Singular solutions in gels: Cavitation and debodning

该奖项支持首席研究员正在进行的关于凝胶中材料失效的建模、分析和模拟的研究项目。工程设备通常由一种以上的材料组成。它们的互连，以及设备与基板的结合，容易受到环境因素的影响，最终导致它们分解，导致设备故障。据估计，仅在医疗器械行业，产品召回的成本每年就在25亿至50亿美元之间，其中50%归因于材料失效。该项目将研究聚合物材料的断裂，主要关注凝胶。凝胶中的失效机理以及解决它的数学方法与晶体材料中的失效机理有很大的不同。当聚合物材料被放置在液体丰富的环境中，比如植入体内的医疗设备，它往往会因吸收水分而膨胀。不同的材料有不同的膨胀比，这导致应力在结合界面和衬底处积聚。当达到粘合剂的应力阈值时，设备可能会出现脱粘。研究活动将包括建模、分析和数值模拟导致凝胶从底物分离的现象。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，将有一个强大而明确的联系。几名研究生和本科生将参与这项研究。了解材料断裂及其发生和演变，仍然是材料科学的主要挑战之一。从数学的角度来看，在标准力学假设失效的情况下，研究脱粘需要处理奇异解。该项目将开发和应用从变分法、几何测量理论、渐近分析和偏微分方程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。空化研究的主要数学困难是将参考位形中的点转换为空化表面的变形图的注入性损失。测量理论工具的应用，如分布行列式，允许在奇异框架中验证控制方程。该研究将为研究空化和脱粘及其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点，假设是聚合物和流体的混合物。附着在衬底上的凝胶样品的膨胀可能产生奇点形成所必需的力。分层的分析将基于这样的假设：有两个相关参数，几何参数和材料参数，能够根据脱离基材的长度来描述体的最佳能量。一个目标是表明界面处的应力集中与剪切相对应。

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.