the mechanics of crumpled thin sheets simon de villiers, pgp, university of oslo

The Mechanics of Crumpled Thin Sheets

Simon de Villiers , PGP, University of Oslo

Why Crumpled Paper?

• Ubiquitous Phenomena

Why Crumpled Paper?

• Ubiquitous Phenomena • Power law statistics

Why Crumpled Paper?

• Ubiquitous Phenomena • Power law statistics• Self affine geometry

Why Crumpled Paper?

• Ubiquitous Phenomena • Power law statistics• Self affine geometry• Contact area of rough surfaces

Why Crumpled Paper?

• Ubiquitous Phenomena • Power law statistics• Self affine geometry• Contact area of rough surfaces• Convenient in the laboratory

Balance

Crumpled film

Heavy stone table

Computer controlled micrometer screw gauge

Perspex plate

Experimental Setup

Stress Relaxation

0 2000 4000 6000 8000 10000 12000 14000 160000

5

10

15

20

25

30

35

40

45Stress relaxation when 16cm diameter cirle of crumpled Aluminum foil was confined to 3mm

Bal

ance

rea

ding

in g

ram

s(st

ress

)

time in seconds

Stress Relaxation

0 2000 4000 6000 8000 10000 12000 14000 160000

5

10

15

20

25

30

35

40


Bal

ance

rea

ding

in g

ram

s(st

ress

)

time in seconds10

010

110

210

310

410

50

5

10

15

20

25

30

35

40


Bal

ance

rea

ding

in g

ram

s(st

ress

)

time in seconds

Stress Strain Relationship

0.5 1 1.5 2

x 104

0

500

1000

1500

2000

2500

3000

16cm circle of crumpled cellophane compressed 0 .1mm every minute

Time in seconds (incremental strain increase of 0.1mm per minute)

Bal

ance

read

ing

in g

ram

s(s

tress

)

Stress Strain Relationship

0.5 1 1.5 2

x 104

0

500

1000

1500

2000

2500

3000


0.5 1 1.5 2

x 104

10-2

10-1

100

101

102

103




Bal

ance

read

ing

in g

ram

s(s

tress

)

Regularly Folded Paper

5 10 15 20 25 300

100

200

300

400

500

600

700

800

time in seconds (constant strain rate 414µm/s)

Bal

ance

rea

ding

in g

ram

s(st

ress

)

Graph of elastic response with 1.5cm folded sheet against the empty response of the apparatus

1.5cm folded paperno paper

Single Point Forcing Response

500 520 540 560 580 600 620 640 660

10

20

30

40

50

60

70

time in seconds (constant strain rate applied at a single point)

Bal

ance

rea

ding

in g

ram

s

Crumpled sheet pushed at a single point on its top surface

3D Imaging

Summary

• The importance of crumpled disorder in thin sheets.

• Logarithmic relaxation of stress.• Exponential increase of stress with strain.• Linear increase of stress with strain for

ordered/simple systems.

Geological Relevance

• Frictional processes

• Thin elastic sheet dynamics in geological systems

• Rough surfaces

Friction in the Experiment

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 139131719

Amplitude normalized Graph of sizing with number of ridges being varied

norm

aliz

ed s

tres

s

time(s)

0 100 200 300 400 500 600 700 8000

200

400

600

800

1000

1200Positive and negative strainrate averaged Response

Bal

ance

rea

ding

in g

ram

s (s

tres

s)

Time in seconds (constant strain rate 41µm/s)

0 200 400 600 800 1000 1200 1400 16000

200

400

600

800

1000

1200


Bal

ance

rea

ding

in g

ram

s (s

tres

s)

Compression and decompression of a regularly folded sheet

negative strain ratepositive strain rate

Repetition of the Experiment

0 10 20 30 40 50 60 700

500

1000

1500

2000

2500

3000

3500

4000Graph of 8 compressions with a 10 second interval


Bal

ance

rea

ding

in g

ram

s (s

tres

s)

0 5 10 15 20 25 30 350

500

1000

1500

2000

2500

3000

3500

4000

time in seconds (constant strain rate 414µm/s)

Bal

ance

rea

ding

in g

ram

s(st

ress

)

8 crumpled pieces of paper (15cm side square)

the mechanics of crumpled thin sheets simon de villiers, pgp, university of oslo

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