1 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für...

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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik

Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller

Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2010

by

W.H. Müller1), T. Hauck2)

STAMM 2010

Berlin

August 30 – September 2, 2010

Nonlinear Buckling Analysis

of Vertical Wafer Probe Technology

1) 1) Technische Universität BerlinTechnische Universität Berlin Institut für Mechanik - LKM Institut für Mechanik - LKM Einsteinufer 5Einsteinufer 5 D-10587 BerlinD-10587 Berlin

2) 2) Freescale HalbleiterFreescale HalbleiterDeutschland GmbHDeutschland GmbHSchatzbogen 7Schatzbogen 781829 München81829 München

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Outline

Introduction and motivation: Buckling beams and vertical probe card technology

Theoretical approach to non-linear buckling

Finite element approach to non-linear buckling

Comparison with an experiment using a macroscopic beam structure

Prediction and comparison of vertical buckling beam technology experiments

Conclusions

3




Outline






Conclusions

4




Buckling beams and vertical probe card technology I

Objective: Testing the electrical connectivity of dies with buckling needles

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Buckling beams and vertical probe card technology II

• Pads at the periphery of the chip, which will later be wire-bonded. • Each pad is probed by a needle in parallel and at the same time.• It is possible to probe several chips simultaneously or even all chips on a wafer.

probe needles:

pads:

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Buckling beams and vertical probe card technology IIIPrinciple: Dies with pads are pressed against needles in tool

Needles start buckling.

Buckling beam technology guarantees consistent contact pressure

on every point tested, independent of travel.

Optimal tolerance even under changing planarity conditions.

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Buckling beams and vertical probe card technology IV

With applied force: After release of applied force:

Conclusion: This is a completely reversible process, buckling eqns. apply

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Outline






Conclusions

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Theoretical approach to non-linear buckling I

Recall the 4 Euler Cases

Objective: Exact calculation of the displacements w(s) and x(s)

→ non-linear problem

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Geometry non-lin. DE for bending

Theoretical approach to non-linear buckling II

Free-body-diagram

Note: only for Euler case 30F~

sxlxFsFwsM ~

Equilibrium of moments

sins

w

d

d coss

x

d

d

sin

~cos ddd

d

d

d

d

2

1 222

kksss

02 EI

Fk

02 EI

Fk

~~

Differentiation and rearrangement

• Procedure following Timoshenko and Gere “Theory of elastic stability”:

sMs

EI d

d

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Case 4: or (two points of inflection)

Case 3: (pin joint)

(free end) Case 2: (pin joints, symmetry)

Theoretical approach to non-linear buckling III

• Integration for all Euler cases

• Constant of integration, , from boundary conditions

0lM 00 M

0lM

0250 lM . 075.0 lM

0lM

Ckks

kksss

sin~

cos

sin~

cos

222

222

d

d

2

1

dddd

d

d

d

2

1

Case 1:

C

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• Critical load

Theoretical approach to non-linear buckling IV

• Results for Euler case 1 (cf., Timoshenko & Gere;

Theory of Elastic Stability)

0

0

0

02

22

20 00 sinsin

d

2

1

coscos

d

2

1d

kk

sll

s

,sin 200p

20

1 sinsinp

,

sin k

pK

pkl 0

2π

022

01

d1

02 EI

Fk

2

02

c00c π

4

2

π00

pK

F

FpKFF

This implicit relation

allows to compute

the load F required to

achieve an angle 0

for a given length l:

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• Critical load: with: and:

Theoretical approach to non-linear buckling V

• Results for Euler case 1 cont. (hold also for cases 2 and 3)

2

02

c π

4 pK

F

F

2

2

c

π

lK

EIF

i

5012 .,,iK

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• Vertical displacement

Theoretical approach to non-linear buckling VI

• Results for Euler case 1 cont.

cossin 1

2

d

d

0

0

pK

p

l

sw

s

w

• Horizontal displacement

00

0

21

d

dpFpE

pKl

sx

s

x,,cos

200sinp 2

0

1 sinsinp

• Analogous results for Euler cases 2 and 4 due to self-similarity with case 1

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Theoretical approach to non-linear buckling VII

• Euler case 1

• Euler case 2

• Euler case 4

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Theoretical approach to non-linear buckling VIII

• Euler case 3; (i) relation for total length of the beam

1

1

0

0

0

0 02

00 sinsincoscos

d

2

1d

ksl

l

s

FFkk~~ 222

0sinsincoscos

dsin1

1

0

0

0

0 02

0

• (ii) Vertical displacement at pinned end vanishes

• (iii) Moment vanishes at (unknown position of) point of inflection

1

12

sxlx

sw

1

0 02

0

1

dsin

2

1

sinsincoscoskswwith

0

0 02

0

1

1

0

sinsincoscos

dcos

2

1

ksxlx

These three relations allow us to compute (numerically) for a given horizontal push F

(i) the vertical force and

(ii) the angle of deflection at the hinge, 0, and

(iii) the angle of deflection at the inflection point, 1,

for a given beam length l.

In addition the position s1 of the point of inflection

can be obtained.

F~

1s

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Theoretical approach to non-linear buckling IX

• Euler case 3

Deformation pattern:

1s

Current position of point of inflection:

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Outline






Conclusions

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Finite element approach to non-linear buckling (Example)

Buckling Beam Column

• ANSYS finite element code, uniaxial beam elements with consistent tangent stiffness matrix, large deformation option NLGEOM,ON

• Initial displacement with a small perturbation :

• 100 elements "3-D elastic beam" = 2 nodal elements für bending, tension-compression, and torsion, each node with 6DOFs

• A total of 101 nodes and 606 DOFs• Application of displacement in x-direction in 2000 loading steps

Dimensionslength = 520 mmthickness = 0.7 mmwidth = 20 mmYoung’s modulusE = 210 GPa

build-in

pinned

Finitial displacement

F´

ux

6990

π

6990

π0 .

sin.

sin)(l

xl

l

xlxu

F~

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Outline






Conclusions

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Comparison with macroscopic experiment I

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-10

-5

0

5

10

15

20

-200 0 200 400 600

x [mm]

F, F

' [N

]

F F' Experiment

L = 520 mmH = 0.7 mmB = 20 mmE = 210 GPa

Comparison with macroscopic experiment II

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Outline






Conclusions

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Comparison with beam technology experiments I

Force vs. Overtravel

0

20

40

60

80

100

120

0 50 100 150 200

overtravel in µm

forc

e in

mN

2.5 mil beam 2.0 mil beam 3.0 mil beam

beam dimensions: d = 3 and d=2.5 mil, l = 7.8 mm , d = 2 mil, l = 5.33 mm

Note: These are measurements averaged w.r.t. many needles; co-planarity issues, steady increase of deflection, not an abrupt one

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Comparison with beam technology experiments II

Probe force prediction (treated as Euler case 4)

0

20

40

60

80

100

120

140

0 50 100 150 200

Overtravel [µm]

Fo

rce

[m

N]

7.8mm, 3mil

7.8mm, 2.5mil

5.33mm, 2.0mil

Needle L, D

22

c π4l

EIF

118mN

57 mN

50 mN

E = 110000 N/mm²

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The buckling of micrometer size beams used in VI-probe card technology can be treated in closed-form using Timoshenko’s non-linear deflection approach.

All four Euler cases have been analyzed based on non-linear buckling theory.

The results for the Euler cases 1, 2, and 4 are essentially the same due to their self-similarity.

Euler case 3 (one pinned and one clamped end) needs a special treatment.

The Euler length (position of the point of inflection) in case 3 changes from 0.699 l to slightly higher values as the horizontal push increases.

The closed-form solution agrees well with FE results that take large deformation and compressibility of the beam into account.

FE and closed form solutions can both be used to predíct the experimentally observed deformation pattern both for macroscopic as well as microscopic bucking beams

Conclusions

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