Genetic-based approach to achieve higher durability for stone repairs A. Isebaert + L. Van Parys

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Since natural stone is the main construction material for built heritage, its alteration implies a challenge for conservation institutions. Repair of original stone can be performed with replacement stone but also with repair mortars. In both cases, the replacement material has to be compatible in short-term and long-term, to avoid damage or different alteration. In addition to mineralogical considerations, mortar compatibility can be determined through their properties (mechanical, physical) and their appearance. Creating a durable repair mortar by taking into account these parameters can be achieved through mathematical optimization, since genetic algorithms are suited to solve such a multi-objective problem [1].

Natural stones weather due to a combination of processes that catalyst or slow down one another [2,3,4,5,6,7]. This creates complex ever-evolving alteration patterns [8,9]. Part of these processes is due to extrinsic environmental factors, while other processes are caused by the intrinsic characteristics of the stone.
The extrinsic factors include climate, location of the monument (city vs. countryside), the orientation (protected or exposed to rainfall) and the location of the stone in the building (ground-level, protected or protruding) [10]. Water in general is an important catalyst for degradation, since it can transport harmful materials into the stone and cause stress into the stone when it freezes.
But in addition to these more natural causes of deterioration, other causes have appeared since the 19th century, such as pollution of the environment. Weathering due to this and other human interference is sometimes greater than natural weathering [7]. The famous black crusts for example, are caused by a chemical reaction of
deposition of sulphur oxides (combustion) and calcite (stone), creating gypsum (CaSO4. ½ H2O). In the porous gypsum, dirt particles such as carbon settle in, blackening the crust. Regardless of the fact that the specific amount of sulphur dioxide in the atmosphere is decreasing, pollution remains an important cause for degradation [11,12].
Additionally, there are also intrinsic causes for degradation. The composition and properties of the stone determine, for a large part, possible alterations [7,13]. Stones in general were formed through processes in the crust of the earth in a specific environment up until a stable condition was reached. Their exposure to the atmosphere brings them in another environment, leaving them in se in a physic-chemical disequilibrium [7]. Thus, the minerals and matrix of the stone are altered in composition or condition when exposed to the environment, e.g. they oxidise or dissolve [13,14,15]. Secondly, some stones in one and the same wall seem to be more ‘durable’ than others. This difference can be explained by the simple fact that stones have a spatial variability and heterogeneity [14]. As third aspect, the orientation of the rock fabric in the monument is also important to take into account, e.g. the orientation is partially responsible for the compression and flexural capabilities of the stone [14,16,17]. Finally, the appearance and finishing of the stone play a role in the alteration susceptibility as well: rough stones, with a higher surface area, allow more dirt and other particles to attach.

Repair mortar compatibility problems
After determining the deterioration patterns and their causes, restoration interventions are set up to preserve the building. One of the possible interventions is the repair of original stones (through replacement stones or repair mortars). However, the choice for an adequate repair material is of vital importance. Researches have pointed out severe deterioration of stones due to the inadequate use of repair mortars [18,19] or replacement stones [20,21,22].
An ideal repair mortar for natural stone should be durable enough, but self-sacrificing in the long run [23,24]. However, with variable deterioration processes and a heterogeneous material to start with, this isn’t easy to achieve. In the past, incompatible repair mortars were used, causing direct and indirect deterioration. Key aspects associated with the development of an “ideal” mortar are shortlisted here:

1. Perceptive aspects
Repair mortars have to be compatible with the stone in colour and surface roughness, so as not to disturb the general perception of the building. The main problem in terms of perception is designing a mortar with the right colour (in fresh and weathered state).

2. Physical and mechanical aspects
- Water transfer properties. Water plays an important role in the deterioration process; causing stress cracks and fractures, favour biological colonization, erode, dissolve and transport material. Consequently, the water transfer properties of the mortar are important for the durability of the stone.
- Modulus of resistance and/or elasticity. They can be significant in certain cases, when the repair mortar is used for the filling-in of large or structural parts taking into account the form of the filled in part and how it fits into the whole [25]. The rigidity of the material plays an important role and deterioration such as loss or cracks can occur to both repair mortar and stone.
- Thermal response. High temperature differences on sun-faced walls cause the minerals in stones to expand and contract. Consequent cycles of temperature differences cause internal stress, leading to detachment, deformation or cracking of mortar or stone [25].

3. Chemical aspects
Some repair mortars contain, create or attract harmful materials in the stone, such as salts that crystallize inside or outside the stone [25]. Deterioration patterns vary from efflorescence to cracks and scaling due to sub-florescence. Repair mortars can also be made with organic polymers, which are susceptible for biological organisms. Their presence will therefore speed up biological colonization on the stone [13].

A grand challenge lies in the perceptive compatibility (on the long run), and a large part of the compatibility problem lies in the physical-mechanical aspects where several objectives have to be considered simultaneously. Adherence is an emblematic topic where initial capabilities should be sufficient while water transfer or thermal response shouldn’t imply problems like freeze, crystallization or interface shearing in due course, the relative mechanical behaviour of stone and mortars playing there a fundamental role [26]. However, the care dedicated to mortar formulation should never overshadow the mastery of application. Several cases prove the necessity to use non-corrosive metal where dowels are needed and to finish the mortar as such that allows water to run off easily [27].

Mortar formulation through optimization
A repair mortar should be adapted to each stone specifically, but the theoretical ideal mortar isn’t achievable through current methods , literature pointing out three main paths towards an acceptable compatible repair mortar. The first one relies on commercial mixes for given types of stones: the task consists in pointing out which product could be used or adapted to meet the required standards [28,29,30,31,32,33]. The second approach builds up the mortar from scratch, aiming at repair mortar composition which is as close to the stone as possible: the same mineralogical composition is sought for and the binder is adapted to the demands of the stone [34,35,36,37]. In practice, tuning operations required by both these approaches are expensive in time and resources and justify the more recent third approach that tries to develop a modular system, where known base ingredients lead to the composition of a repair mortar for a specific stone [38]. This opens the way to computer-aided decision, but it is also merely valid for the researched components.
The current research starts from the same approach but takes benefit of mathematical tools. The perceptive, mechanical and/or physical properties of the desired mortar are linked with data gathered from the components, the system remains open: once the general framework has been developed, the method can be applied by any end-user who feeds the problem with data collected on the materials he intends to use.
Genetic algorithms are used for multi-objective problems. They rely on the natural selection principle. The Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II) allows multi-objective optimization. To obtain a mortar whose properties are close to the set targets, this intuitive approach can be used: many mortar formulations are the potential solutions x = [x1, x2,…, xn]T of the problem, expressed through n design variables that are the types and relative proportions of mortar constituents (type and quantity of sands, cement and water). A function is defined for each aspect the formulation takes into account (e.g. colour function and strength function), and this function will express the gap between the targeted value and the value corresponding to each individual. This implies one should already have an idea of the desired value for each of the variables (i.e. given proportion of sands, cement and water). Therefore, general mixing laws, based on available theories, are established which guarantee the universality of the method [39,40]. The tool will then create an initial population by a randomized set of values for each variable, with N individuals. This population of mortar recipes will then evolve, through a selection and replacement process, keeping the best individual. Then, crossing over and mutation processes combine the genes of these selected individuals. Through the evaluation process, the value of objective functions for each individual is calculated, considering simultaneously the whole of objectives (e.g. minimizing the gap between the strength/colour value of an individual and the targeted strength/colour). Through these evolutions and under certain conditions mainly associated with the initial population, it has been proven that the population converges towards an optimal formulation meeting the constraints and approaching as close as possible to the targets.

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