Singular solutions in gels: Cavitation and debodning

凝胶中的单一解决方案：空化和脱粘

• 批准号: 1616866
• 负责人: Maria-Carme Calderer
• 金额: $ 30万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1616866&HistoricalAwards=false
• 关键词: Singular; solutions; gels; Cavitation; debodning

This award supports the ongoing research program of the Principal Investigator on modeling, analysis, and simulation of material failure in gels. Engineering devices generally consist of more than one material. Their interconnections, as well as the binding of the device to substrates, are susceptible to environmental agents that ultimately cause them to break down, leading to device failure. It is estimated that the cost of product recall in the medical device industry alone ranges between $2.5 and $5 billion annually, 50% of which is attributed to material failure. This project will investigate material fracture in polymeric materials, with a main focus on gels. The mechanism of failure in gels as well as the mathematical methods to address it are very different from those in crystalline materials. Upon placement of a polymeric material in a fluid-rich environment, such as an implanted medical device in the body, it tends to swell by absorbing water. Different materials have different swelling ratios, which cause stress to build up at the bonding interfaces and substrates. Upon reaching the stress threshold of the adhesive, the device may experience debonding. The research activities will consist of modeling, analysis, and numerical simulation of the phenomena that causes a gel to break apart from its substrate. There will be a strong and clear connection with laboratory experiments and industrial applications, especially those related to the medical device industry. Several graduate and undergraduate students will be involved in the research.Understanding material fracture, its initiation and evolution, remains one of the main challenges in materials science. From the mathematical point of view, the study of debonding requires dealing with singular solutions, in cases when standard mechanical assumptions fail. This project will develop and apply combined tools from calculus of variations, geometric measure theory, asymptotic analysis, and free boundary problems for partial differential equations to provide a mathematical description of how microdefects present in the material may evolve into cavities that grow and cause the material to detach from a substrate. The main mathematical difficulty in the study of cavitation is the loss of injectivity of the deformation map that transforms a point in the reference configuration into a cavity surface. The application of tools from measure theory, such as the distributional determinant, allows validation of the governing equations in the singular frameworks. The research will provide a unified framework to study cavitation and debonding, and the evolution from the former to the latter. The project will focus on studying singularities that form in gels, assumed to be mixtures of polymer and fluid. Swelling of a gel sample attached to a substrate may produce the necessary force for singularities to form. The analysis of delamination will be based on the conjecture that there are two relevant parameters, geometric and material, able to describe the optimal energy of the body in terms of the length of detachment from the substrate. One goal is to show that stress concentrations at interfaces correspond to shearing.

该奖项支持主要研究者正在进行的关于凝胶材料失效建模、分析和模拟的研究计划。 工程器械通常由一种以上的材料组成。 它们的互连以及器件与衬底的结合容易受到环境因素的影响，最终导致它们分解，从而导致器件故障。 据估计，仅医疗器械行业的产品召回成本每年就在25亿至50亿美元之间，其中50%归因于材料故障。 本项目将研究聚合物材料中的材料断裂，主要关注凝胶。 凝胶中的失效机制以及解决它的数学方法与结晶材料中的失效机制非常不同。 在将聚合物材料放置在富含流体的环境中时，例如在体内植入的医疗装置，其倾向于通过吸收水而膨胀。 不同的材料具有不同的膨胀率，这导致应力在结合界面和基底处积聚。 在达到粘合剂的应力阈值时，装置可能经历剥离。 研究活动将包括建模，分析和数值模拟的现象，导致凝胶从其基板分开。 与实验室实验和工业应用，特别是与医疗器械行业相关的应用，将有强烈而明确的联系。 一些研究生和本科生将参与这项研究。理解材料断裂，它的起源和演变，仍然是材料科学的主要挑战之一。 从数学的角度来看，研究脱粘需要处理奇异的解决方案，在标准的力学假设失败的情况下。 该项目将开发和应用变分法，几何测量理论，渐近分析和偏微分方程的自由边界问题的组合工具，以提供材料中存在的微缺陷如何演变成生长并导致材料从衬底分离的空腔的数学描述。 在空化研究中的主要数学困难是将参考配置中的点转换成空腔表面的变形映射的注入性的损失。 从测量理论的工具，如分布行列式的应用，允许验证的奇异框架中的控制方程。 本文的研究将为研究空化和脱粘以及从空化到脱粘的演变提供一个统一的框架。 该项目将重点研究凝胶中形成的奇异性，假设凝胶是聚合物和流体的混合物。 附着在基底上的凝胶样品的溶胀可以产生形成奇点所必需的力。 分层的分析将基于这样的推测，即存在两个相关参数，几何参数和材料参数，能够根据从基底分离的长度来描述主体的最佳能量。 一个目标是表明在界面处的应力集中对应于剪切。