FROM WRINKLES TO CREASES - ON STABLE LOCALIZED DEFORMATION SOLUTIONS IN HIGH SYMMETRY STRUCTURES USING GROUP THEORY
Nicolas Triantafyllidis 1,2
1Solid Mechanics Laboratory (CNRS-UMR 7649) & Department of Mechanics,
École Polytechnique, Palaiseau 91128, France
2Aerospace Engineering Department & Mechanical Engineering Department,
The University of Michigan, Ann Arbor, MI 48109-2140, USA (emeritus)
We are motivated by the celebrated Biot problem of surface instability in a hyperelastic half-space under axial compression and in particular the evolution, from an initially unstable bifurcated short wavelength, periodic deformation pattern to a stable solution involving a localized deformation. This feature is shared by other problems in mechanics, where a high initial symmetry (due to translational invariance) leads to a complex bifurcation pattern. Understanding the evolution of bifurcated solutions from a periodic pattern (wrinkling) to a highly localized one (crease) is the goal of this work.
Due to the complex structure of these nonlinear problems, we first study the behavior of an inextensible infinite Euler-Bernoulli beam subjected to a compressive axial force and connected to a nonlinear (polynomial) elastic foundation. We use group-theory methods to follow all bifurcated equilibrium paths in a systematic way and explore the emergence of stable localized solutions.
We next proceed with the analysis of the nonlinear hyperelastic half-space problem under axial compression, using different energy densities. To avoid the arbitrariness of the critical wavelength associated with the homogeneous half-space, we consider a half-space with functionally graded properties as well as a finite thickness layer bonded atop a half-space. The same group-theoretic approach is used to find the bifurcated solutions, from the initial unstable periodic wrinkles to a stable crease. A continuation method using the layer/half-space initial shear moduli ratio is also proposed as an alternative to seek the creased solution of the homogeneous half-space problem.
Work is collaboration with: S. Pandurangi, T. Healey (Cornell), and R. Elliott (U. Minnesota)
Seminar to be presented at Northwestern University, Evanston IL, USA on April 2, 2020
Short Bio of N. Triantafyllidis:
Prof. Triantafyllidis has obtained his Ph.D. in Engineering from Brown University in 1980. The same year he joined the faculty of the Aerospace Engineering Department at the University of Michigan in Ann Arbor, MI, USA starting as an Assistant Professor and reaching the rank of Full Professor in the Departments of Aerospace Engineering and Mechanical Engineering & Applied Mechanics. He is currently an emeritus Professor of the University of Michigan. In 2009 he moved to France to become CNRS Director of Research in the Solid Mechanics Laboratory (LMS) and a Professor of Mechanics at the Ecole Polytechnique.
Prof. Triantafyllidis’ research is in the areas of stability in solids and structures, structural mechanics and nonlinear continuum mechanics. Over the years he has worked in sheet metal forming problems, large flexible space structures, composites and cellular materials, the geomechanics of the earth’s crust and stability issues in homogenization problems. More recently he worked on the thermomechanical stability of shape memory crystals. His recent interests are in multiphysics – i.e. problems involving interactions between mechanical, electromagnetic and electronic effects in solids – with applications in magneto-elasticity, electromagnetic forming electric motors, strain effects in semiconductors, flexible photovoltaics and thin film electronics.
