08 1 Mechanical Properties

7/31/2019 08 1 Mechanical Properties

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1. MECHANICAL CHARACTERIZATION OF

MATERIALS. TENSILE PROPERTIES

1.1 Introduction

1.2 Stress and Strain. Tensile tests1.3 Stress State

1.4 Elastic Deformation and Plastic Deformation

1.5 Elastic Properties of Materials

1.6 Tensile Properties

1.7 Elastic Recovery. Strain Hardening

1.8 True Stress/True Strain Curve. Necking Criterion



1. MECHANICAL CHARACTERIZATION OF

MATERIALS. TENSILE PROPERTIES

TOPIC’S OBJECTIVES

- Concepts of stress and strain

- Define the state of stress in a point of a solid

- Introduce the Hooke’s law in three dimensions

- Describe the tensile tests- Define the parameters that describe the mechanical

behavior of materials



- To assure performance, safety and durability of devices,instruments and structures

- The knowledge of the mechanical properties provides the basis for preventing failure of materials in service

• Why must the mechanical properties of materials be known?

1.1 INTRODUCTION

• How are determined the mechanical properties ofmaterials?

- Mechanical characterization, i.e. studying of their deformation and cracking



- Any change in the material that induces the lost or worsening of its structural capabilities

- Deformation and fracture

1.1 INTRODUCTION

• What is the failure of a material?



1.1 INTRODUCTION

DEFORMATION

Time Independent■Elastic

■Plastic

FRACTURE

Static Loading■Brittle ■Ductile

■ Enviromental

■ Creep Rupture

Fatigue: Cyclic Loading

■Low cycle ■High cycle

■ Fatigue crack growth

■ Corrosion fatigue

MATERIALSFAILURE

Time Dependent■Creep



1.2 STRESS AND STRAIN

Dashed lines represent the shape before deformation, and solid line afterdeformation.

o A

P =σ

o

o

o

o

l

l

l

l l Δ=

−=ε

Engineering stress:

Engineering strain:

P

P P

P

Shear stress: o A

F

=τ

Shear strain: a

δ θ γ == tan

Tensile test Compression test

Shear deformation

a

δ



1.2 STRESS AND STRAIN TENSILE TESTS

Tensile test machine

6 mm

3 mm

2 mmt =0.5 – 1.5 mmt

6 mm

11 mm

1.5 mm

35 mm

Standard specimens for tensile tests



Tensile Specimens and ApparatusTensile Specimens and Apparatus




Tensile Test ConceptTensile Test Concept




Specimen

Clamp

Crosshead & Load Cell

Clamp



During the Tensile TestDuring the Tensile Test



Elongation

F o r

c e




Elongation


F o r

c e



Results and AnalysisResults and Analysis

Nominal strain

N o

m i n a l s t r e

s s




6061-T651

Aluminum

Cold Rolled

1018 Steel

Copper

C2600 Brass,

half hard

Annealed

1018 Steel S t r e s s (

M p a )

Strain (mm/mm)

1.3 STRESS STATE



θ θ σ

θ

θ τ

θ σ

θ

θ σ

θ

θ

cossin

cos

sin

cos

cos

cos

//

2

===′

===′ ⊥

o

o

A

F

A

F

and

A

F

A

F

is the applied stress

σ’ is the normal stress acting on the plane pp’

τ’ is the resolved shear stress in the specific direction p-p’

o A

F =σ

// F F F rrr

+= ⊥

F

F

F // F ⊥

Aθ

Ao

θ

O

1.3 STRESS STATE STRESS COMPONENTS



FnZ

FtX

F

ΔΑ

Y

Z

FtY

k ji A

F

A

F

A

F

A

F s

k F j F i F F

nztytxnZ

A

tY

A

tX

A A

tZ tY tX

rrr

rrrr

r

rrrv

σ σ σ ++=Δ

+Δ

+Δ

=Δ

=

++=

→Δ→Δ→Δ→Δlimlimlimlim

0000

The stress state at a point of a given plane is define by two stress components tangent to the plane, tx and tx , and one component normal to the plane,

nz !!

1.3 STRESS STATE STRESS COMPONENTS



The state of stress at a point is completely defined when

the stress components are known on three mutually perpendicular planes

X

Y

Z

σ zx

σ zz

σ zy

σ yx

σ yz

σ yy

σ xx

σ xz

σ xy

Stress component notation:

•The first subscript is the direction of thenormal to the plane, and the second thedirection of the stress component.

•A normal stress is positive if the

direction of the unit normal vector andthe direction of the stress component areboth in the positive direction or both inthe negative direction of the coordinatesystem.

•Tensile stresses are defined as positiveand compressive stresses are negative.

3 sr

1 s

r

2 sr

componentsstressshear are jiijij≠≡ τ σ



1.4 ELASTIC DEFORMATION



For shear forces :

G the shear modulus

Elastic strain is produced by small reversible changes in the equilibrium interatomic spacing

⇒

⎪⎪⎭

⎪⎪

⎬

⎫

⎟ ⎠ ⎞⎜

⎝ ⎛ ⎟

⎠ ⎞⎜

⎝ ⎛ =

+=⇒=

ee x x

ee

e

e

d dx

dxdF

A E

x x x x

x-xε

ε

ε

1

e

eee

x x

x x x

dx

dF

Ad

dx E

A

F

d dx

dxd

d d E

⎟ ⎠

⎞⎜⎝

⎛ ⎟ ⎠

⎞⎜⎝

⎛ =⇒

⎪⎪⎭

⎪⎪⎬

⎫

=

⎟ ⎠ ⎞⎜

⎝ ⎛ ⎟

⎠ ⎞⎜

⎝ ⎛ =⎟

⎠ ⎞⎜

⎝ ⎛ =

1

ε σ

ε σ

ε σ

e

e

xdx

dF

A

x E ⎟

⎠

⎞⎜⎝

⎛ =

x, Interatomic distance

Repulsión force

Attraction force

xe

F ,

F o r c e

e xdx

dF ⎟ ⎠ ⎞

⎜⎝ ⎛ Fig. 1.7. Interatomic force as a function of the interatomic spacing.

γ τ G=

A is the cross-sectional area of material per atom

1.4 PLASTIC DEFORMATION



Elastic limit

N O M I N A L S T R E S S

NOMINAL STRAIN Plastic ElasticεEεp

Total Strain

loading unloading

Fig. 1.8. Stress-strain curve showing elastic and plastic deformation

I I E E

ε σ ε ε ε +=+=

Plastic deformation ⇔ bond breaking betweenneighbor atoms and reforming bonds between new

neighbor atoms ⇒ slip process; dislocation motion

Time independent → plastic strain

Time dependent → creep strain

ε I Inelastic

strainElastic limit

N O

M I N A L S T R E S S

NOMINAL STRAIN Plastic ElasticεEεp

Total Strain

loading unloading

Fig. 1.8. Stress-strain curve showing elastic and plastic deformation

I I E E

ε σ ε ε ε +=+=

Plastic deformation ⇔ bond breaking betweenneighbor atoms and reforming bonds between new

neighbor atoms

Time independent → plastic strain

Time dependent → creep strain

ε I Inelastic

strain

Slip Process

&

Formation and motion ofdislocations

σ

1.5 ELASTIC PROPERTIES OF MATERIALS



z x

E ε

ν σ −=

A homogeneous and isotropic material is subjected to an axial stress σ x

Y

Z

Xd o

Ao

l o

Y

Z

Xd o

A

l

d

O

E

σ x =E ε x

E/ ν

ε x ,ε y ,ε z strain

σ x

s t r e s s

O

ε y , ε z

ε x strainν

σ x

Fig. 1.9. Longitudinal extension and transversal contraction

oo

o x

l l

l l l

Δ=

−=ε

oo

o z y

d

d

d

d d Δ=

−== ε ε

z x

E ε

ν σ −=

z y x

E E

ε ν ε ν σ −=−= x x

x

z

x

y

E ε σ ε

ε

ε

ε ν

⎪⎭

⎪

⎬

⎫

=

−=−=−=⇔

strainallongitudin

ncontractioltransversa ratiosPoisson'

⇒



1.5 ELASTIC PROPERTIES HOOKE’S LAW FOR 3 D



σ yy

Consider an isotropic body under a general stress state

RESULTING LONGITUDINAL STRAINSTRESS

X-direction Y-direction Z-direction

σ xx

σ yy

σ zz

xxε yyε

X

Y

Z

σ zx

σ zz

σ zy

σ yx

σ yz

σ xx

σ xz

σ xy

zzε

E zzνσ

−

E xxσ

E

yyνσ −

E zzνσ

−

E xxνσ

− E

xxνσ −

E

yyνσ −

E

yyσ

E

zzσ

zzε

Shear stresses σ xy = σ

yx , σ

yz = σ

zy and σ

zx = σ

xz produce only shear strains

given by

,,,GGG

yz

zy yz xz

zx xz

xy

yx xy

σ ε ε

σ ε ε

σ ε ε ======

1.5 ELASTIC PROPERTIES HOOKE’S LAW FOR 3 D



These equations, taken together, are the generalized Hooke’s law for a

isotropic material

( )[ ]

( )[ ]

( )[ ]

,,,

1

1

1

GGG

E

E

E

yz

yz yz xz

xz xz

xy

xy xy

yy xx zz zz

zz xx yy yy

zz yy xx xx

σ ε γ

σ ε γ

σ ε γ

σ σ ν σ ε

σ σ ν σ ε

σ σ ν σ ε

======

+−=

+−=

+−=

⎟⎟⎟⎟⎟⎟⎟

⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎜

⎝

⎛

⎟⎟⎟

⎟⎟⎟⎟⎟⎟

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎜⎜⎜⎜⎜⎜

⎜⎜⎜⎜

⎝

⎛

−−

−−

−−

=

⎟⎟⎟⎟⎟⎟⎟

⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎜

⎝

⎛

zx

yz

xy

zz

yy

xx

zx

yz

xy

zz

yy

xx

G

G

G

E E E

E E E

E E E

σ

σ

σ

σ σ

σ

ν ν

ν ν

ν ν

ε

ε

ε

ε ε

ε

100000

01

0000

001

000

0001

0001

0001

or



1.5 ELASTIC PROPERTIES Relationship between E, G and



( ) ⎟ ⎠ ⎞⎜⎝ ⎛ +=⇒⎪⎭

⎪⎬

⎫

=

⎟ ⎠

⎞⎜⎝

⎛ +==

Δ

22

1

12

2

1

2a F

E aad

a

F

E d d

d

o

oo ν δ

ν δ

Now, ⎟ ⎠

⎞⎜⎝

⎛ === 2

1

a

F

GGa

τ δ γ

E G

)1(2 ν +=⇒



1.6 TENSILE PROPERTIES Tensile strength



• Ultimate tensile strength, or tensile strength ↔ the maximum stress in

stress-strain curve.

• Necking ↔ formation of a small constriction or neck in the specimen.

• Fracture strength ↔ stress at the fracture point

Engineering stress-strain curve showing the ultimate tensile strength and the fracture point.

σ uts

Strain

S t r e s s

Uniform strain

Necking; UTS point

Strain at the neck

1.6 TENSILE PROPERTIES - curves



• Ultimate tensile strength ranges from 40 MPa (Mg alloys) to 3000 MPa (W

alloys).• For design purposes, the yield strength is used instead of the tensile

strength.

• Fracture strength are not normally specified for engineering designpurposes

1.6 TENSILE PROPERTIES Ductility



Engineering stress-strain curve for brittle and ductile materials

• Ductility is the capability of a material to sustain plastic deformation before

fracture• Ductility is quantitatively expressed as either percent elongation or

percent reduction in area at fracture

100% ×⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −=

o

o f

l

l l EL

100% ×⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛ −=

o

f o

A A A RA

1.6 TENSILE PROPERTIES Resilience



• Resilience is the capability of a material to store elastic energy during

loading, and to realease it during unloading.

• Resilience is measured by the resilience modulus U r

Representation of the resilience modulus

E d U

y

y yr

y

22

12

0

σ ε σ ε σ

ε

=== ∫

1.6 TENSILE PROPERTIES Toughness

Th h i h bili f i l b b



• Thoughness is the capability of a material to absorb energy up to fracture

• For static loading conditions, that is, at low strain rate, toughness may be determined from a tensile stress-strain curve up to fracture.This toughness is referred to as tensile toughness.

f

uts y f

d ε

σ σ

ε σ

ε

20

+

≈∫

Strain

S t r e s s

utsσ

ε f

Tensile Toughness:

0.002

σ y 2

)( uts y σ σ +



1.8 T- T CURVE Graphical Interpretation of Necking Criterion

At th i t f i l d i t bilit i t i



• At the point of maximum load appears instability in tension

(non-homogeneous deformation) and it satisfies: criterion! Necking T

T

T

d

d σ

ε

σ =

ε T

σ T

1

ε T,uts

σ T,uts

Determination of the point of necking at maximum load in the true stress/true strain curve



1.8 TRUE STRESS/TRUE STRAIN CURVE

• Above the necking onset, true strain can not be determined l ( ) f h d d f



⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=⇒⎪⎭

⎪⎬

⎫

=⇒=

=−

+=+=

i

oT

iioo

o

i

o

oiT

A

A

l Al ActeV l

l

l

l l

ln

ln)1ln()1ln(

ε

ε ε

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=i

o

i

oT

D

D

A

Aln2lnε

gas ln(1+ ) from the measured strain , because deformation is

not uniformly distributed any more.

• Now,

• For cylindrical specimens of diameter D,

• The formation of a necked region introduces triaxial stresses that make difficult to determine accurately the longitudinal

tensile stress from the onset of necking until fracture occurs



1.8 T- T CURVE True stress at maximun load

If A i h i l i l d h

TRUE TENSILE STRENGTH



• If Au is the cross-sectional area at maximun load, then

• If ε T,uts is the true strain at maximum load, also called true uniform strain , then

• Also,

⇒⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=⇒⎪⎪

⎭

⎪⎪

⎬

⎫

=

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =⎟⎟

⎠

⎞⎜⎜⎝

⎛ =

uts

utsT

utsT

uts

u

outsT

u

o

o

uutsT

A

A

A

A

l

l

σ

σ

ε σ σ

ε ,

,

,

,

ln

lnln

uts

u

outsT

o

uts

u

utsT

A

A

A

P

A

P

σ σ

σ

σ

=⇒

⎪⎪⎭

⎪⎪⎬

⎫

=

=

,

max

max,

→True stress at maximum load

utsT eutsutsT

uts

utsT

utsT ,

,

,

, lnε

σ σ σ σ ε =⇒⎟⎟

⎠ ⎞⎜⎜

⎝ ⎛ =

True strain at maximum load

or true uniform strain

→True stress at maximum load

08 1 Mechanical Properties

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