Variational modeling of plane-strain hyperelastic thin beams with thickness-stretching effect

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In the present work, we develop a new Euler–Bernoulli finite strain beam model for soft thin structures subject to stiff constraint in the width direction. The beam model assumes plane-strain deformation and accounts for the thickness-stretching effect, which is very significant for soft structures under large strain. With the Euler–Bernoulli kinematic hypothesis and incompressibility assumption, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. Based on the new beam model, analytical formulae are given for uniform stretching and pure bending, which cannot be accurately described by any finite strain beam model with the rigid cross-section assumption. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with movable or immovable ends. The developed beam model is expected to benefit the modeling and simulation of soft robots and soft devices.