Geometrically Nonlinear Stress-Deflection Relations for Thin Film/Substrate Systems With a Finite Element Comparison

1994 ◽  
Vol 61 (4) ◽  
pp. 872-878 ◽  
Author(s):  
C. B. Masters ◽  
N. J. Salamon
Keyword(s):  
Thin Film ◽  
Finite Element ◽  
Ritz Method ◽  
Spherical Shape ◽  
Free Edge ◽  
Boundary Region ◽  
Film Stress ◽  
Geometrically Nonlinear ◽  
Nonlinear Stress ◽  
Film Substrate

A new higher order geometrically nonlinear relation is developed to relate the deflection of a thin film /substrate system to the intrinsic film stress when these deflections are larger than the thickness of the substrate. Using the Rayleigh-Ritz method, these nonlinear relations are developed by approximating the out-of-plane deflections by a second-order polynomial and midplane normal strains by sixthorder polynomials. Several plate deflection configurations arise in an isotropic system: at very low intrinsic film stresses, a single, stable, spherical plate configuration is predicted; as the intrinsic film stress increases, the solution bifurcates into one unstable spherical shape and two stable ellipsoidal shapes; in the limit as the intrinsic film stress approaches infinity, the ellipsoidal configurations develop into cylindrical plate curvatures about either one of the two axes. Curvatures predicted by this new relation are significantly more accurate than previous theories when compared to curvatures calculated from three-dimensional nonlinear finite element deflection results. Furthermore, the finite element results display significant transverse stresses in a small boundary region near the free edge.

Practical Application of a Geometrically Nonlinear Stress-Curvature Relation

1991 ◽  
Vol 239 ◽  
Author(s):  
Christine B. Masters ◽  
N. J. Salamon
Keyword(s):  
Strain Energy ◽  
Film Stress ◽  
Silicon Substrates ◽  
Substrate Thickness ◽  
Geometrically Nonlinear ◽  
Practical Application ◽  
Total Strain Energy ◽  
Initial Film ◽  
Nonlinear Stress ◽  
Film Substrate

ABSTRACTA recently developed geometrically nonlinear stress-curvature relation based on a minimization of the total strain energy, which predicts a bifurcation in shape as the magnitude of intrinsic film stress increases, is discussed in this paper. It is compared with the linear theories of Stoney and Brenner & Senderoff for a thin molybdenum film on silicon substrates with various thicknesses. Although the ratio of film to substrate elastic modulus is only 2, Stoney's equation generates significant error for this film/substrate system and the Brenner & Senderoff relation should be used for calculating initial film stress when plate deflections are small. When deflections exceed approximately half the substrate thickness the Brenner & Senderoff equation produces over 10% error and consequently, the nonlinear stress-deflection relation should be used to relate plate curvatures to initial film stress.

The effect of material anisotropy on the mechanics of a thin-film/substrate system under mechanical and thermal loads

2021 ◽  
pp. 108128652110312
Author(s):  
E. Nart ◽  
Y. Alinia ◽  
M. A. Güler
Keyword(s):  
Thin Film ◽  
Finite Element ◽  
Thermal Stresses ◽  
Stress Singularity ◽  
Film Stress ◽  
The Finite Element Method ◽  
Leading Role ◽  
Principal Directions ◽  
Soft Thin Film ◽  
Film Substrate

In this study, the stress analysis for an orthotropic thin film bonded to an orthotropic elastic substrate is addressed using both the analytical and finite element methods. The analytical method employs the integrodifferential formulation with the aid of membrane assumption. Utilizing the finite element method, the effect of orientation of the material principal directions are studied. The loading scenarios include a temperature gradient imposed on the film and a far-field uniaxial tension on the substrate. The results of current study indicate that the ratio of the film to the substrate stiffness plays a leading role in the film stress distribution. For the mechanical loading applied to the substrate, a soft thin film attached to a relatively stiffer substrate is preferred. The film can tolerate the induced thermal stresses as it is bonded to a softer host structure. The rotation angle of material orthotropy directions significantly affects the stress singularity near the film edges up to a certain extent.

Stresses in a Multilayer Thin Film/Substrate System Subjected to Nonuniform Temperature

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
X. Feng ◽  
Y. Huang ◽  
A. J. Rosakis
Keyword(s):  
Thin Film ◽  
Field Measurements ◽  
Film Stress ◽  
Effect Of Temperature ◽  
Multilayer Thin Film ◽  
Nonuniform Temperature ◽  
Curvature Invariant ◽  
Full Field ◽  
Curvature Measurements ◽  
Film Substrate

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to uniform film stress and system curvature states over the entire system of a single thin film on a substrate. By considering a circular multilayer thin film/substrate system subjected to nonuniform temperature distributions, we derive relations between the stresses in each film and temperature, and between the system curvatures and temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part, which reflects the effect of temperature of other points on the location of scrutiny. We also derive relations between the film stresses in each film and the system curvatures, which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. The interfacial shear tractions between the films and between the film and substrate are proportional to the gradient of the first curvature invariant, and can also be inferred experimentally.

scholarly journals The Effect of Stress on the Nanomechanical Properties of Au Surfaces

1996 ◽  
Vol 440 ◽  
Author(s):  
J. E. Houston
Keyword(s):  
Thin Film ◽  
Residual Stress ◽  
Local Level ◽  
Maximum Shear Stress ◽  
Critical Role ◽  
Film Stress ◽  
Indentation Modulus ◽  
Nanomechanical Properties ◽  
Average Grain Size ◽  
Film Substrate

AbstractStress in thin films plays a critical role in many technologically important areas. The role is a beneficial one in strained layer superlattices where semiconductor electrical and optical properties can be tailored with film stress. On the negative side, residual stress in thin-film interconnects in microelectronics can lead to cracking and delamination. In spite of their importance, however, surface and thin-film stresses are difficult to measure and control, especially on a local level. In recent studies, we used the Interfacial Force Microscope (IFM) in a nanoindenter mode to survey the nanomechanical properties of Au films grown on various substrates. Quantitative tabulations of the indentation modulus and the maximum shear stress at the plastic threshold showed consistent values over individual samples but a wide variation from substrate to substrate. These values were compared with film properties such as surface roughness, average grain size and interfacial adhesion and no correlation was found. However, in a subsequent analysis of the results, we found consistencies which support the integrity of the data and point to the fact that the results are sensitive to some property of the various film/substrate combinations. In recent measurements on two of the original substrate materials we found a direct correlation between the nanomechanical values and the residual stress in the films, as measured globally by a wafer warping technique. In the present paper, we review these earlier results and show recent measurements dealing with stresses externally applied to the films which supports our earlier conclusion concerning the role of stress on our measurements. In addition, we present very recent results concerning morphological effects on nanomechanical properties which add additional support to the suggestion that near-threshold indentation holds promise of being able to measure stress on a very local level

Extension of Stoney’s Formula to Arbitrary Temperature Distributions in Thin Film/Substrate Systems

2006 ◽  
Vol 74 (6) ◽  
pp. 1225-1233 ◽  
Author(s):  
Y. Huang ◽  
A. J. Rosakis
Keyword(s):  
Thin Film ◽  
Field Measurements ◽  
Film Stress ◽  
Effect Of Temperature ◽  
Curvature Invariant ◽  
Full Field ◽  
Polar Components ◽  
Temperature Distributions ◽  
Curvature Measurements ◽  
Film Substrate

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to nonuniform and nonaxisymmetric temperature distributions, we derive relations between the film stresses and temperature, and between the plate system’s curvatures and the temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part that reflects the effect of temperature of other points on the location of scrutiny. Most notably, we also derive relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. Finally, it is shown that the interfacial shear tractions between the film and the substrate are related to the gradients of the first curvature invariant and can also be inferred experimentally.

A study of the mechanics of microindentation using finite elements

1992 ◽  
Vol 7 (3) ◽  
pp. 618-626 ◽  
Author(s):  
T.A. Laursen ◽  
J.C. Simo
Keyword(s):  
Thin Film ◽  
Finite Element ◽  
Contact Area ◽  
Contact Stiffness ◽  
Surface Profile ◽  
Critical Examination ◽  
Actual Contact ◽  
Actual Contact Area ◽  
Straight Line ◽  
Film Substrate

In this paper the finite element method is used to explore the mechanics of the microindentation process. In the simulations discussed, aluminum and silicon are investigated both in their bulk forms and in thin film-substrate combinations. Among the quantities readily computed using this approach and given in this paper are hardness (computed using actual contact area), contact stiffness, effective composite modulus, and surface profile under load. Importantly, this investigation builds on previous work by providing a more critical examination of the amount of pileup (or sink-in) around the indenter in the fully loaded configuration, as well as the variation of the actual contact area during indenter withdrawal. A key conclusion of this study is that finite element simulations do not support the widely used assumption of constancy of area during unloading (for either bulk materials or thin film systems). Furthermore, the amount of pileup or sink-in can be appreciable. The implication of these findings is that for many situations the commonly used straight-line extrapolation of a plastic depth may render an estimate for the contact area that is quite distinct from the actual area. This assertion is demonstrated herein through comparison of hardnesses calculated using actual contact area with values calculated using the straight-line extrapolation of plastic depth.

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