1 INTRODUCTION

The structural analysis of multilayer works of art such as paintings or photographs is complicated by the uncertainty of the mechanical properties of the individual material components and by the physical interactions between the layers. The mechanical properties are affected by temperature, relative humidity (RH), and the rate at which forces are applied to the material. For materials responding to changes in temperature and relative humidity that occur within slow but measurable time intervals, the mechanical and dimensional properties must be determined under equilibrium and quasi-equilibrium conditions. However, once the correct mechanical properties of the individual components in the object have been determined, comprehensive numerical models can be developed. The benefit of a numerical solution is that the magnitude and direction of the internal stresses as well as the dimensional distortions occurring nonuniformly in the object can be accurately evaluated. Structural failure can also be readily predicted when the calculated stress levels exceed the measured breaking stresses of the individual constituent materials. Thus, the information provided by a numerical model can be used to reassess the environments in which museum objects are exhibited, stored, and transported. In addition, numerical analysis can help evaluate proposed structural and chemical conservation treatments prior to the actual treatment.

This paper expands on research presented in an earlier paper (Mecklenburg et al. 1993) and discusses the development of the mathematics, the measurements of the mechanical properties of some typical materials used in the construction of paintings and photographs, and the use of these measurements in numerical analysis. It is important to recognize that the direct measurement of stress in most cultural objects subjected to changes in temperature and RH is simply not possible. Direct stress measurements are generally valid for objects that can be considered a single homogeneous material. This is not the case for multicomponent structures like paintings and photographs, where individual materials reside in discrete regions or layers. Previous attempts to measure the effects of RH on the stress development in paintings have been made by measuring the forces developed at the edges (Berger and Russell 1986, 1988). Although such measurements give general indications that stresses are developing within the object, they have inherent limitations because they cannot quantify either the distribution or the magnitude of the stresses developed within the individual layers. One layer under severe stress, for example, can be masked by the response of the other layers.

Consider a simple two-component system composed of a paint film adhered to a rigid glass support. When the relative humidity is lowered, the glass will not respond to the desiccation, while the paint film will attempt to contract due to a loss of absorbed water. Since the paint is restrained by its adhesion to the glass support, and the glass dimensions are unchanged by decreasing relative humidity, strain gauges attached to the object will record no change in dimension and, hence, no development of stress. However, internal stresses will clearly exist within the paint and locally within the glass near the paint-glass interface. On the other hand, if the paint film is separated from the glass, stress-strain relationships can be determined for the individual paint component, but the results will not account for any interactions taking place when the paint is adhered to other layers. Thus, the only way to develop a reasonably clear picture of the total stresses in complex objects is by the application of numerical techniques.

One of the most powerful analytical tools for structural analysis is the digital computer using the technique of finite element analysis (FEA) (Cook 1974). FEA readily lends itself to complex structures composed of multiple materials. The fundamental principle of the method allows one to “subdivide” a complicated structure into a “finite” number of simpler substructures called “elements” that can be assigned individual material properties (Przemienicki 1968). These elements are connected at points called “nodes” and are mathematically relatively easy to manipulate. In general, an “element stiffness matrix,” [k], is computed for each element and reflects the geometry and material properties assigned to it. All of the elements are then mathematically assembled to construct the “model structure.” This results in the “structure stiffness matrix,” [K]. This matrix represents the generalized structural behavior of the object to be analyzed. In fact, the general equation to be solved is

Fig. .

The forces acting on any object can be a result of external causes such as pressure, impact, and vibration, or they can develop internally due to changes in temperature, RH, and chemical composition, for example, by long-term loss of solvents (Mecklenburg and Tumosa 1991a) and volatile components (Michalski 1991). Thus, the factors that induce stress can originate externally or internally to the structure and will act to distort the structure physically. If sufficiently high, the stresses cause cracking and splitting of the structure.

In general, the more elements used, the more accurate the FEA model. The displacements of the nodes represent the theoretical distortions of the structure and are the primary solution of the program. The resultant stresses and strains of each element are then computed from the displacements at the element nodes. In this way, one can get an accurate estimate of the stresses and forces acting throughout the entire structure. Experimental verification of the mathematical model can be accomplished by comparing the computer results to the displacements of actual objects subjected to vibration, impact, or, in this study, changes in temperature and RH. Since objects made of cultural materials can exhibit large displacements with changes in temperature and relative humidity, the commercial FEA software must have “large displacement” capability. The program should also be able to handle both elastic and plastic material properties. These are not unusual requirements for today's programs. The program used at the Conservation Analytical Laboratory, Smithsonian Institution is ANSYS, developed by the Swanson Analysis Systems, Inc. It is an exceptionally powerful program and has excellent subroutines for the construction of the model, or preprocessing. For analyzing and interpreting the results, the postprocessing systems are quite useful. These preprocessing and postprocessing capabilities are essential to rapid and efficient use of FEA.