A Novel Method to Measure Poisson's Ratio of Thin Films
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0
-
• 0
strain•Plane
Euler buckling 10-
h10X
2000;
-Y (v02)P
00V
P
SEuler
-600
-400 -200 Reduced prestress S0
-4nr
0
200
Fig. 1 Reduced center deflection of an infinitely long plate as a function of the reduced prestress &o and pressure P. In the hatched area the plate is in a state of plane strain. The translationalinvariance is broken at the v-dependent ripple transition line, where longitudinal buckling occurs.
"3S
< Srippie(V) : When P is lowered to a critical pressure Pcr, the plane-strain deflection of the plate becomes unstable at a critical deflection OcrO. A longitudinal ripple with reduced critical wavelength ýcr appears. The physical wavelength Xcr is related to ýcr through ,cr = )icr/a. Both Pcr and Xcr are functions of 30 and v. The ripple transition can be understood as follows. Under the load P, the stress components ax and are increased from the prestress (O, due to the elongation of the structure in the y-direction. In the plane strain region P > Pcr' the stresses are related by (Txx(P) - (TO= v{Uyy(P) -a 0o (1) At Pcr the longitudinal stress .xxis sufficiently compressive for the plate to become unstable in the longitudinal direction. At 1P1< Pcr the ripple buckling profile develops. According to Eq. 1 the ripple transition depends on v. Measurement of Pcr, wcrO, or Xcr for a long plate with known dimensions a and h and material 0Yand D therefore allows to determine its Poisson's ratio v. An example of a ripple transition is shown in Fig. 2. It shows measured longitudinal deflection profiles of a sample under various pressures. At the highest pressure the ripple has disappeared. In Fig. 1 the corre-
sponding 10 lies in the plane strain region above the ripple transition line. At a critical pressure Pcr= 20.6 kPa the ripple transition line is crossed and a sinusoidal ripple with amplitude A occurs. As the load is reduced A increases.
THEORY In the plane strain region the profile of the plate is independent of x and given by [5,6] Wps(Y) = "Pa4(fI _ (X)2}4 I cosh((/2)-cosh((y/a)'S,1 w~()=2DS1J4
a
42J
sinh(,fS=/2)
(2)
)(2
where the reduced stress S = ayyha2/D is defined by [5] + 16
2 (O) 2( 3S- 9,SsinhTS= + (48 +_23)(sinh(,f//2)) ) 2 (4J.•sinh (,S/2) - 8(sinh(•S/4)) )2
28
(3)
f/,
-••
30 •f.• 30
M
-21.5 kPa -20.6 kPa -20.3 kPa 20.0 kPa 19.4 kPa 18.5 kPa
7A,
-15.5 kPa
.
20
11.0 kPa
II"
SIII
J6.4
-
kPa
10
0 -2000
-4000
2000
0
4000
x [PM] Fig. 2 Longitudinalprofiles of a buckled long plate as a function of differential pressure.The aspect ratio b:a is 10:1. The plate becomes unstable at Pcr = 20.6 kPa and buckles longitudinallyfor P wcrO, U is minimal at q = 0. This means that the Hessian matrix M(a,h,%,SO,wpsO,v) with elements h91 v) = 1na 'o "po =0' Mmn(a, h,L, 30~, WpO, V) = 1
~
Uelt(a, h,
(9)
30, D,wpsO v, q)
is positive definite for all X, i.e., Det(M) > 0. At wpso = Wco, the plate becomes unstable for a specific Xcr M is then singular at X'cr but positive definite for all other X, and therefore Det(M(a,h, XLcr,30, WcrO, v)) = •Det(M(a,h, X,30,
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