Magneto-Active Elastomers: Homogenization, Instabilities and Relaxation

磁活性弹性体：均质化、不稳定性和松弛

• 批准号: 1613926
• 负责人: Pedro Ponte Castaneda
• 金额: $ 30万
• 依托单位: University of Pennsylvania
• 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 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613926&HistoricalAwards=false
• 关键词: Magneto; Active; Elastomers; Homogenization; Instabilities

This award supports the research program of the Principal Investigator on the mathematical modeling of soft composite materials responsive to magnetic fields. Magneto-active elastomers (MAEs) are composite materials consisting of nearly rigid, magnetically susceptible particles embedded in a soft, magnetically insensitive elastomer matrix (a rubber-like material). MAEs exhibit field-dependent strains (changes in length) and changes in stiffness. However, the strains that have been achieved experimentally to date are still relatively small (on the order of 1%). The reason for these small strains can be traced back to the nature of the forces between the particles. Large particle concentrations are required to generate strong forces, but large concentrations also lead to large overall stiffness for the composite material, which, in turn, tends to reduce the overall strain. Generating large actuation strains and stresses in MAEs for successful application as "artificial muscles" requires novel strategies. This project will be concerned with the possible excitation of a particular instability in a certain class of MAEs that will allow the generation of large strains by means of externally applied magnetic fields. The work will result in novel and highly efficient multi-scale, multi-physics modeling techniques of broad application.The development of constitutive models for MAEs undergoing such field-generated instabilities will require the use and appropriate generalization of several powerful and sophisticated mathematical tools. First, nonlinear homogenization methods will be developed to obtain estimates for the macroscopic "pre-bifurcation" response. For this purpose, a partial decoupling of the magnetic and mechanical energies will be implemented by means of a variational statement involving a purely magnetic problem in the deformed configuration, as determined by the unknown particle rotations, and a purely mechanical problem with prescribed torques on the particles. The average particle rotations will then be obtained by minimizing the total magneto-elastic energy of the system. The resulting constitutive model is expected to lose strong ellipticity, and to lead to the development of domain mesostructures, when the magnetic field and mechanical loading conspire to generate sufficiently large compression along the long axes of the fibers, which can in turn be relieved by collective rotation of the fibers within their respective domains. Making use of multi-layered structures, the rank-1 convexification of the magneto-elastic energy will be computed, thus leading to an upper bound for the "quasi-convexification" or "relaxation" of the energy in the "post-bifurcation" regime. Attempts will then be made to show that the rank-1 convexification is polyconvex and therefore quasi-convex. The resulting models will be used to explore the parameter space of microstructural variables (e.g., fiber volume fraction and aspect ratio) for enhanced magnetostriction and other coupled magneto-elastic properties (e.g., field-dependent moduli).

该奖项支持对磁场响应的软复合材料的数学建模的主要研究者的研究计划。 磁性弹性体（MAES）是复合材料，该复合材料由嵌入在柔软的，磁性不敏感的弹性体基质（橡胶状材料）中的几乎刚性，易感易感的颗粒组成。 MAE表现出依赖场的菌株（长度变化）和刚度变化。 但是，迄今为止在实验上实现的菌株仍然相对较小（以1％为1％）。 这些小菌株的原因可以追溯到粒子之间力的性质。 需要较高的颗粒浓度来产生强力，但是较大的浓度也导致复合材料的总体刚度较大，这反过来又倾向于减少整体应变。 作为“人造肌肉”的成功应用，在MAES中产生大型的致动菌株和压力需要新颖的策略。 该项目将与某种类别的MAE中的特定不稳定性可能激发，这将允许通过外部施加的磁场产生大型菌株。 这项工作将导致广泛应用的新颖且高效的多尺度多物理建模技术。经历此类现场生成的不稳定性的MAE的本构模型的开发将需要使用并适当地概括几种强大而复杂的数学工具。 首先，将开发非线性均质化方法，以获得宏观“生成前”反应的估计值。 为此，将通过未知的粒子旋转确定的变形构型中的变异语句来实现磁和机械能的部分脱钩，并涉及变形构型中的纯磁问题，以及在粒子上规定的扭矩的纯机械问题。 然后，将通过最小化系统的总磁弹性能来获得平均粒子旋转。 预计所得的本构模型将失去强大的椭圆形，并导致域中结构的发展，当磁场和机械载荷共同沿着纤维的长轴产生足够大的压缩时，这又可以通过在其各自域内的纤维的集体旋转来缓解。 利用多层结构，将计算磁弹性能的等级-1凸化，从而导致“准化合物”状态中能量的“准分子化”或“放松”的上限。 然后将尝试证明rank-1凸率是polyconvex，因此是准分子。 所得模型将用于探索用于增强的磁截图和其他耦合磁弹性属性（例如，依赖于场依赖的模量）的微结构变量（例如，纤维体积分数和纵横比）的参数空间。