6. Mechanical properties of solids6. Mechanical properties of solids

H. S. Leipner

1. Basics of elasticity 2. Deformation experiments3. Microscopic processes of plastic deformation 4. Hardening mechanisms5. Brittle fracture

hsl 2002 – Physics of materials 6 - Mechanical properties 2

1. Basics of elasticity

• External force: Change in the shape of a body• Experience: Change in the shape is proportional to the force (if small)

F F

Tension

0 0

F lEA l

E

⊥ ∆=

=σ ε

�

�

Length l0, l, difference ∆l = l – l0 Cross section A0

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Shear

• Force parallel (tangential) to the face A0 results in shear• Usually, shear modulus G < (Young modulus), materials can be

easier sheared

Shear

0 c

F xGA d

G

∆=

=τ γ

�

Thickness of the crystal dc, shift ∆xShear angle tan γ = ∆x/dc

F

F

E�

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• Considering a small cubic volume element in a solid, the total stress state can be described by the forces perpendicular and parallel to the faces of the cube

• On each face, three stresses: 1 normal σii, 2 shear σii (i ≠ j; i, j = x, y, z)• All together nine components of the stress

Stress tensor

Stress in the solid

Stress tensor

xx xy xz

yx yy yz

zx zy zz

=

σ σ σ

σ σ σ

σ σ σ

σ

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Stress tensor

• Stress tensor is symmetrical, σij = σji (rotational equilibrium)• Magnitude of the individual components depend on the orientation of

the coordinate system• A special coordinate system can always be found, where there are only

normal stresses

• Positive normal stress as tension• Hydrostatic pressure is the average normal stress,

1

2

3

=

σ 0 0

0 σ 0

0 0 σ

σ

1 2 31 ( )3

p = + +σ σ σ

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Strain

• Initially, the strain was introduced as uniaxial tension/compression ε or shear γ

• Generally, the elastic deviation of the shape of the solid can be expressed as a strain tensor

• εii elongations, εij shear (i ≠ j)• Strain tensor also symmetrical

xx xy xz

yx yy yz

zx zy zz

=

ε ε ε

ε ε ε

ε ε ε

ε

0 ( )xx yy zzV V

V− = + +ε ε ε

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Stress–strain relationship

• Stress as force per unit area of surface; consider orientation of the surface and direction of the force

• Uniaxial tension σ = ε, shear τ = Gγ

• Special cases of Hooke’s law• Relation between stress and strain tensors

• Expression of 9 equation like

• C has 34 = 81 components Cijkl (4th rank tensor)

=σ Cε3

, 1ij ijkl kl

k lCσ

== ∑ ε

E�

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Elastic constants C

• In praxi, number of constants is reduced due to symmetry• For isotropic solids only two parameters (e.g. G and Lamé constant )

• In cubic crystals, three constants are needed

σxx = 2Gεxx + Σεii

σyy = 2Gεyy + Σεii

σzz = 2Gεzz + Σεiiσxy = 2Gεxy, σyz = 2Gεyz, σzx = 2Gεzx

λ�

λ�

λ�

λ�

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Elastic moduls

• Other constants to be used: Young’s modulus , Poisson’s constant ,and bulk modulus

• Poisson’s constantElongation in x-direction connected with reduction of cross section

εyy = εzz = − εxx

l2 (1 ), ,2(l ) 3(1 2 )

EE G KG

ν ν

ν

�� �� �

�= + = =

+ −

K�E� ν�

ν�

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Measurement of elastic constants

• Propagation of ultrasound in the crystal in certain crystallographic directions

HF generator Amplifier

Quartz

SupportSample Excitation and reflected

pulse

• Sound velocity (longitudinal wave, [001] propagation direction in the crystal)

sl

E�=

ρv

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2. Deformation experiments

• Uniaxial tension, compression with constant straining rate

���� Homogeneous ���� Change of the sample cross section• Torsion

���� No geometrical hardening ���� No homogeneous deformation• Indentation

���� Easy to perform ���� Difficult interpretation

���� Local probe

• Creep test with constant load• Fatigue experiments with cyclic loading

• Variation of the temperature

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Dynamic deformation experiment

Dynamic testing [Gottstein:98 ]; TensTest_1 [Russ:96]

Load cell

Dilatationmeasurement

Sample

Movablecross head

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Engineering stress–strain diagram

Plot of the applied stress versus the strain or elongation

[Bohm 1992] Engineering stress-strain curve

Engineering strain andengineering stress:

e p0 0

0

1l ll lFA

∆= = − = +

=

ε ε ε

σ

εF

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Parameters of the stress–strain curve

• σH Proportionality limit• σE Elasticity limit• σA Anelasticity limit• σ0.2 % 0.2 % offset yield• σS Upper yield point• σS’ Lower yield point• σB Failure stress• σF Fracture point

Yield drop σS − σS’ related to a low initial dislocation density

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True stress and strain

• Engineering stress–strain diagram defined in analogy to definition of elastic values

• Large plastic deformation related to changes in dimensions which must be taken into account

• True strain τt and true stress σt

0

t

0t

0 0

d

d ln ln ln(1 )l

l

dll

l ll ll l l

=

+∆= = = = +∫

ε

ε ε

0t

0

t (1 )

AF FA A A

= =

= +

σ

σ σ ε

0 0

0

01

l A lAA lA l

=

= = + ε

The volume does notchange during deformation

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Tensile test

Macroscopic effects of dislocation motion in single crystalline metals plastically deformed in a tensile test

[Kleber 1990]

Salami

β tin

Bismuth

Zinc

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3. Microscopic processes of plastic deformation

The motion of dislocations is the elementary process of the plastic deformation of crystals.

Salami

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Orientation relations

Transformation of measured σ, ε to the values of the slip system τ (resolved shear stress) and a (slip)

a = εp/ms τ = msσ

(at the beginning of deformation)

Ψ angle between the normal of the slip plane and direction of external force,

θ angle between slip direction and

direction of external force

Φ FΨ

Slip direction Normal

of the

glide

plan

eA0

θ

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Single and multiple slip

• Schmid factor ms = cos θ cos Ψ (orientation factor)• Max. 0.5, decreases with progressing deformation • One slip system with the highest ms: single slip

• Slip sets in in a slip system having the highest Schmid factor

• If ms decreases to the orientation factor of another slip system:

simultaneous activation of both slip systems

• Multiple slip or cross slip

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Hardening curve

Resolved shear stress τ versus slip a. Distinct regions I, II, and III are to be recognized. The strain rate da/dt is constant.

[Bohm 1992] Hardening curve

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Regions of plastic deformation

I Above crit. shear stress τc (yield stress), inset of the plastic deformation in the easy glide mode, low hardening coefficientθh = dτ/da

II Almost linear function, high θh

III Parabolic shape, decrease in θh (dynamic recovery)

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Orowan equation

• Relation between slip a and dislocation motion• One dislocation gliding across the glide plane (area A) causes slip

• Nd dislocations moving over a distance of dx give a slip of

• Density of mobile dislocations ρ = Nd/Ad; slip

d ˆda bt

ρ= v

ˆ /a b d=

dˆd (d / ) /a N x A b d=

ˆd da b xρ=

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Dislocation characteristics in region I

7X0312 Deformed Si

TEM image of dislocations in silicon plastically deformed in a single-slip orientation up to a strain of 2 %

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Dislocations in region II

Dislocation interaction in GaAs plastically deformed in region II

7AA5975 Deformed GaAs

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Dislocation cell structure

Deformed Mo

TEM of the dislocation structure in Mo plastically deformed up to ε = 12 %[Luft et al. 1970/Bohm 1992]

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Dislocation multiplication

S18-90 Frank Read source

• Radius of curvature depends on resolved shear • Critical bow out for r = l/2 ( )

• Further steps are the formation of a kidney-shaped loop and the annihilation of dislocation segments with the same Burgers vector but opposite line sense

Frank–Read source

ABl =ˆ /Gb lτ ≈

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4. Hardening mechanisms

• Work hardening• Solution strengthening• Precipitate strengthening• Martensite hardening

• Hardening in small-grained polycrystals

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Work hardening

Based on the interaction of dislocations

• Applied stress for one dislocation to pass other (parallel) dislocations

ˆCGbτ ρ≈

distance of dislocations d ≈ ρ−1/2, C = 0.1 … 1

Intersection with (nonparallel) forest dislocations

[Vollertsen, Vogler 1989] Forest dislocations

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Solution strengthening

Based on the interaction of dislocations with foreign atoms• Paraelastic interaction: Different atomic radius• Interaction with the strain field of the dislocation

[Vollertsen, Vogler 1989] Impurity dislocations interaction

hsl 2002 – Physics of materials 6 - Mechanical properties 30

Cottrell atmosphere

[Vollertsen, Vogler 1989] Cottrell atmosphere

• Increase in the yield stress• Tear-off the impurity atmosphere

connected with a certain value of the strain (Lüders strain)

• Low strain rates (dislocation velocities): impurities can follow dislocation motion by diffusion

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Precipitate strengthening

Based on the interaction of dislocations with incoherent or coherent precipitates

• Bypass of dislocations (Orowan mechanism)• Cutting of precipitates

hsl 2002 – Physics of materials 6 - Mechanical properties 32

Cutting of precipitates

Formation of an antiphase boundary by cutting of an ordered Ni3Al particle by a perfect dislocation

[Haasen 1985]

Precipitate cutting; NiAl precipitates

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5. Brittle fracture

• The breaking stress experimentally much smaller than that required to pull two planes of atoms apart

• (theoretical breaking stress)• Reason is the existence of microcracks, which may propagate under load

⊥⊥⊥⊥ ⊥

⊥⊥⊥

Formation of microcracks by the pile-up of dislocations

σpile = nσ

theoˆ0.1Eσ ≈

hsl 2002 – Physics of materials 6 - Mechanical properties 34

Propagation of microcracks

• Assumed that a solid always contains microcracks• Complicated stress distribution at the tip of the crack

K21 dl

σ σ

= +

2l

2d

Maximum stress at the tip

0theo

ˆ ˆ10

E Ea

γσ = ≈

Experimental failure stress:

0B

B

theo

Ed

ad

γσ

σ

σ

=

=

hsl 2002 – Physics of materials 6 - Mechanical properties 35

Stress concentration

A shallow groove in the surface of a brittle material leads to a concentration of the stress lines at the tip of the groove. When a force is applied on either side of the groove, the groove becomes deeper, and the stress concentration increases.

[Turton::00] Crack propagation

F F

hsl 2002 – Physics of materials 6 - Mechanical properties 36

Stress–strain curve of a brittle material

Strain

Stre

ss

εB

σB

Deformation behavior of a completely brittle material. The failure stress εB is usually less than 1 %.

hsl 2002 – Physics of materials 6 - Mechanical properties 37

Summary

• Description of elastic parameters of solids in terms of linear elasticity theory

• Proportionality between stress and strain• Plastic deformation of crystals based on the motion of dislocations• Different mechanisms may influence the mobility of dislocations and

hence give rise to the hardening of the material• Processes of brittle fracture determined by the existence and propagation

of microcracks