Optimal rectangular crack pattern based on constructal, fracture saturation, and energy minimization theories for painting on wood

The knowledge of how the craquelure happens and their pattern on historical objects especially paintings are interested in the field of cultural heritage. In this paper, optimal rectangular patterns are calculated based on current theories for a fully anisotropic model of wood and an anisotropic model of gesso annealing shrinkage. First, the constructal theory is applied which is responsible for the crack growth at the beginning periods of drying the produced painting. As the constructal theory is based on greater access to the drying currents that flow through the crack pattern, the compromise of two mechanisms of diffusion of moisture content and advection drying by fluid flow through the cracks detect the optimal scale of rectangular patterns. A rectangular two-dimensional solid is dried by the diffusion mechanism and advection boundary condition. The resulted shape is found by the intersection of two limits of many cracks (dense crack by high advection) versus a few cracks analytically. Then a parametric study is done to find the optimal distance between blocks of crack islands numerically. Second, the fracture saturation mechanism in long-term climate changes is applied to find the possible rectangular pattern which can appear based on the fact that positive stress should exist in the middle of the crack island to create a new crack. Finally, the strain energy density approach for various fracture strengths is examined to predict the optimal rectangular shape. The numerical results are compared with existing experimental results and previous works demonstrate the capability of the above-mentioned models. The method and results could be generalized for the other types of craquelure patterns and structural conservators.