楊氏係數測量實驗. stresses in solids the level of stress required to obtain a given...

楊氏係數測量實驗

Stresses in Solids

• The level of stress required to obtain a given

deformation ※Tensile stress( 伸長應力 )， Tensile strain ( 伸長應變 )

and Young’s modulus ( 楊氏模數 )

※Shear stress( 剪應力 ) ， Shear strain ( 剪應變 ) and

Shear modulus ( 剪力模數 )

※ Volume stress( 體積應力 ) ， Volume strain ( 體積應變 )

and Bulk modulus ( 體積彈性模量 )

Tensile Strain ( = Fractional change in length

= 0

L

L

應變)

2 Normal force N

Tensile stress ( S ) ( )= = Area m

F

A應力

Tensile stress( 伸長應力 )， Tensile strain ( 伸長應變 ) and Young’s modulus ( 楊氏模

數 )

Tensile stress Young's modulus ( Y ) =

Tensile strain

0

/ Y =

F A

L/L

F⊥

F⊥

F⊥ F⊥

Stress 應力 -strain 應變 relationship

線性範圍

Young’s modulus ( 楊氏模數 ) ， Shear modulus ( 剪力模數 ) ，Bulk modulus ( 體積彈性模量 )

Tensile stress Young's modulus ( Y ) =

Tensile strain

0

/ Y =

F A

L/L

光槓桿系統 (Optical Lever )

雷射光源

直尺

平面鏡

腳尖

圓柱狀金屬栓 ( 鎖住鋼線 )

鋼線

L0

P

A B

C

A,B

C

L0∆d

tan2

tan2 2

d

d

Dd

D

D

2

因 很 小

sin L

P

tan2 d

D

sin L

P

P dL

d

2

=

L

D P

2D Mg

A,B

C

Tensile stress Young's modulus ( Y ) =

Tensile

strain

2

0

0

/ Y =

Mg / r

F A

L/L

L/

P dL =

2D

L

其中

• End of Lecture

Shear stress( 剪應力 )， Shear strain ( 剪應變 ) and Shear modulus ( 剪力模數 )

Shear stressShear modulus ( S ) =

Shear strain

/

/tF A

Sx h

2

Tangential force NShear stress = =

Area mtF

A

Shear strain = Fractional change in length

= x

h

Volume stress( 體積應力 ) ， Volume strain ( 體積應變 ) and Bulk modulus ( 體積彈性模量 )

2

force NVolume stress ( change in pressure ) =

Area m

P = nF

A

Voulme strain = Fractional change in volume

= V

V

Volume stressBulk modulus ( B ) =

Volume strain

> 0 Generally < 0 /

- P VB

V V P

Note : Compressibility ( k ) = 1 / B

線性範圍

Shear stress( 剪應力 )， Shear strain ( 剪應變 ) and Shear modulus ( 剪力模數 )

Shear stressShear modulus ( S ) =

Shear strain

/

/tF A

Sx h

2

Tangential force NShear stress = =

Area mtF

A

Shear strain = Fractional change in length

= x

h

血液動力學中的剪力

3

4 Q

R

Volume stress( 體積應力 ) ， Volume strain ( 體積應變 ) and Bulk modulus ( 體積彈性模量 )

2

force NVolume stress ( change in pressure ) =

Area m

P = nF

A

Voulme strain = Fractional change in volume

= V

V

Volume stressBulk modulus ( B ) =

Volume strain

> 0 Generally < 0 /

- P VB

V V P

Note : Compressibility ( k ) = 1 / B

Young’s modulus ( 楊氏模數 ) ， Shear modulus ( 剪力模數 ) ，Bulk modulus ( 體積彈性模量 )

Stresses in Fluids• Normal stress ( pressure ) compress or expand the fluid particle witho

ut changing its shape

※ Bulk modulus ( 體積彈性模量 )• Tangential or shearing stress shear the fluid particle and deform its shape ※ Viscosity ( 黏滯力 ) The viscosity of a fluid measures its ability to resist a shear stress.

The Nature of Fluids

• The fluids cannot support Tensile stresses and Shear stresses .

• The fluids flow and deform continuously and permanently under Shear stresses .

Young’s modulus ( 楊氏模數 ) ， Shear modulus ( 剪力模數 ) ，Bulk modulus ( 體積彈性模量 )

Shear stress ( rate of change of shear angle

But

So

coefficie

tu tyuy

剪應 )力

Δu duτ = μ = μΔy dy

nt of viscosity

Shear stress = FA

velocity of top face relative to the bottom faceu

u

dudy

ddt

Viscosity Shear Stress ≡ Change in momentum in bulk fluids

Viscosity Stresses tend to decrease the velocity of the flow on the high speed side of the layer, increase the velocity on the low speed side.

黏滯應力 黏滯應力 ( ( Viscosity Stress )

Momentum exchange by molecular mixing

A shear layer near a solid wall

Velocity profile in the region near a solid surface.

u

yyx

d

d

du

Growth of a boundary layer along a stationary flat plate.

Boundary layers

Viscous effects particularly important near solid surfaces.

Laminar and Turbulent Flow

1

2

DvReynolds number e =

D = diameter of the pipev = velocity of the flow

= density of the fluid = viscosity of the fluid

2v Dynamic pressureDve = =2 v Viscous stress

D

R

R

Viscous force

Inertia force

Figure 1.22 The no-slip condition in water flow past a thin plate. Flow is from left to right. The upper flow is turbulent, and the lower flow is laminar. With permission, Illustrated Experiments in Fluid Mechanics, (The NCMF Book of Film Notes, National Committee for Fluid Mechanics Films, Education Development Center, Inc.,1972).

Viscous Internal Flows

τ(y)

Laminar flow of viscous fluidsof viscous fluids in a circular pipe

dpp+2

dpp-2

dx

D ω

d rd x

D

2

τ, P

2 2

w

1 πD 1 πD Dp - dp - p + dp = τ 2π dx

2 4 2 4 2

w w

dp 4 D dp= - τ τ = -

dx D 4 dx

shear stress at the wallwτ

dr

r

v

τ dr dr τ dr drF = τ + 2π r+ dx- τ - 2π r- dx

r 2 2 r 2 2

τ

=τ 2π drdx+ 2π rdrdxr

p

pdx pdxF = p- 2π rdr- p+ 2π rdr

x 2 x 2

p=- 2π rdrdx

x

0 v pF F

0 , at ( )2

Du r no slip

0 , at 0 ( )du

r symmetrydr

1

rp

x r r r r

dr

rd

rdx

dp 1

anddp

Kdx

Kdr

rd

r

1

1

2

2cK

rr

r

cKr 1

2

But, du

dr

Boundary conditions

21

2

ln4

cru K r c

21 2 0 , and 16c c D K

22KD 2ru= 1-

16μ D

22D dp 2ru= - 1-

16 dx D

u

max

Q Av

dQ u r 2 rdr

u r

u u r 0

π

22

2

D dp 2r= - 1-

16 dx D

D dp-

16 dx

Flow rateFlow rate

r

dr

r

D/2 dr

r D/2 r D/2

r 0 r 0

r D/2 r D/23

r 0 r 0

dQ (2 r dr)

π Q dQ rdr

π r dr r dr

π

22

22

2

2

D dp 2r- 1-

16 dx D

D dp 2r- 1-

8 dx D

D dp 4-

8 dx D

2 2

r D/2

r 0

D D

8 16

Poiseuille's Law

R=D/2 ;

1 u r 2 rdπ

4

41 2 1 2

1 DQ

128

P P P P1 RQ

8 L L

V

其中

dp-dx

dp-dx

A r

2Q 1 D

32

dp-dxA

Figure 1.17 A long flat plate moving at constant speed in a viscous fluid. On the left is shown the velocity distributions as they appear to a stationary observer, and on the right they are shown as they appear to an observer moving with the plate.

Surface TensionCohesive forces≡attractive forces between molecules of the same type

2

Tensile ForceSurface Tension =

Length

N J

m m

F =σ ( 2 l )

Surface Tension

2 22

Nsurface tension ( )m

r p r

pr

• (a) Drop

• (b) BubbleEquilibrium (a) drop and (b) bubble, where the excess pressure is balanced by surface tension.

∆p

2 44

r p r

pr

2 22

Nsurface tension ( )m

r p r

pr

Capillarity Adhesive forces≡attractive force between molecules of the different type

Pa

=

2

For water on clean glass

For mercury in a glass tube

upward force weight of the liquid column(due to surface tension)

2 cos2 cos

090 < 0 ( depression )

r g r

hgr

h

h

θ

Figure1.25 Angle of contact. (a) Free surface shape of water and mercury in glass tubes. (b) A wetting, and a non-wetting liquid.

Figure 1.26 A drop of liquid squeezed between two glass plates.

• Home Work• 流體力學• chapter 1 Introduction 19, 21, 22, 26, 27, 28, 29, 31• chapter 9 Viscous Internal Flows 35, 36, 38 (a), 39• chapter 29 NMR 24, 25, 27, 29, 30, 31, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 44