Wedged (cutting) deformation of worksheet

and its stress analysis with the slab method

The 9th lesson

L8

くさび刃の基本的な形状

http://en.wikipedia.org/wiki/File:Microtome-knife-profile.svg

のみ，たがねの形状，二段刃 (多段くさび刃)

片刃 (side beveled)

一段刃(くさび刃)

対称刃 (symmetric wedge)

これらの対称刃もある

くさび(wedge)

堅い木材や金属で作られたV字形または三角形の道具. 一端を厚く,

もう一端に向かってだんだん薄くなるように作られている.切れ目を作

る，あるいは，隙間に打ち込むための形状である. その用途として

物を割る(切断する)

物と物とが離れないように圧迫する

というまったく異なる目的がある.

裁断機と断裁機

一般的に裁断機という言葉は、布・皮革などの素材や印刷物を含む紙全般を裁ったり、抜き型でプレスして型抜き加工したりする機械を指す言葉として広く使われる。これに対し断裁機という言葉は紙を直線的に切り離す機械に限定される。--

決められた寸法（長さ）を合わせて紙・プラスチックなどを切断するための機械のこと。製本の加工工程の一つで多く利用されているほか、合成樹脂や、カーボンシート等の切断加工にも広く利用される。

(http://ja.wikipedia.org/wiki/断裁機)

Observation of cutting process of several work

sheets (introduction for pushing cut)

1) Paperboard … orthotropic, laminated, quasi-

plastic (non-metallic)

No necking, de-lamination, surface breaking

v.s. last bursting

2) Resin sheet … isotropic, plastic unstable-

sticky, fragile for cracking

3) Aluminum sheet … isotropic, plastic

Necking at the lower layer, wedge

indentation at the surface layer

Structure of laminated plies in case of

paperboard

Observation of surface breaking at near the

first peak load fC1

Cutt

ing d

irec

tion

(2) Previous

stage of fC1

(1) Post stage

of fC1

Sectional view of white-coated paperboard

Stroke, x

Lin

e fo

rce,

f fc1

fc2

(1)(2)

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1Indentation depth d/t

Lin

e fo

rce

f

k

N/m

C,w=91m

N,w=91m C,w=51m

N,w=51m

C,w=12m

N,w=12m

n = 350 g/m2

V = 1 mm/min

C:coated,N:non

Direction: CD

fC1

fC2

[ ]

Lower crosshead

Upper crosshead

Blade

Load cell

Holder

Paperboard

Counter plate

V : Feed velocity

Cross direction

b

wDetail of blade tip

b

wDetail of blade tip

White-coated paperboard

(recycled paper)

d1/t =50%

d2/t =85%

Pushed cutting test

Need

theoretical

estimation

models

1) Paperboard … orthotropic, laminated,

quasi-plastic (non-metallic)

No necking, de-lamination, surface

breaking v.s. last bursting

2) Resin sheet … isotropic, plastic

Necking at the lower layer, wedge

indentation at the surface layer

Video demonstration

White-coated paperboard (0.42mm),

A 42 deg. center bevel blade

Resin sheet [PC] (0.5mm) mounted on underlays,

A 42 deg. center bevel blade

Elementary analysis of the second stage by the slab method

The contact surface generates the in-plane tensile force and the out-of-plane upsetting force.

The wedge friction and the counter plate friction are considered.

From the principle of Saint-Venant, the projected area with the contact surface is only considered as a free body which is used for the equilibrium of the cutting forces.

The equilibrium of applied forces are locally satisfied with the contact surface zone

if there is not any external loads else on the body.

’p’c (t-d)s

p’c

pb

pbd

tc

pbcos(/2) pbsin(/2)

pbpbcos (/2)

pbsin(/2)pb

t-d

Wedge

Counter plate

Sheet material

Line force

f

A trapezoidal slab element which is subjected to a line force

p: the pressure on the contact surface of blade

p’: the pressure on the counter plate

s : the separation stress

1- tan(/2) s = ｛ ―――――― -' ｝ ―――――

tan(/2) + t-d )

- f = -'

t

s = l(,,') f / (t-a)

’p’c (t-d)s

p’c

pb

pb

= (,,') f / (t-d)

Converting efficiency of wedge separation force

2(

s L

Relationship between the in-

plane tensile stress s and the

line force f

…(A)

…(B)

t-d = L(,,') f / s,

smaller as the line force f is smaller, while it is smaller as the

friction coeff. of counter plate and the burst strength s are higher.

Since the residual height of work sheet is

1- tan(/2) s = ｛ ―――――― -' ｝ ―――――

tan(/2) + t-d )

- f = -'

t2(…(B)

L(,,’)

Final burst condition is estimated…

f max∝ apex angle “t-d“ is small for a keen angle

A high tensile strength sheet has a small “t-d“ .

-0.5

0

0.5

1

1.5

2

0 40 80 120 160

Tip angle [°]

Fac

tor

L(

,,

')

=0.0,'=0.0

=0.45,'=0.35

=0.45,'=0.0

=0.15,'=0.0

=0.0,'=0.35

楔分力の変換効率に及ぼす角度と摩擦の影響

Effect of apex angle and friction on efficiency of

wedge separation force

L=40%

L=120%

TIP 400HV, Nc=5rpm

TIP 400HV, Nc=5rpm

Experimental response of cutting resistance

Material: a white-coated paperboard, =350 g/m2

Second peak, final burst

dC2/t = 85%~ fC2= 25 kN/m

~

t = 0.424 mm

d

= 0.45 '=0.35

ThicknessFriction coeff.

blade

c/p

=42°

From Eq.(A), we get

s = 0.642081x25

0.848x(1-0.85)

= 126.195 = 3.15sB

sB = 38MPa

In-plane tensilestrength (MD)

sY =26MPa

=4.85sY

sB≒40 MPa ,

= 0.45, '=0.35,

Half angle: /2 = 21°, Thickness of work: t = 0.424 mm,

Burst point: d / t = 0.85, f = 25 kN/m

s /sY ≒4.85

Case study, experimental result

Material: a white-coated paperboard, =350 g/m2

Burst strength (stress) is very high !

In-plane tensile strength:

Friction coeff. (wedge-sheet):

Friction coeff. (c/p – sheet):

Plastic restriction by distortion flow around the notched zone

must be considered. Why is this stress too high ?

p/2

d2

t

YYkkkk ssp

s 96.23

2571.22

2120 =

=

==

Tensile deformation with a sharp notch:

Orowan’s solution

By using the sliding

field theory which

will be explained in

the next lecture, this

restriction resistance

is estimated as 2.96sY.

sY≒26 MPa ,

= 0.45, '=0.35,

Half angle: /2 = 21°, Thickness of work: t = 0.424 mm,

Burst point: d / t = 0.85, f = 25 kN/m

s /sY ≒4.85

Case study, experimental result

Material: a white-coated paperboard, =350 g/m2

Burst strength (stress) is very high !

In-plane tensile strength:

Friction coeff. (wedge-sheet):

Friction coeff. (c/p – sheet):

Plastic restriction by distortion flow around the notched zone

must be considered.

By replacing sY to 2.96 sY, the ratio of burst stress and

the yield strength becomes s / (2.96sY) = 1.64

sB=38MPa

Since sB/sY=1.461, s/(2.96sB) = 1.12 Final burst is estimated from the tensile strength

From a slip-line theory model…

The paperboard has…

Important and useful results

• Final burst stress s at the second peak point

( fC2 , dC2) is estimated from the tensile

strength sB : s = 3sB

• Experimental paperboard cutting says that

dC2/t = 85%, dC1/t=50~60%, while the Slip-line

theory (frictionless) shows that dC1/t =64%,

dC2/t =90%.

~

These information were derived from the slab method, and the experimental observation.

Today’s home work,1

1- tan(/2) s = ｛ ―――――― -' ｝ ―――――

tan(/2) + t-d )

- f = -'

t2(

When we consider the specified trapezoidal free body,

verify that this equation is correct. Consider the

equilibrium of all the forces in the horizontal direction.

…(B)

Today’s home work,2

’p’c’(t-d)s

p’c’

pb

pb

c’

c

When we consider a new free

body as shown here, derive the

relationship between the

separation stress s and the line

force f.

Brief answer

The result is the same as the

case of the trapezoidal free

body.

Assumed to

be a free

surfacenc > c’ > c, n>1