4. Practical Examples for mild steel Aside: Type of ...140.126.122.189/upload/1052/B27204A201612302049281.pdf · 4. Practical Examples for mild steel : Aside: Type of Plastic Deformation

4. Practical Examples for mild steel :Aside: Type of Plastic Deformation :(a) Slip－Plastic deformation indep. of time duration of the applied

load.(b) Creep－Plastic deformation which continues to increase under a

constant stress, i.e. , a time-depend plastic deformation.

III.1. Ductile Matl.－A large strain is sustainable, e.g. , st , Al , Al-alloy ,

Mo ,Zn .DFN :

(i)

Behavior εσ −

) % 100 ( reduction area

) % 100 ( elongation %Ductility

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=

≡

0

0

0

0

%A

AAL

LL

f

f

(ii) Yielding strength ( ) = The strength that will induce a specified“ permanent set ”, usually, 0.05 ~ 0.3 %. ~ offset method ~

Why using offset methods ? , but some don’t !

Yet, note that Physical property of the matl.Ex.s:

(iii) Assumption in elementary mech.

( yielding point ) ( proportional limit )

(iv) Typical tested speci. Fig.s :Fig. 2.10 & 2.12 ( B & J )

yS

yielding const have metals Some σ−Q

.... lpepyyS σσσ ===

≠yS

2. Brittle Matl. No yielding , i.e. , (Gray) cast iron, concrete,－At a micro-crack, appears first and then rapidly spreads !－No well-defined tensile . Instead, is used !－Usually, better compression resistance, compared to tension, e.g. ,

⇒fσ

fσ .)( avgfσ

－Most Matl.s may exhibit ductility & brittleness, e.g. , high C-content St. become brittle : Temp.

IV. Other Important DFN & Observations* Strain hardening or Working Harding.

(i) , due to strain-hardening.

⇒ ductility ⇒↑sbrittlenes ⇒↓

12 yy σσ >

(ii) The non-overlapping Mechanical Hysteresis, due to heat or energy loss.

* Ultimate Strength , The Max. Stress developed in a matl. beforefracture.

* Poisson’s Ratio:(i) Why “－” ?

and vice versa.(ii) and “ dimensionless ”.!

≡

=..tuσ

allongitudinlateral εεν −= on)(contracti (tension) a ,Physically .. latlong εε ⇒Q

; for Constant eσσν <=

2.2 Elastic Strain Energy－Loading induced deformation Energy storing in the body.

I. Expression ?? Recall,

The work done by

* Consider a differential body under :

The strain energy stored in the body is,

⇒energy. absorb to Capability Energy Strain ≡∴

221 2

0

δδδ ⋅

===== ∫ ∫FkxdxkxdFUF

xxσ

( ) ( ) dVdxdydzdU xxxxxxxx εσεσ21

21

=⋅=

⇒

dVUV xxxx∫= εσ

21

* Due to

For a general state of stress in which all 6 arepresent, the total

DFN : Strain energy density = u = s.e. per unit volume =

:xyτ

( ) ( ) dVdydxdzud xyxyxyxy γτγτ21

21

=⋅= ∫=∴V xyxy dVu γτ

21

⇒ components−σ

( ) dVUV zxzxyzyzxyxyzzyyxx ∫ +++++= γτγτγτεσεσεσ

21

VU

; Single E

u xxxx 22

1 2σεσσ ==⇒∴G

u xyxyxyxy 22

1 2τγττ ==⇒ Single

⇒

General state of stress

II. Strain-Energy Related Modulus

= The ability of the Matl. to absorb energywithout any permanent damage.

⇒zxzxyzyzxyxyzzyyxxu γτγτγτεσεσεσ +++++=

[ ] [ ]222222

21

21

zxyzxyzyx GEτττσσσ +++++=

Resilience of Modulus o1

Eu pl

plplr 221 2σ

εσ ==

Toughness of Modulus o2

curve. under AreaTotal εσ −=tu

= The strain energy density of the matl. right before fracture.

Ex. As right, the C－content in steel may change tr uu &

.2.3 Other Topics - Briefing

I. Pure shear and Shear Modulus－ In addition to prev. shearing force

ex., pure shear can also be induced by torsion, e. g. , a shaft or tube under twisting.

－

Typically, it is similar to in tension, i.e. , as shown.For elastic shearing ;G = Shear Modulus of Elasticity = slope of

to be proven later !

II. Fatigue－ A material break-down due to cyclic stressing.－ Practical ex.s : Connecting rods & cranshafts of engines, turbine

blades, railroad wheels & axles ; transmission shaft in bending.－ Feature : The applied And the initial failure is due to micro-cracks

at over-stressed local areas.

: curve γτ −εσ −

⇒ γτ G=

)1(2 νγτ

+=−

E