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

城域数据管理系统-0807543本项目由三部分组成。在第一部分中，我们的目标是发展一种群论方法，通过它可以系统地推导出描述晶体织构对弱织构多晶材料材料张量的影响的显式表达式。证明了一个表示定理，将问题归结为求旋转群的不可约张量基的问题。给出了一种生成固定张量空间不可约张量基的方法。该方法同时给出了适用于所有织构和晶体对称性的表示公式。文中还给出了声弹性和塑性方面的重要应用实例。第二部分介绍了用超声波检测地下残余应力的无损检测方法。所讨论的材料体被建模为垂直不均匀的、预应力的、各向异性的半空间。发展了Stroh公式的推广，将瑞利波的频散与应力和/或织构的垂直非均匀联系起来。在第三部分中，发展了解释薄板超声共振频谱中共振位移和线形的现象学理论。所讨论的金属薄板被模拟为一种质地、主要是弹性的材料，具有很小的幂定律粘度。通过对基本初边值问题的分析，试图得到关于共振位移、线形和衰减的频率依赖性的信息。构成多晶材料(例如，锻造和铸造金属合金)的晶粒通常具有较好的取向(或晶体织构)，这些取向(或晶体织构)强烈影响聚集材料的机械行为(例如，板材的成形性)。本项目的第一部分详细介绍了定量描述晶体织构如何影响多晶材料的非弹性和塑性变形行为的公式。第二部分源于工业上对无损测量技术的需要，该技术允许监测金属零件表面处理引起的亚表面残余应力的保留，以提高疲劳失效和应力腐蚀开裂的寿命。这种非破坏性技术的出现也使新兴的表面强化技术(如激光喷丸、低塑性抛光)更具成本效益，并改善了表面处理过程的质量控制。第三部分帮助为开发超声波技术奠定数学基础，以便在生产线上监测板材的织构和材料微结构(以及它们的可成形性)。

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.