3.4 HOOKE’S LAW - KOCWcontents.kocw.net/KOCW/document/2015/hanyang/hanseog... · 2016-09-09 · 3.4 HOOKE’S LAW • Modulus of Elasticity (Hooke’s Law) • Strain Hardening

3.4 HOOKE’S LAW • Modulus of Elasticity (Hooke’s Law)

• Strain Hardening

E

3.5 STRAIN ENERGY

• Modulus of Resilience

• Modulus of ToughnessEntire area under the stress-strain diagram

Eu pl

plplr

2

21

21

Copyright ©2014 Pearson Education, All Rights Reserved

EXAMPLE 13.1

EXAMPLE 13.1

EXAMPLE 3.2 The stress–strain diagram for an aluminum alloy that is used for making aircraft parts is shown in Fig. 3–19. If a specimen of this material is stressed to 600 MPa, determine the perma-nent strain that remains in the specimen when the load is re-leased. Also, find the modulus of resilience both before and after the load application.


EXAMPLE 3.2 (cont)

• When the specimen is subjected to the load, the strain is approximately0.023 mm/mm.

• The slope of line OA is the modulus of elasticity,

• From triangle CBD,

Solutions

mm/mm008.0

100.7510600 96

CDCDCD

BDE

GPa 0.75006.0

450E


EXAMPLE 3.2 (cont)

• This strain represents the amount of recovered elastic strain.

• The permanent strain is

• Computing the modulus of resilience,

• Note that the SI system of units is measured in joules, where 1 J = 1 N • m.

Solutions

(Ans) MJ/m 40.2008.060021

21

(Ans) MJ/m 35.1006.045021

21

3

3

plplfinalr

plplinitialr

u

u

(Ans) mm/mm 0150.0008.0023.0 OC


EXAMPLE 13.1

EXAMPLE 13.1

3.6 POISSON’S RATIO

long

latv


EXAMPLE 3.4

A bar made of A-36 steel has the dimensions shown in Fig. 3–22. If an axial force of P = 80kN is applied to the bar, determine the change in its length and the change in the dimensions of its cross section after applying the load. The material behaves elastically.


EXAMPLE 3.4 (cont)

• The normal stress in the bar is

• From the table for A-36 steel, Est = 200 GPa

• The axial elongation of the bar is therefore

Solutions

mm/mm 108010200100.16 6

6

6

st

zz E

Pa 100.16

05.01.01080 6

3

AP

z

(Ans) m1205.11080 6z

zz L


EXAMPLE 3.4 (cont)

• The contraction strains in both the x and y directions are

• The changes in the dimensions of the cross section are

Solutions

m/m 6.25108032.0 6 zstyx v

(Ans) m28.105.0106.25

(Ans) m56.21.0106.256

6

yyy

xxx

L

L


3.7 SHEAR STRESS-STRAIN DIAGRAM• Strength parameter G

Shear modulus of elasticity or the modules of rigidity• G is related to the modulus of elasticity E and Poisson’s

ratio .• Only two material constants are required for isotropic

materials.

G

vEG

12


EXAMPLE 3.5 A specimen of titanium alloy is tested in torsion and the shear stress– strain diagram is shown in Fig. 3–25a. Determine the shear modulus G, the proportional limit, and the ultimate shear stress. Also, determine the maximum distance d that the top of a block of this material, shown in Fig. 3–25b, could be displaced horizontally if the material behaves elastically when acted upon by a shear force V. What is the magnitude of V necessary to cause this displacement?


EXAMPLE 3.5 (cont)

• By inspection, the graph ceases to be linear at point A. Thus, the proportional limit is

• This value represents the maximum shear stress, point B. Thus the ultimate stress is

• Since the angle is small, the top of the will be displaced horizontally by

Solutions

(Ans) MPa 504u

(Ans) MPa 360pl

mm 4.0mm 50

008.0rad 008.0tan dd


EXAMPLE 3.5 (cont)

• The shear force V needed to cause the displacement is

Solutions

(Ans) kN 270010075

MPa 360 ; VVAV

avg


EXAMPLE 13.1

EXAMPLE 13.1

3.8 FAILURE OF MATERIALS DUE TO CREEP AND FATIGUE

• Creep: Change of time-dependent deformation or stressdue to time under the constant temperature

- Creep: change of time-dependent deformation under the constant stress and temperature


Stage I: transient creepStage II: steady creepStage III: accelerate creep


- Creep LimitThe maximum stress among the stresses having steady creep rate 0 at a specified temperature.

- Creep StrengthThe maximum stress a given material can withstand in a given time without exceeding a specified quantity of creep (ex.: 0.1%/1000h).



- Stress relaxation: change of time-dependent stress under the constant strain and temperature



- Fatigue Test: Rotation bending fatigue tester


weight

specimen motor


- Fatigue: S-N curve


elS : Endurance limitor fatigue limit

: Fatigue strengthat N cycles

hS

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3.4 HOOKE’S LAW - KOCWcontents.kocw.net/KOCW/document/2015/hanyang/hanseog... · 2016-09-09 · 3.4 HOOKE’S LAW • Modulus of Elasticity (Hooke’s Law) • Strain Hardening