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

本项目由三部分组成。在第一部分中，目标是发展一种群论方法，通过该方法可以系统地推导出描述弱织构多晶材料的晶体织构对材料张量影响的显式表达式。证明了表征定理，将问题简化为寻找旋转群的不可约张量基的问题。给出了给定张量空间生成不可约张量基的方法。该方法提供了一次适用于所有纹理和晶体对称性的表示公式。对声弹性和塑性应用有重要意义的具体实例进行了充分的计算。第二部分是利用超声测量法对地下残余应力进行无损检测。所讨论的材料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。发展了Stroh形式的扩展，将瑞利波的频散与应力和/或结构的垂直非均质性联系起来。在第3部分现象学理论是发展的解释共振移位和线形在超声共振光谱的金属板材。所讨论的金属板被建模为具有小幂律粘度的有纹理的、很大弹性的材料。通过分析基本的初边值问题，试图得出共振位移、线形和衰减的频率依赖性的信息。构成多晶材料的晶粒(例如：例如，锻造和铸造金属合金)通常具有优先取向（或晶体织构），强烈影响聚集体材料的机械行为（例如，钣金的可成形性）。本项目的第1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部分源于工业对非破坏性测量技术的需求，该技术允许监测由金属零件表面调节处理引起的地下残余应力的保留，以增强其抗疲劳失效和应力腐蚀开裂的寿命。这种非破坏性技术的可用性也使得新兴的表面增强技术（例如，激光喷丸，低塑性抛光）更具成本效益，并改善了表面调理过程的质量控制。第3部分为超声波技术的发展奠定了数学基础，用于监测生产线上金属板的纹理和材料微观结构（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.