Elastic, Transversely Isotropic Film Indented by a Punch; Application to Multilayered Thin Films
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Substrate
MODEL Film
The present analysis limits itself to squarez=-h shaped indenters (shape-effects are weak [2-6]; L see reference [4] for extension to 3-sided indents). Analysis proceeds through three steps: (i) diss placements from traction a, =-a. cosarcosby FIGURE I acting on the surface are calculated; (ii) displacements beneath a uniformly-loaded, squareshaped area are obtained by using Fourier transforms; and (iii) these displacements are averaged. Results from this approach differ by a few percent from those obtained using a rigid punch assumption [3]. Transversely isotropic solids have 5 independent elastic constants: C11, C33, C44, C13 and C12 [7]. Here, the x3-direction corresponds to the axis of symmetry (z-axis, Figure 1). C4, is the out-of-plane shear modulus. Results of the present analysis do not depend on C12.
687
Mat. Res. Soc. Symp. Proc. Vol. 356 01995 Materials Research Society
To proceed with step (i) above, consider strain functions V•, based on which displacements U in the transversely isotropic film may be determined as:
Here, CC is a constant and +P Z-y O(2)
ax
az
ay
P
P is another constant. a and are chosen as C,1 /(C,3 +C44) and (C,, - C4) / (C,3 + C.), respectively, so that the equations of equilibrium simplify to a single
where
equation entailing even powers of the partial derivatives:
{Lc+~~J~c+][jy +~>C ~5
4
}V2....=0 (3)
-ý +
(C13 +CI 4,
z
We focus our interest on two strain functions satisfying Equation 3. The first is: V, ={Ae -'I
+ Be+'lIZ' + Ce-r-UZ + De+fl-UZ
}cosaxcosby
(4a)
where
SC.42 +1C13-(1 1 C44 )2
+ -V[C442
+1-C11C33(1
+-C44)2 ]2
-C(C, 3(C)3
-C442C,,I
-4C
CC
(4b)
33
and u=(a 2 -b 2 )1/2 .
(4c)
This strain function is used when the quantities 1± are real.. Otherwise: V= {eRxuz [A'cos(RYuz) + B'sin (Ryuz)] +e-Ruz [C'cos(Ryuz) + D' sin(Ryuz)]}cosaxcosby .... (5a) is used, where R, = Rcos(4/2), Ry = Rsin( /2), andR= 4VC,,/C and
688
3
(5b)
2 2 2 C•C332 -[C4 2 +CIC33 -(C, 3 +C ) ] ]4Can [C44 +Cl1C 3 -(C +C44) 2 3 ] 13
We approximate the elastic properties of the substrate as being isotropic (to learn about the indentation response of an elastically anisotropic half-space, see work by Vlassak and Nix [8]). A Love's strain function [9] is used for the substrate: (6)
V = (F + Gvz) cosaxcosby e-uz,
such that the displacement field inside the substrate is given by
U = 2(l- v)iV2V1 - (aav.
(7)
Here, v is the Poisson's ratio of the substrate. Next, the constants A, B, C, D (or A , B', C' D) F, and G are evaluated subject to boundary conditions: o= = -(o cosaxcosby (Y = 0 a,, (film) = a, (substrate) ay, (film) = o, (substrate) U, (film) = U, (substrate)
at z = -h. at z = -h.
U,(film) = U,(substrate)
(8a) (8b)
at z = 0.
(8c)
atz = 0. at z = 0.
(8d) (8e)
atz = 0.
(8f)
Stages (ii) and (iii) of the calculation are detailed in reference [4], resulting in a compliance:
4= -f/
4
sin (pcosO)sinm(psinO)G(2h)dýpd
The function G(x) differs depending upon whether or not
-C, ,,4C_(Ae
=C C,193 ) -=
+x+Be-l l
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