 111 1 3 111 2 3 100( ₃2 1₃) 1 A

100 et 111: deformation measurement along 2 crystalline axes

simplification of local Helmholtz energy (cubic symmetry)

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+ free deformation associated with phase transformation: defined at the variant or phase scale ✏tr_↵ = 0 @ ✏✏1112 ✏✏1222 ✏✏1323 ✏13 ✏23 ✏33 1 A total inelastic deformation

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At scale ↵ of cubic symmetry _!

Gibbs free energy final expression

g (T↵, ~H↵, ↵) = h↵ T↵s_↵0 + ⇢↵c↵p T↵ T_↵0 + T↵ln(T 0 ↵ T_↵ ) +K1( ₁2 ₂2 + ₂2 ₃2 + ₃2 ₁2) + K2( ₁2 ₂2 ₃2) µ0MsH~↵.~↵ 1 2 ↵ : C 1 : ↵ ↵ : ✏I↵

Localization procedures: loading L (T , , ~H)

L _{! L}g ! L' ! L ! L↵

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Stochastic modeling, localization, homogenization

Stochastic modeling, localization state parameters:

volume fraction of domains, inside a variant, inside a phase, inside a grain. magnetization orientation of a domain family

constitutive equation: Boltzmann function (at equilibrium solution of stochastic problem) f↵ = P P ↵ exp ( A.g↵) P ' P P ↵ exp ( A.g↵) P ↵ exp ( B.g↵) P P ↵ exp ( B.g↵) exp ( C .g↵) P ↵ exp ( C .g↵)

Local Gibbs free energy minimization to get the magnetization orientation ~↵ = min(g↵(~ , T↵, ~H↵, ↵)) variant fraction f = P P ↵ exp ( A.g↵) P ' P P ↵ exp ( A.g↵) P ↵ exp ( B.g↵) P P ↵ exp ( B.g↵) phase fraction f' = P P ↵ exp ( A.g↵) P ' P P ↵ exp ( A.g↵)

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Identification, homogenization Physical parameters and ODF Identification (A, B, C )

anhysteretic initial susceptibility DSC

anhysteretic mechanical loading

Boundary e↵ects

demagnetization field (surface e↵ect)

initial configuration (inherited from forming process)

Averaging operations ~ M = X ' f'M~ ' = X ' f' X f ~M = X ' f' X f X ↵ f↵M~ ↵ s = X ' f's' = X ' f' X f s = X ' f' X f X ↵ f↵s↵ ✏ = X ' f'✏' = X ' f' X f ✏ = X ' f' X f X ↵ f↵✏↵

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Some results Anisotropy RD TD 101 102 103 0 4 8 12 16

H (A/m) -Log format

M (10 A /m ) experimental calculated RD TD 5 - 20 - 10 0 10 20 - 5 0 5 10 15 20 εµ l ongi tuna l (ppm ) experimental calculated RD TD - 20 - 10 0 10 20 - 40 - 30 - 20 - 10 0 M (10 A/m) ε (ppm ) µ t ra ns ve rs al experimental calculated RD TD 5 M (10 A/m)5 (a) (b) (c) (d)

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0 5000 10000 15000 0 4 8 12 16 H (A/m) 0 5000 10000 15000 0 4 8 12 16 M (1 0 A/ m) 5 −180MPa unloaded 90MPa 180MPa H (A/m) (a) (b) −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −8 −6 −4 −2 0 2 4 6 8 10 ε (p p m) μ // −180MPa unloaded 90MPa 180MPa −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −8 −6 −4 −2 0 2 4 6 8 10 −180MPa unloaded 90MPa 180MPa (c) (d) M (10 A/m)6 M(10 A/m)6

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Some results

Constant magnetic field, variable stress

−100 −80 −60 −40 −20 0 20 40 60 80 100 −50 0 50 100 ε µ (p p m) σ (MPa) / longitudinal exp / model / transversal exp / model

(a) −100 −80 −60 −40 −20 0 20 40 60 80 100 −5 0 5 10 15 20 25 30 35 d M/ d σ (kA/ m/ MPa ) σ (MPa) experiment modelling (b)

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1 _INTRODUCTION

2 MAGNETIC BEHAVIOR

Typical scales

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Multiscale coercive field description (Hauser’s model [24]) Micromagnetics for smart AM magnetic structures [25]

NDE of AM structure by magnetic/mechanical inspection [24]

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Micromagnetics for smart AM magnetic structures [25]

NDE of AM structure by magnetic/mechanical inspection [24]

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