unit 1 lecture 1_def_v1 3@page

8/7/2019 UNIT 1 lecture 1_DEF_v1 3@page

1/22


2/22

Machine Design Unit 1 - Lecture 1: Introduction to material behavior

Page 2 2009 Politecnico di Torino

4

Introduction to material behavior

Tensile specimens

Basic definition of stressTensile test conditions

Basic definition of strain

Stress-strain curve and material properties

Examples of stiffness properties

Example of strength properties andmacrostructure of the fractured surface

Ultimate strain


Tensile specimens

6

Specimen geometry (1/5)

Fillet

Head

The test specimen is composed of a sampleof the material in question and is constructedin the shape of a slender, cylindrical cross-section bar.

LG: gage length

LG


3/22


4/22



10

Specimen shape (5/5)

Example of plane specimen

11

Specimen cross-sections

Area of the undeformed

cross-section: AO

12

Proportional specimen (1/2)

Proportional specimen

LO = 5d, rounded to the nearest integermultiple of 5mm

LO + d/2 < LG LO + 2d

d

Lc

Lo


5/22



13

Proportional specimen

, rounded to thenearest integer multiple of 5mm

Proportional specimen (2/2)

2

oA = d

44

5.65=5.0

LG

LO

oo A65.5L =

ooGoo A5.2LLA5.1L ++


Basic definition of stress

15

Stress on cross-section

Average stress

o

F

A =

F

F

Ao


6/22



16

Average and local stress (1/2)

Infinitesimal force dF on the

infinitesimal area dAo

Force on area Aoh

b

dF

F oLocal

dA

dF=

oAverage

A

F=

17

Average and local stress (2/2)

dF

h

b

If the local stress is thesame on eachinfinitesimal area of thecross section then the

stress is uniformlydistributed over thecross-section:

dAo

Ao

Local = Average

oLocal

dA

dF=


Tensile test conditions


7/22



19

Testing machine

mobile crossbar

base

grip heads

load cell

columns

specimen

20

Specimens clamping

AA

circularspecimen

Planespecimen

Sect. A-A

wedge grip

21

Test rate

The force is assumed to be applied slowly andthen maintained at a constant level. The limitsto the rate of loading are:

s

N/mm30

t

6

2

For steel

For aluminums

N/mm10

t

2

2


8/22


9/22



25

Infinitesimal element within the specimen

Along the specimen gage length the stresses

and the strains are constant over anycross-section

OdF dA =

dx

26

Transverse deformation (1/5)

Material undergoes both axial and transversedeformation. (Here the tensile case is shown)

27

h

dx

In the linearly elastic range every infinitesimalvolume within the gage length undergoes thesame deformation



10/22



28


h

b

)1(b

)1(h

)1(dx +

dx

Isotropic material

29


h

b

dy

dzdx

This is true for any element belonging the cross-

section

dx

30


Contraction

( )

o

o

L

LL

dx

dx1dx

+=

Extension( )1dx +

dz

dy

dx

( )1dz

( )1dy


11/22



31

Generally speaking

y

x

z

y1dydy +

( )x1dxdx +

( )z1zddz + zy ==


Strain-stress curve and material properties

33

F- ductile material (1/4)

FyFy,low

F

Fu

local plasticdeformation

uniform plasticdeformation

Ductile material with yielding

fracture

elasticdeformation


12/22



34


FY

F

elastic deformation: whenthe load is removed thematerial returns to itsoriginal state

Yield load: FY

35


Fu

F

Uniform plastic deformation:there is a permanent uniformdeformation upon removal ofthe load

Ultimate load: Fu

36


F

FfFracture

Local plastic deformation(necking)

Fracture load: FF


13/22



37

F- ductile material without yielding (1/2)

F

%0.2

Fu

Fp0.2 Fracture

Local plasticdeformation

Uniform plasticdeformation

38

F- ductile material without yielding (2/2)

Offset yield load: Fp 0.2

%2.0 %2.0

Fu Fu

Fp 0.2 Fp 0.2

Offset strain

= 0.2%

39

F- brittle material

FractureF

Fu

elastic deformation


14/22



40

From F- to - (1/2)

FFu

Fy

Su

Sy

Unlike Force-strain curve F-, the stress-straincurve - does not depends on the area of thecross-section but only on the strain

SY or y= yield strengthSu or u= ultimate strength

41

From F- to - (2/2)

The indicated decrease in the stress level betweenultimate stress and fracture is due to the fact that theundeformed original area is used for computing . Thestress computed using the actual area is called truestress whereas the conventional stress computedusing the original area is called engineering stress.

FFu

Fy

Sy

Su

42

elasticdeformation

- curve for ductile materials 1/2

A material that behaves in ductile mannerexperiences large amount of strain before fracturing.The elastic range is much smaller then the plasticrange. A realistic plot scale is the following

F

~0,10,5% ~1025%

plastic deformation


15/22



43

- curve for ductile materials 1/2

0.2%

Su Su

Sy

Sp0.2


Linearly elastic materials

45

Linearly elastic deformation

For many commonengineering materialsthere is a portion of theforce-strain curve that islinear. The force isproportional to strain

Fp0.2

F

%2.0

KF =

Proportional limit


16/22



46

Modulus of elasticity (1/2)

0.2%

Sp0.2

The constant ofproportionality is calledmodulus of elasticity orYoungs modulus

HOOKEs LAW:ut tensio sic vis

= E

47

Modulus of elasticity (2/2)

Steel 2 105 0.3

Cast iron 1 105 1.8 105 0.27

Titanium 1.2 105 0.3

Aluminum 7 104 0.3

Material properties (E, ) for some selected

metallic materials

E - N/mm2

48

The maximum allowable stress on steel is

about = 1000 N/mm2

The corresponding strain is

The area of the deformed section is:

Then is justifiable to define the conventionalengineering stress as:

( ) ( ) ( )

( ) ( )

2

O

O O O

A dy 1 dz 1 A 1

A 1 2 A 1 0,003 A 0,997

= =

= =

Order of magnitude for strain

0,005E

=

=

OF/A =


17/22




Strength of selected materialand macroscopic characteristics of failure

50

Strength of selected materials

Material (minimum values)A%S

u

MPa

Sy

STEEL - Structural(UNI EN 10025)

STEEL annealed(UNI EN 10083)

CAST IRON

Gray

CAST IRONSpheroidal

262222

181111

9

---

1772

370500700

100200290

230320420

---

600850

10001250

400580800

1050

360430510

235275355

S 235S 275S 355

C 30C 60

41Cr436NiCrMo3

G10G20G30

Gs370-17Gs500-7Gs700-2

MPa

51

Characteristics of ductile failure (1/4)

Adjacent fractured parts of a specimen, from awelded plate, placed together.

necking

welding

{

failure


18/22



52

Characteristics of ductile failure (2/4 )

Plastic lips

53


Ductile failure oninclined cross-section

Adjacent fractured parts of aspecimen from rolled plate placedtogether.

54

Thin rolled plate, plastic flow before failure

Detail of the plastic flow



19/22



55

Characteristics of brittle failure

Specimen from cast aluminum,after failure

Fracturecross-section


Strain at fracture

57

Uniform plastic deformation

Within gage length everycross-section behaves inthe same way

Uniform plastic deformation

Elastic deformation


20/22



58

Permanent deformation at Rm

Sy

m

Uniform permanentdeformation:it is a distinctivefeature of the material

but

not standardized anddifficult to measure

59

Localized plastic deformation

A%: permanentstrain after fracture

u o

o

L LA% 100

L

=

%

Localized plasticdeformation

A%

Sy

60

Necking and proportional specimens (1/5)

Initial shape

At fracture

Lo

L

Up to = Su

Lu


21/22



61


Lu

Uniform deformation dueto maximum stress Su

Strain due tonecking

aS

( ) smof a1LL ++

u o s

m

o o

L L aA% 100 100 100

L L

= = +

62


Depending onmaterial

Depending also on cross-section shape and size

(with shape restrictionaccording to standard)

M.J. Barba, Mem. Soc. Ing. Civils,

Pt. 1, p. 682, 1880

u o sm

o o

L L aA% 100 100 100

L L

= = +

S Oa K A=

63


To compare strain measurements afterfracture of specimens with different size theyneed to be proportional; indeed, as:

S Oa K A=

sm

o

aA% 100 100

L= +

O

m

O

AA% 100 K

L

= +


22/22


64

In order A% being indicative of a material

proprietythat is

to compare strain measurements after failure ofspecimens with different size

the specimens need to be similar; fromwhich:

O

m

O

AA% 100 K

L

= +

O OL 5,65 A =


unit 1 lecture 1_def_v1 3@page

Documents