George C. Johnson

A discussion of stress rates in finite deformation problems

Abstract It has recently been shown that the finite elastic-plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed, while the solution for isotropic hardening gives a monotonically increasing stress. This paper analyzes this response on the basis of continuum mechanical descriptions of the problem. Three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress. In light of the equations for the evolution of the yield surface, the hypoelastic solution to the simple shear problem for each of the three stress rates is presented. It is shown that the use of the Jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution. An alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress.

Post-CMOS integration of germanium microstructures

Polycrystalline germanium (poly-Ge) microstructures have been fabricated on standard CMOS wafers. Conventional low pressure chemical vapor deposition (LPCVD) and rapid thermal annealing (RTA) processes were used to achieve low-resistivity (2.3 m/spl Omega/-cm) tensile poly-Ge structural films, with a thermal budget which is compatible with Al (2% Si) metallization. The CMOS circuitry was passivated with low-temperature oxide and amorphous Si; the latter served as a mask against HF during the microstructure release etch. Comb-drive microresonators with integrated trans-resistance amplifiers were used to demonstrate feasibility of this integration strategy. Preliminary measurements on test structures indicate that poly-Ge has promising material properties. Its fracture strength is 2.2 GPa +/- 0.4 GPa, which is comparable to that of poly-Si. Clamped-clamped lateral resonator test structures have quality factors in vacuum as high as /spl sim/30,000. For the process conditions used in this work, the residual stress of as-deposited poly-Ge is -79 MPa (compressive); RTA shifts the stress to 203 MPa (tensile). Deposition and annealing conditions have yet to be optimized to minimize stress.

Micromechanical structures for thin film characterization

Micromechanical structures designed for material characterization through analysis of their lateral vibrations are reported. The structures are made of doped LPCVD (low-pressure chemical vapor deposited) polycrystalline silicon and consist of beams supporting a rigid mass which is used to drive as well as sense the oscillatory motion. Amplitude and phase responses with respect to frequency are measured by an amplitude modulation technique and are used with the corresponding mechanical model to estimate various electromechanical properties. Youngs modulus for this material is estimated to be 174+or-10 GPa, with the film supporting a tensile residual stress of 10+or-2 MPa. It is shown that some structures exhibit significant nonlinearities for moderate lateral displacements. The observed response curves are well modeled by Duffings equation for a stiffening spring.<<ETX>>

Effective elasticities of short-fiber composites with arbitrary orientation distribution

Abstract The object of this study is the derivation of the effective elasticities of short-fiber composites with an arbitrarily specified orientation distribution of ellipsoidal fibers. The developed formalism is sufficiently general to include the cases of arbitrary material symmetry of both the fiber and the matrix. Fiber interactions and fiber geometry are accounted for by means of a generalized Mori-Tanaka assumption. The orientation distribution is described by a probability density function, the expansion of which in a series of generalized spherical harmonics permits considerable reductions in the computational work.

Stress in undoped LPCVD polycrystalline silicon

The effect of processing conditions on stress in undoped LPCVD polycrystalline silicon films was investigated. Films were deposited at temperatures between 605 and 700 degrees C and pressures from 300 to 550 mtorr, with varying silane flow rate and deposition time. Low temperatures produced tensile films, while temperatures greater than about 620 degrees C resulted in compressive stress. Film thickness and deposition pressure also affect the stress rate. By removing film layers with a plasma etch, the stress profile through the film thickness was determined. Tension results when silicon atoms deposit in the amorphous state and subsequently crystallize, but thermal stress, due to differences in expansion coefficient between the film and substrate, is ruled out as significant contributor to the film stress. By superposing the crystallization stress and a compressive intrinsic stress component that decreases with temperature, the observed stress trend is obtained.<<ETX>>

Acoustoelastic response of polycrystalline aggregates exhibiting transverse isotropy

Acoustoelasticity is an ultrasonic technique which has been used for the determination of active and residual stresses in common structural materials. This paper examines the effect of texture on the acoustoelastic response in polycrystalline bodies. In particular materials which are transversely isotropic aggregates of cubic crystals are studied. The second- and third-order elastic constants of the polycrystal are derived from the elastic properties of the constituent crystals, and the crystalline orientation relative to the bodys symmetry axis. The acoustoelastic relations between velocity and deformation are then presented for the aggregate. Finally, evaluation of the acoustoelastic response for several ideal textures using data for aluminum single crystals shows that the response is highly dependent on the texture.

Stress and Microstructure in Lpcvd Polycrystalline Silicon Films: Experimental Results and Closed Form Modeling of Stresses

Characterization of undoped polycrystalline silicon films indicates that correlations exist between stress and microstructure. Films of thickness between 0.5–3.6 μm were deposited onto SiO 2 -covered single crystal silicon wafers between 605 and 700°C using low pressure chemical vapor deposition (LPCVD). The average in-plane film stress and the stress gradient through the film thickness were determined from wafer curvature measurements, and film microstructure was studied with cross-sectional TEM. Films deposited near 605°C exhibit overall tensile stresses that result from an amorphous to crystalline phase change. At deposition temperatures exceeding 630°C, a columnar grain structure evolves out of a transition region of small grains at the SiO 2 interface. The columnar films are compressive, with the source of compression linked to the region of small grains. Stress is modeled using a closed form solution ihat considers a linearly elastic contracting ellipsoidal inclusion near the surface of a half space. Several applications of the stress model are discussed.

Precision of lattice strain and orientation measurements using high-energy monochromatic X-ray diffraction

A systematic framework for estimating the uncertainty associated with measurements of finite stretch and orientation of a crystalline lattice using monochromatic X-ray diffraction is presented. A hierarchical method is implemented, in which uncertainties in the locations of diffraction peaks are communicated to the lattice stretch and rotation parameters by using the classical method of weighted least squares. This enables the uncertainty of the lattice stretch and rotation parameters to be estimated from a single full rotation scan. This method is applied to diffraction data obtained from a ruby single crystal as an idealized case for validation, and an example application is demonstrated by analyzing a strained and plastically deformed polycrystalline titanium alloy, β21S. For the ruby single crystal, it was possible to attain average uncertainties for lattice orientation and strain that were found to be comparable to standard statistical analysis of repeated measurements. For the titanium alloy, a single grain was analyzed, and a precision of 0.03° for lattice orientation and 100–250 × 10−6 for lattice strain components was obtained. The basic framework of the uncertainty analysis is generally applicable, although specific results are unique to monochromatic X-ray diffraction experiments.

Statistical characterization of fracture of brittle MEMS materials

The fracture of brittle MEMS materials is often characterized by ultimate strength measures such as the maximum stress or strain in an element at failure. It has been known for many decades that a better way to characterize the strength of a brittle material on the macro-scale is to make use of statistical measures. This is due to the nature of brittle materials in which failure occurs when a critically sized flaw exists in the region that is under tensile stress. The distribution of flaws is often random, so the strength of a brittle material can only be properly characterized by statistical measures. Working with MEMS devices, where the site scale is small, it becomes even more important to use a statistical approach. Doing so can explain two observed effects. First, there is an apparent size effect on the strength of the material. The larger the structure that is under a given stress, the larger the region where a critically sized flaw may exist, resulting a higher probability of failure. Second, two identical beams with different stress states, loaded to the same maximum stress can have dramatically different average strengths. In this paper, Weibull statistics are used to characterize the strength of one MEMS material-- polycrystalline silicon. The relevant statistical measures are obtained from the fracture of a large number of cantilever beams. It is shown that, for this material, the average failure strength of a beam loaded in uniaxial tension should be on the order of 40% lower than the average strength of identical beams loaded in cantilever bending.

Investigation of Texture and Stress in Undoped Polysilicon Films

Undoped LPCVD polysilicon films deposited on thermal oxide and prepared under various process conditions have been investigated for their texture and stress characteristics. Pole figures measured by X-ray diffraction in reflection geometry were used to determine the orientation. distribution function which provides a quantitative description of the texture. Stresses were determined using wafer curvature measurements. Both texture and stress show substantial variation with deposition condition. Textures typically exhibit axial symmetry and appear to be correlated with the sign and magnitude of the stress. Under certain deposition conditions, the stress varies considerably along the tube, with the stress in the wafers at the upstream end being tensile and at the downstream end being compressive.