No abstract

Preventive conservation is conductive to the long-term preservation of works of art. In order to realize the avoidance of damages in advance, risk management as well as foresighted thinking is required. The application of the method of engineering mechanics for preventive conservation is at the very beginning of its development. This article is a contribution to this still very young field. Generally, sensitive artworks combine all properties of complex mechanical structures. Oil paintings on canvas, for instance, are asymmetric, multiple curvilinear structures made of aged anisotropic compound materials with cracks and other damages. Due to their popularity, some artworks travel a lot, and during the exhibition and storage, they are always exposed to shocks and vibrations, therefore the protection of sensitive paintings is a demanding task, the solution of which requires the multidisciplinary cooperation especially in the context of engineering mechanics with its many specializations. The subject of the presented research is an artificial aged painting dummy in the simplest conceivable composition. This paper aims to describe the mechanical behavior of this test object, which is the basic requirement for the development of technological protective measures. The concept of the digital twin, known from Industry 4.0, is used to solve this task. This article focuses on the design of a virtual painting dummy that has the same static and dynamic behavior as the investigated real test object. Therefore, the deflection of the real dummy in lying position as well as the curvature of its standing position without and with dynamic excitations have been measured. The advantage of the analytical and Finite Element Analysis (FEA) models presented are their practicability and quick realizability at fair correlation. The concept presented offers a potential way to assess and finally reduce the risk for original paintings during various transport, exhibition, and storage situations with the help of virtual objects.

Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

Classical plates theories, like Kirchhoff's plate theory [1], are based on kinematical a‐priori assumptions. Avoiding these assumptions, we derive from the three‐dimensional theory of linear elasticity by means of Taylor‐series expansions the quasi two‐dimensional problem. This problem consists of infinitely many partial‐differential equations (PDEs) written in infinitely many displacement coefficients. With the consistent approximation approach we arrive at solvable hierarchical plate theories. By using the modular structure of the displacements coefficients (modularity), we obtain from these generic‐plate theories the complete‐plate theories, whose results fulfill the strong form of the local equilibrium conditions and the Neumann boundary conditions on the upper and lower face of the plate (local conditions) a‐priori. Furthermore, we show that every variable of the complete‐plate theories can be calculated.

By the use of Taylor series expansion, we arrive from the three-dimensional theory of linear elasticity at the exact twodimensional plate problem which consists of infinitely many equations and unknowns. Applying the approximation approach of Vekua or the consistent-approximation approach to this exact problem leads to hierarchical PDE-systems. For the latter approach a modular structure of the displacement coefficients is assumed and proved for several orders. With this modularity we show that "fully" reducible PDE-systems, so called complete plate theories, can be obtained by using a special schema. The results of the complete plate theories coincide with the classical theories and also fulfill several local conditions. When proving the existence of "fully" reducible PDE-systems for an N th order complete plate theory, Hilbert-type matrices appear. We calculate the determinant for these matrices and present a sketch of evidence.