## Differential Equations Seminar

#### Dr. Ehsan Ban, University of Pennsylvania

Monday, December 11, 2017 · 11 AM - Noon

**Title:**Effective Elasticity of Random Fiber Networks and Composites

**Abstract:**The mechanical response of a variety of human-made and natural materials can be modeled using elastic random fiber networks and composites. For instance, fiber networks can be used to explain the influence of fibrous tissues on the invasiveness of cancer cells, and random composites can be used to explain the stiffness of mineralized tissues. We use the reciprocal theorem of elastic structures to show that the overall elastic modulus of a fiber network is always a concave function of the elastic modulus of a single fiber. Together with a series expansion, this result shows that fiber networks made from non-identical fibers become softer with increasing the contrast in the stiffness of fibers. We extend our analysis to random composites made by the coalescence of homogeneous isotropic linear elastic phases whose deformation is governed by the Navier-Lamé differential equation. We treat a single phase as an isotropic linear elastic domain subjected to Neumann boundary conditions. Our analysis makes use of an orthogonal decomposition of the displacement of each homogeneous phase. We show that the overall stiffness of a composite is always a concave function of the elastic modulus of an inhomogeneity embedded inside the composite. Therefore, random composites become softer with increasing microstructural heterogeneity: The gain in overall stiffness from the stiffer constituents is smaller than its loss due to the softer phases. This result is supported using the series expansion of the composite stiffness in the space of the elastic moduli of the individual phases.