Material Science - LMT Cachan - ENS Cachan

Material Science 

Mechanisms at the origin of the deformation of materials 

Polymers 

B. Fayolle, C. Creton 

Objective: To give answers to the following questions: 

What are the basic deformation mechanisms in polymers, metals and nanomaterials ? 

What are the relationships between microstructure and mechanical properties ? 

Sylvie Pommier (ENS Cachan - LMT) 

Metals and alloys 

S. Pommier, V. Favier 

What clues does it provide for modeling ? 

Nano materials 

B. Devincre, S. Forest 

Véronique FAVIER (Arts et Métiers ParisTech- PIMM) 

1

Introduction – Why ? 

Why are we interrested in the deformation mechanisms of materials ? 

� to design new materials for new problems 

Lighter cars 

� Increase material strength 

more secure cars 

� increase material ductility 

2


3

Elongation (%) 


Tensile strength (MPa) 

4


5


� design a material considering its entire life 

It includes : 

•Production, 

•forming processes, 

•assembly processes, 

•operating, 

•recycling 

A geometrical defect created during metal forming may induce a 

premature failure in operating conditions 

6

� We need to open the black box 


in order to understand the mechanisms 

7



in order to model mechanisms and simulate 

8

vacancy 

intertitial 

Introduction – Material scales 

substitution 

Macroscopic scale (~ mm) 

Scale of computational mechanics 

Phases (~10 µm) 

Scale of metallurgy 

Grain (~µm) 

Scale of cristallography 

Coherent domain 

(~10 nm) 

Scale of diffraction 

Crystallographic lattice 

(~ Å) 

Scale of « Transmission Electronic Microscope » 

9

Introduction – Material scales 


in order to 

understand, 

model, 

characterize, 

and simulate 

material behaviour 

at various scales from nanometer to meter 

10

1 Basic mechanisms: crystals 

nuclear 

electronic 

total 

A B 

E = − + m n 

r r 

A material tends to 

crystallize to 

minimize its energy 

11

1 Basic mechanisms : crystals 

Body centered cubic (BCC) 

For metals and alloys 

Face centered cubic (FCC) 

hexagonal compact (HC) 

12


Miller indices for planes and directions 

(orthotropic systems) 

OA = x A .a = a/h, 

OB = y B .b = b/k 

OC = z C .c = c/l 

Be careful : if the system is not cubic [h,k,l] is not orthogonal to (h,k,l) 

OM = x. 

a. 

e + y. 

b. 

e + z. 

c. 

e 

x 

a, b, c interatomic distances 

Equation of the plane 

hx + ky + lz = 1 

Index : (h,k,l) 

Direction : [h,k,l] 

y 

z 

13


Most metals and alloys are 

polycristalline 

14


X-Ray diffraction 

15

BRAGG’s law 

Sinθ 

= 


nλ 

2d 

16

BRAGG’s law 

Sinθ 

= 


nλ 

2d 

17


Keep the angle θ constant 

And vary 

the diffraction vector g 

versus 

the vector n normal to the sample 

Incoming 

X-Ray beam 

diffracted 

X-Ray beam 

18

Stereographic projection 


19

Stereographic projection 


20


Stereographic projection : pole figures 

21


Metal forming : the material is not isotropic 

DL 

22

1 Basic mechanisms: elasticity 

nuclear 

electronic 

total 

A B 

E = − + m n 

r r 

23


nuclear 

electronic 

total 

Elastic behaviour : asymptotic behaviour of atomic bonds 

around r=ro 

24

551 

482 

413 

DL 


loading / rolling directions angle 

Youngs modulus 

loading / rolling directions angle 

25

2 

Evidences of plastic 

deformation in metals 

26

2 Evidences of plastic deformation in metals 

σ 

ε p 

Non recoverable deformation 

inelastic – plastic deformation 

ε 

Hanriot, 1993 (from S. Forest) 

Slip lines, Slip bands 

Single slips 

Crystallographic slip 

27

2 Evidences of plastic deformation in metals 

CROSS SLIP 

28

3 

crystallographic slip, 

slip systems 

29

A slip system consists in : 

•A slip plane 

•A slip direction 

A B 

E = 

− + m n 

r r 

3: crystallographic slip 

30

structure 

FCC 

BCC 

HCP 

Slip planes 

{111} 

{110}, {112}, 

{123} 

{0001}, {1010} 


Slip directions 

 

 

 

Examples of 

metals 

Al, Fe γ, Cu, Ni, 

Au, Ag 

Fe α, Nb, Mo 

Mg, Ti, Zn, Zr 

α, Be 

31

structure 

FCC 

BCC 

HCP 


{110}, {112}, {123} 

basal 

pyramidal 

Slip planes 

{111} 

{0001}, {1010} 

Slip directions 

 

 

 

Examples of metals 

Al, Fe γ, Cu, Ni, Au, Ag 

Fe α, Nb, Mo 

Mg, Ti, Zn, Zr α, Be 

prismatic 

32

3: crystallographic slip : slip systems in FCC crystals 

33


34


35


36


37

3: crystallographic slip : slip systemms in FCC crystals 

Octaedra 

38

4 

When is a slip system active ? 

Yield strength 

39

10 

µm 

When is a slip system active ? 

One, two, or more slip systems can be activated 

10 

µm 

Single crystal of aluminium 

40

τ 

s 

When is a slip system active ? Schmid law 

For any stress tensor 

s s 

= Rij 

ij 

τ σ 

1 

R n m n m 

2 

( ) 

s 

ij = i j + i j 

Schmid tensor 

( n) 

⋅ m 

= σ 

Resolved shear stress on the slip system (s) 

n 

n 

χ 

F 

m Elastic if 

θ 

F 

τ 

m 

τ = σ 0 cos θ cosχ 

Schmid factor 

τ < τ c 

Plastic if 

τ = τ c 

Schmid criterion 

41

When is a slip system active ? Tensile strength for single crystals 

n 

χ 

F 

θ 

F 

τ 

Elastic if 

τ < τ c 

Plastic if 

τ = τ c 

Schmid criterion 

m 

Yield strength σ o 

τ = σ 0 cos θ cosχ 

cos θ cosχ 

τ = 0.5× 

σ 

max 0 

⇒ σ = 2τ 

y c 

42

When is a slip system active ? Yield stress of polycrystals 

Austenitic stainless steel 316L 

σ 0 

σ 0 500 µm 

43


τ σ sinθcosθ = 

0 

Austenitic stainless steel 316L 

σ 0 

τ = τ when θ=45° ⇒ τ = 0.5× 

σ 

max max 0 

⇒ σ = 2× τ 

= 2× k 

y critical( macroscopic ) 

σ 0 500 µm 

44


⇒ σ = 

2τ 

y c 

⇒ σ = 2× τ 

= 2× k 

y critical(macroscopic) 

45

When is a slip system active ? Scale transition from grain to polycrystal 

σ 

b) The Sachs model (1928) 

σ τ 

Heterogenous stress and strain fields in polycrystals !! 

The stress field is assumed uniform and one single slip system per grain is active. 

Uniaxial tensile test 

a) The static model (Batdorf and Budiansky, 1949) 

The stress field is assumed uniform and 

Uniaxial tensile test 

σlocal,grain ≠σ 

y = 2.24 c 

Isotropic F.C.C 

( s ) s ( ) 

Max R σ = Max R σ = τ 

grains,slip systems ij ijlocal,grain grains,slip systems ij ij c 

σ = 2τ 

y c 

σlocal,grain = 

σ 

46

When is a slip system active ? Scale transition from grain to polycrystal 

σ 

d) The Taylor-Lin Taylor Lin model (Lin, 1957) 

Heterogenous stress and strain fields in polycrystals !! 

local ,grain E 

ε ≠ 

c) The Taylor model (1938) 

The elastic deformation is neglected (and so internal stresses) and 

the strain field is assumed uniform. 

Uniaxial 

tensile test 

σ = 3.06τ 

y c 

Isotropic F.C.C 

The strain field is assumed uniform and one single slip system per grain is active. 

Many other scale transition models … see course on Heterogeneous media : large scale behaviour 

47

Tensile 

tests 

iron single 

crystals 

After the onset of plastic yielding ? 

HARDENING IS A KEY PHENOMENON 

48

To summarize 

•Plastic deformation in metals 

•Slip systems (plane and directions) 

•Schmid law in single cristal (critical shear stress on slip systems) 

•Various models for polycristals 

49

Material Science - LMT Cachan - ENS Cachan

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