Computational modelling of metal soap formation in historical oil paintings: the influence of fatty acid concentration and nucleus geometry on the induced chemo-mechanical damage

Abstract

Metal soap formation is one of the most widespread degradation mechanisms observed in historical oil paintings, affecting works of art from museum collections worldwide. Metal soaps develop from a chemical reaction between metal ions present in the pigments and saturated fatty acids, which are released by the oil binder. The presence of large metal soap crystals inside paint layers or at the paint surface can be detrimental for the visual appearance of artworks. Moreover, metal soaps can possibly trigger mechanical damage, ultimately resulting in flaking of the paint. This paper departs from a recently proposed computational model to predict chemo-mechanical degradation in historical oil paintings, as presented in Eumelen et al. (J Mech Phys Solids 132:103683, 2019). The model describes metal soap formation and growth, which are phenomena that are driven by the diffusion of saturated fatty acids and proceed by a nucleation process from a crystalline nucleus of small size. This results into a chemically-induced strain in the paint, which may promote crack nucleation and propagation. The proposed model is here used to investigate the effects of saturated fatty acid concentration and initial nucleus geometry on the amount of chemo-mechanical damage generated. Numerical simulations show that both factors have a marginal influence on the growth rate of the metal soap crystal, but play a significant role on the extent of fracture induced in the paint.

Chemo-mechanical model for degradation of oil paintings by amorphous and crystalline metal soaps

“…The fracture properties of the paint and crystallised metal soap are derived from the stress-strain diagram reported in Fuster-López et al ( 2016), leading to an ultimate tensile strength 𝑡 u 1 = 2.2 N/mm 2 and a mode I fracture toughness 𝐺 I,c = 0.418 N/mm. From the simulation results presented in Eumelen et al (2021), it can be observed that the fracture process in the painting is indeed dominated by mode I cracking events. Therefore, the fracture properties in mode II are expected to only have a minor influence on the computational results, and, for simplicity, are taken equal to the properties under mode I conditions, i.e., 𝑡 u 2 = 𝑡 u 1 = 2.2 N/mm 2 and 𝐺 II,c = 𝐺 I,c = 0.418 N/mm.…”

“…This modelling strategy was originally proposed in Xu and Needleman (1994), and allows for the robust simulation of crack patterns at arbitrary locations and in arbitrary directions. It naturally includes the effects of crack bifurcation, crack branching and crack coalescence, as previously demonstrated for applications related to historical paints (Eumelen et al, 2019(Eumelen et al, , 2020(Eumelen et al, , 2021, wood (Luimes et al, 2018;Scheperboer et al, 2019;Luimes and Suiker, 2021), polymers (Tijssens et al, 2000), fibrous composites (Cid Alfaro et al, 2010a,b;Geng and Suiker, 2019) and cementitious materials (Scheperboer et al, 2021). The interface elements are characterised by the interface damage model proposed in Cid Alfaro et al (2009).…”

“…The shape of the nuclei is taken as circular, with an initial radius equal to one tenth of the layer thickness, 𝑟 0 = 𝑙∕10 = 1.5 μm. The finite element discretisation of the geometry is similar to that used in Eumelen et al (2021), whereby, after a mesh convergence study, the element size close to the metal soap nuclei was set to 0.2 μm, and at the corners of the geometry to 0.5 μm, as required for obtaining mesh-independent numerical results. Accordingly, the computational domain is discretised with 27, 030 continuum elements.…”

“…To the best of the authors' knowledge, for the chemical growth strain associated to metal soap formation no experimental value has been reported in the literature. Hence, the value of 𝜀 g * = 0.1 pragmatically adopted for the plane-stress analyses presented in Eumelen et al (2019Eumelen et al ( , 2020Eumelen et al ( , 2021 is adjusted for the current, plane-strain analysis by requiring that the effective, in-plane chemical growth strain in both 2D models is the same, which, in accordance with Equation (26) in Eumelen et al (2019), is the case for 𝜀 g = (1 + 𝜈 𝑠 )𝜀 g * = (1 + 0.3) ⋅ 0.1 = 0.077. Note hereby that the choice of representing a 3D paint layer by a 2D plane-stress or plane-strain model is somewhat arbitrary, but, in principle, leads to comparable results for the in-plane chemo-mechanical behaviour of the paint layer.…”

“…which, with 𝜀 g * = 0.1 (Eumelen et al, 2019(Eumelen et al, , 2020(Eumelen et al, , 2021 and an original paint volume of 𝑉 paint,0 = 1 m 3 , leads to 𝑉 paint = 1.3 m 3 . Combining this result with Eqs.…”