Some Studies in the Mathematics and Mechanics of Textured and Prestressed Polycrystalline Materials

This proposed project comprises three parts. In Part 1, the objective is to develop a group- theoretic method by which explicit expressions that delineate the effects of crystallographic texture on material tensors of weakly-textured polycrystalline materials can be derived systematically. A representation theorem will be proved which reduces the problem to that of finding irreducible tensor bases of the rotation group. A method will be developed to generate irreducible tensor bases for a given tensor space. Specific instances important for applications in acoustoelasticity and plasticity will be ifilly worked out. Part 2 concerns nondestructive inspection of subsurface residual stress by simultaneous Rayleigh-wave and surface P-wave measurements. The material body in question will be modeled as a vertically heterogeneous, prestressed, anisotropic half-space. An extension of the Stroh formalism will be developed to relate the dispersion of the Rayleigh wave to the vertical heterogeneity of stress and/or texture. A higher-order ray method will be used to examine the effects of the aforementioned vertical heterogeneity on surface displacement pertaining to the surface P-wave. In Part 3 a phenomenological theory will be developed for the interpretation of resonance shifts and line shapes in ultrasound resonance spectroscopy of sheet metals. The sheet metal in question will be modeled as a textured, largely elastic material with a small power-law viscosity. An attempt will he made to draw information on resonance slufts, line shapes, and frequency dependecne of attenuation through analysis of the basic initial-boundary-value problem. Broader Impacts. The method developed in Part 1 of the project will deliver representation formulas at once applicable to all texture and crystal synimetries. This will open up new opportu- nities for work on materials with lower crystal symmetries for which constitutive relations showing explicit effects of texture are hitherto unavailable. Part 2 arises from the industrial need for a non- destructive measurement technique which allows monitoring of the retention of subsurface residual stress induced by surface conditioning treatments on metal parts for lifetime enhancement against fatigue failure and stress corrosion cracking. The availability of such nondestructive technique will also allow emerging surface-enhancement technologies (e.g., laser-peening, low plasticity burnish- ing) to be more cost effective and will improve quality control of surface conditioning processes. Part 3 constitutes a leg of one of the P1's long-term research goals to develop an ultrasonic tech- nique for in-line monitoring of texture and material microstructures in sheet metals (and hence of their formability). Two graduate students per year will participate in this project.