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

ManDMS-0807543 该项目包括三个部分。 在第一部分中，我们的目标是发展一种群论方法，通过这种方法可以系统地导出晶体织构对弱织构多晶材料的材料张量的影响的显式表达式。 证明了一个表示定理，将问题归结为寻找旋转群的不可约张量基。 本文给出了一种生成张量空间不可约张量基的方法。 该方法提供的表示公式，一次适用于所有的纹理和晶体对称性。对声弹性和塑性中的重要应用给出了具体的例子。 第2部分是关于用超声波测量法无损检测表面下残余应力的。 问题中的物质体被建模为一个垂直非均匀的、预应力的、各向异性的半空间。 扩展的斯特罗形式主义发展到涉及到的Rayleigh波的频散的verticallheterogeneity的应力和/或纹理。 在第三部分中，发展了解释金属薄板超声共振谱中共振位移和谱线形状的现象学理论。 金属板的问题是作为一个纹理，很大程度上是弹性材料与小幂律粘度建模。 试图通过对基本的初边值问题的分析，得出有关共振位移、线形和衰减频率依赖性的信息。 构成多晶材料的晶粒（例如，锻造和铸造金属合金）通常具有强烈影响聚集体材料的机械行为的优选取向（或晶体学织构）（例如，金属板的可成形性）。 本项目的第一部分提供了定量描述晶体结构如何影响多晶材料非弹性和塑性变形行为的公式。 第2部分是由于工业上需要一种无损测量技术，这种技术可以监测金属零件表面处理引起的表面下残余应力的保留情况，以提高金属零件的寿命，防止疲劳失效和应力腐蚀开裂。 这种非破坏性技术的可用性还允许新兴的表面增强技术（例如，激光喷丸、低塑性抛光），以提高成本效益，并改善表面处理工艺的质量控制。 第3部分有助于奠定数学基础的超声波技术的发展，用于监测纹理和材料的微观结构在金属板材（因此，他们的可成形性）的生产线。

ManDMS-0807543 This project comprises three parts. In Part 1, theobjective is to develop a group-theoretic method by whichexplicit expressions that delineate the effects ofcrystallographic texture on material tensors of weakly-texturedpolycrystalline materials can be derived systematically. Arepresentation theorem is proved that reduces the problem to thatof finding irreducible tensor bases of the rotation group. Amethod is developed to generate irreducible tensor bases for agiven tensor space. The method delivers representation formulasat once applicable to all texture and crystal symmetries. Specific instances important for applications inacoustoelasticity and plasticity are fully worked out. Part 2concerns nondestructive inspection of subsurface residual stressby ultrasonic measurements. The material body in question ismodeled as a vertically heterogeneous, prestressed, anisotropichalf-space. An extension of the Stroh formalism is developed torelate the dispersion of the Rayleigh wave to the verticalheterogeneity of stress and/or texture. In Part 3 aphenomenological theory is developed for the interpretation ofresonance shifts and line shapes in ultrasound resonancespectroscopy of sheet metals. The sheet metal in question ismodeled as a textured, largely elastic material with a smallpower-law viscosity. An attempt is made to draw information onresonance shifts, line shapes, and frequency dependence ofattenuation through analysis of the basic initial-boundary-valueproblem. The crystal grains that constitute polycrystalline materials(e.g., wrought and cast metallic alloys) often carry preferredorientations (or crystallographic texture) that strongly affectmechanical behavior of the aggregate material (e.g., theformability of sheet metals). Part 1 of this project deliversformulas that describe quantitatively how crystallographictexture affects the behavior of a polycrystalline material inelastic and plastic deformations. Part 2 arises from theindustrial need for a nondestructive measurement technique thatallows monitoring of the retention of subsurface residual stressinduced by surface conditioning treatments on metal parts forlifetime enhancement against fatigue failure and stress corrosioncracking. The availability of such nondestructive techniquealso allows emerging surface-enhancement technologies (e.g.,laser-peening, low plasticity burnishing) to be more costeffective and improves quality control of surface conditioningprocesses. Part 3 helps lay the mathematical foundations for thedevelopment of an ultrasonic technique for monitoring of textureand material microstructures in sheet metals (and hence of theirformability) on the production line.