By Helina Torv, Kerstin Kracht, Alexandra Jeberien
In conservation gap-fills are ideally reversible and used only for reducing mechanical instabilities in the object. However, the exact locations where support is needed and the strength and stability of these areas are rarely elucidated. Furthermore, the strength of the gap-filling materials is often an unknown. Until today, conservation has by-in-large lacked scientific methods for determining where gap-fills for structural stability are needed and what their optimal characteristics should be; the solution has relied on the experience of the conservator. As a result the strongest adhesives, on metals usually epoxies, are often selected for gap-fills, the fear being that reversible adhesives are too weak.
The main reason for this lack of design methods is the difficulty in calculating the structural strength of an object with complex geometries. Since the 1940s a computer-aided simulation technique called finite element analysis (FEA) has been developed in engineering to study the overall behaviour of structures. Here the FEA is offered as a tool to assist in the selection of a gap-fill strategy for an archaeological cauldron.
THE ROMAN BRONZE CAULDRON
This Roman bronze cauldron (diameter ca 30 cm, height 20 cm) was excavated in 1999 in Apensen, Lower Saxony (Germany). The thin metal walls of the vessel (less than 0.5 mm) are fragmented, severely corroded and have been cleaned, but before further conservation treatment can be carried out, the bronze fragments need reinforcement, which leads to the question of gap-fill methods. Here we test the hypothesis that weaker, but reversible, thermoplastic adhesives—instead of epoxy resins—can be used to reinforce the cauldron.
WHAT IS FINITE ELEMENT ANALYSIS?
Finite element analysis is a computer-aided simulation technique which, in this context, enables the mechanical behaviour of the cauldron to be analysed numerically. In pre-processing, the geometric model of the cauldron is divided into shell and volume elements. Shells are thin two-dimensional elements, which can have a curvature and behave like plates and discs. Volume elements have an extension into the three spatial directions. The nodes of these elements, for instance at the position of the vertexes, can move into the three spatial directions and rotate around the spatial axes. Typical examples are tetrahedrons and hexahedrons. The discretized geometry model is called a mesh. Properties such as density and material behaviour are assigned to each element. Finally, differential equations are applied to each element which describe the stress-strain behaviour under the considered load case. This leads to a system of equations which is solved numerically. In post-processing, for example, the displacements, strains or stresses of the individual nodes are assembled and animated as a whole.
DESIGN OF THE STUDY
Gravity cannot simply be turned off. Therefore, our main investigation focused on analysing the stability and the possible filling of gaps in the cauldron under its own weight. The influence of gravity was assessed using displacement and von Mises yield criterion. Handling, transport and the museum environment can lead to additional dynamic loads due to vibration of the object, and dynamic loads can sometimes be more damaging than static ones, leading to fatigue fractures.
A criterion for the evaluation of measures are the natural frequencies of the cauldron. These are changed by restoration treatment and changes in storage. Natural frequencies are a system property which gained notoriety through the resonance phenomenon. If an object is stimulated to the beat of its natural frequencies, this results in resonance which is a force-enhancing effect that can lead to damage. Padding materials and springs can be used to isolate an object from resonance vibration. For this, the cauldron should be stabilized to avoid natural frequencies below 45 Hz.
Modal analysis is a general method used to determine the modal parameters: natural frequencies, eigenmodes and modal damping of an object. This method was also applied to the cauldron to investigate which gaps require fills in order to reduce natural frequency below 45 Hz.
To evaluate the necessary extent of the reinforcement, four scenarios were examined:
I- Cauldron without any reinforcement
II- The largest crack between bottom and walls being reinforced
III- In addition to scenario II, the bottom cracks being reinforced
IV- All holes are reinforced
We expected the comparison of these four scenarios, in gravitational study and in modal analysis, would inform the extent to which the gap-fills would affect the stability of the cauldron.
We used photogrammetry to capture the object’s geometry. The gaps were cut into the model using information from texture and colour. Because of the limitations of the 3D-programmes, the narrow cracks were also modelled as gaps. The metal walls of the cauldron consist of approximately 10% tin-copper alloy. The thickness of the bronze wall, which we measured to be 0.5 mm, corresponds to the average thickness of the metal core together with the corrosion surface. The bronze walls were modelled with shell elements. The iron border no longer has a metal core and is now iron oxide, but because of its thickness, we modelled it with solid elements. For the characterisation of the gap-fill material, we used the bronze wall thickness of 0.5 mm, density of 1 g/mm3 and Young's modulus 2000 N/mm2 distinctive to the stronger thermoplastic adhesives like Paraloid B44. The gap-fills were also modelled as shell elements. We performed the gravitational study with the assumption that the cauldron will be supported under its base. In a modal analysis we studied the free geometry.
The biggest dead load improvement—a three-fold change in parameters—was seen using scenario II, when the biggest gap between the bottom of the vessel and walls was closed. The cauldron has two critical bridge areas, one between the bottom and walls (filled in scenario II) and the other in the wall itself (filled in scenario IV), that are particularly stressed without gap-fills. Closing the surrounding area of the bridges can almost eliminate the stresses in the bridges, while closing the other voids and cracks has minimal effect on the gravitational load.
The 45 Hz limit was already exceeded with the closing of the largest gap (scenario II). In scenario IV, the cauldron would be stabilized against transport vibrations of trucks. The modal analysis also showed that the material around larger gaps tends to vibrate independently. The areas of the cauldron that are surrounded by loss or damage on three sides are particularly at risk. Thus, it would be important to secure these areas.
According to the FEA, fill materials with Young's modulus above 2000 N/mm2 and thickness above 0.5 mm are suitable for supplementing the cauldron. However, the material is physically somewhat flexible, which is evidenced by the displacements at the largest crack. These displacements are about 0.5 mm, which are invisible to the human eye.
RECOMMENDATIONS FOR CONSERVATION
The results of the finite element analysis show that it is possible to reinforce the Roman bronze cauldron using thermoplastic adhesives with a modulus of elasticity greater than 2000 N/mm2. Our results led us to conclude that an adhesive bond with a thickness of 0.5 mm is sufficient to ensure static stability and to shift the natural frequency above 45 Hz. In order to achieve both goals, a bonding of the large crack is sufficient as a minimal treatment approach. The modal analysis has shown that closing additional cracks, or even all cracks, will contribute to the vessel stability in case of dynamic excitations; this treatment would prevent individual surfaces from vibrating freely thus minimising the risk of fractures at the loaded joints. Closing the main bottom crack successfully would cancel all natural vibrations up to a frequency level of 113 Hz. However, even greater stability can be achieved by closing all cracks since natural frequencies are not below 534Hz.
The FEA model of the vessel was created to accurately correspond to the real object, but simplifications had to be made. The different wall thickness and degree of corrosion had to be combined and standardised in order to obtain results. While these differences should not have a large impact on the results from an engineering perspective, the object studied is unique in material and structure and therefore unpredictable to some degree. One result the bond join properties investigation (executed on the test rig in combination with the FEM) showed was that it is mainly the reinforcement of the geometry, and less the material, of the bond that matters. Thus, the choice of adhesive in this instance can be made with more consideration for reversibility and compatibility with the material.
In summary, the FEA was useful in planning the conservation treatment of the Roman bronze cauldron, however, it is unclear to what extent the results actually correspond to reality since we did not carry out control measurements of the vibration behaviour of the cauldron. Measurements like the experimental modal analysis are therefore recommended for further work.
Link to Helina’s master’s thesis:
Helina Torv is a graduate of the Berlin University of Applied Sciences (HTW Berlin), Conservation of Archaeological Objects program. In her master's thesis she worked on the application of FEA in conservation for the selection of gap fillers for a copper cauldron. She is currently working as an archaeological conservator with the excavation company Denkmal 3D.
Prof Kerstin Kracht (TU Berlin) is a vibration technology and continuum mechanics engineer and has been applying and sharing her expertise in vibration and shock prevention in the field of art and cultural heritage preservation for more than fifteen years. Kerstin studied physical engineering and completed her PhD at the Technische Universität Berlin in 2011, investigating the vibration behaviour of oil paintings depending on age.
Alexandra Jeberien is a professor at HTW Berlin’s conservation and restoration programme. She is a trained conservator for archaeological objects and holds a master’s degree as well as a doctoral degree in cultural heritage preservation. While her teaching focusses on object restoration techniques, her research interests include preventive conservation. Recent projects dealt with climate and pollution control for museums and risk management applications.
(See all the images and video in the October-November 2022 "News in Conservation" Issue 92, p. 18-23)