Buckling of an elastic fiber confined to a thin elastic membrane

When an elastic beam is embedded in a deformable matrix which is then compressed, the beam may undergo ``traction-based buckling''. This buckling is a result of the friction forces between the beam and the matrix and is common in biological and technological systems like living cells containing microtubules, electrodes embedded in tissue, fiber-reinforced composites and stretchable, wearable electronics. Traction-based buckling typically results in periodic buckles with amplitude and wavelength determined by the geometry and stiffness of the beam and matrix. Work has been done on systems where the thickness of the embedding matrix is semi-infinite and much softer than the embedded beam, resulting in buckles with no preferred orientation. In this work we explore systems where an elastic fiber is adhered to a thin, compliant membrane. By altering the stress profile in the membrane and the thickness of either the membrane or the fiber, we are able to continuously control the orientation of the buckles with respect to the plane of the membrane.