Size-dependent instabilities in highly deformable strain-gradient materials

Buckling, wrinkling, and period doubling are commonly observed in soft materials and are often leveraged for functional and design purposes. However, accurately predicting these instabilities at small scales requires accounting for size effects – particularly the emergence of strain gradients – which are absent in classical continuum theories due to their lack of intrinsic length scales. Although these instabilities have been extensively studied using classical theories, such approaches may not fully capture their size-dependent onset and evolution. Moreover, numerical modeling of size effects in highly deformable materials remains largely unexplored. Here, we develop a variational-based computational framework to study mechanical instabilities in hyperelastic strain-gradient materials using isogeometric finite element method and demonstrate the emerging size effects at finite-sized soft domains at large deformations. Additionally, we derive an analytical solution for size-dependent buckling of slender beams, explicitly accounting for Poisson’s ratio, to verify our framework. Through a nonlinear stability analysis of beams and film–substrate systems, we quantify the onset and pattern formation for size-dependent beam buckling, bilayer wrinkling, and period-doubling instabilities—commonly observed and leveraged in soft material applications. Our simulations reveal that incorporating the length scale parameter not only delays the onset of these instabilities but also alters the post-instability pattern evolution. Notably, stiffer strain gradients can shift the instability nature from supercritical to subcritical, including sudden jumps in the equilibrium response once the critical threshold is crossed, and can even suppress certain instabilities entirely. We further demonstrate how the spatial variation of strain changes across buckled configurations and evaluate how instabilities can be used to induce large strain gradients. Our simulations also reveal pronounced stress localization near boundaries, highlighting the critical role of higher-order effects in small-scale soft materials. These insights offer a computational foundation for further understanding, designing, and actively controlling strain-gradient phenomena in soft materials.