Departmental colloquium: Casey Rodriguez

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Casey Rodriguez, University of North Carolina

Scratching the surface beyond classical elasticity

Abstract. Classical elasticity describes how solids deform under applied forces, but it has no built-in material length scale. This limits its ability to capture behavior at length scales comparable to those of a material’s internal structure. In this talk, I will focus on how such effects influence deformation near a solid’s boundary and how we can incorporate them into a new mathematical theory of elasticity. I will introduce models in which a solid’s boundary resists stretching, bending, and spatial variations in stretching. I will then explain how these models emerge from a thin three-dimensional non-classical elastic layer whose energy accounts for spatial variations in strain. As an application, I will consider cracks, whose faces form part of the solid’s boundary. Classical linear elasticity predicts singular stresses that become infinite at a crack tip. We will see how accounting for boundary elasticity can yield finite stresses, and conclude with open mathematical questions, including those devoted to understanding fracture and predicting crack growth.