Parametric model for localized hygro-mechanical buckling and post-buckling in historical paper

A parametric model is presented for predicting localized buckling and post-buckling behavior in historical paper subjected to hygroscopic in-plane compression, commonly referred to as cockling. The model is based on the von Kármán plate theory for circular plates and captures geometrically nonlinear, imperfection-sensitive responses under both mechanical and hygroscopic loading. Building on the formulation for mechanical loading, moisture-induced eigenstrains are treated as deformation-controlled inputs, enabling closed-form expressions to be derived for the critical moisture change initiating buckling and for the corresponding post-buckling response. The framework explicitly accounts for boundary rotational stiffness representative of cockled regions within a larger sheet, geometric imperfections, transient moisture diffusion, and viscoelastic stress relaxation. Analyses suggest that localized buckling occurs only when moisture variations persist long enough for a steady-state profile to develop, while short-lived humidity fluctuations are unlikely to trigger instability. Material aging is accounted for by integrating the model with a multiscale degradation framework from the literature, allowing the assessment of long-term paper stability under realistic microclimatic conditions. The results demonstrate that the hygroscopic expansion coefficient remains essentially constant over most of the paper’s lifetime under museum- and archive-like microclimates, allowing the initial value to be used as a reliable and conservative parameter in stability assessments. By combining modeling results with moisture transport considerations, the study derives quantitative criteria linking allowable moisture variations, characteristic length scales, and exposure durations to acceptable out-of-plane deflections. These criteria are presented in a form suitable for practical application, offering conservators and collection managers a mechanistically grounded tool for defining environmental limits that mitigate or prevent cockling in paper-based cultural heritage.