Eigenstresses in Anisotropic Films
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EIGENSTRESSES IN ANISOTROPIC FILMS FERDINANDO AURICCHIO AND MAURO FERRARI Department of Civil Engineering, University of California, Berkeley, CA. 94710; * also Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA. 94710. ABSTRACT A closed-form solution for a macroscopically homogeneous, fully anisotropic layer subject to non-uniform through-thickness eigenstrain is presented, and employed in determining the three-dimensional deformation and stress states of a thermally loaded ceramic film with microstructure-induced macroscopic anisotropy. The resultant stress field is compared with those that could be deduced by experimental determination of the curvature and the classical structural theories. INTRODUCTION Anisotropic films appear with increasing frequency in the current technology, in areas ranging from magnetic recording devices to thermochemical coatings, micromechanical systems and coated-fiber composites. The anisotropy to be considered in the structural analysis may be either the result of bulk material properties (single crystal structures) or be of macroscopic nature, resulting from a preferred orientation of the constituents grains or of the defect distribution. By design or by the nature of their service environments, such films are commonly subjected to non-mechanical forcings, including thermal, hygroscopic and transformation loads. For the stresses resulting from these actions, the collective term 'eigenstresses' is used, as the corresponding boundary value problems are formally identical (see next section). In this work, an analytical solution for the problem of a fully anisotropic, homogeneous layer subject to a non-uniform through-thickness eigenstrain field is used to compute the stresses and deformations generated in a ceramic layer during post-deposition cooldown. The anisotropy is here induced by the texture of the intergranular voids. The exact stress state is compared with those that could be deduced by curvature measurements, in conjunction with classic beam, plate and anisotropic plate theories. THE THERMOELASTIC LAYER Let the layer -h
:
x 3• h,
subject to a through-thickness E
with lateral boundary F(x ,x
)
0,
be
thermal strain field
=o 6T(x 3)
where a and 6T denote the thermal
(1) expansion tensor and the temperature
variation fields, respectively. Let the constituent material be macroscopically homogeneous but arbitrarily anisotropic, and the layer be subjected to no external mechanical load. Assuming plane stress and independence of the membrane stresses ra from the in-plane coordinates, equilibrium is automatically satisfied, compatibility reduce to the three equations
and
the
conditions
Mat. Res. Soc. Symp. Proc. Vol. 239. ©1992 Materials Research Society
of
246
S 00 Here,
S ay_
are
6T 33 = 0
,, 3'af +
the components
indices take the values 1 and 2.
of
the
(2)
compliance
Solving (2)
tensor,
for the T4,33'_
and
greek
integrating
twice in x3, and imposing that the resultant forces and moments through any cr
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