Masonry elastic characteristics assessment by thermographic images

In the present paper, the elastic mechanical characteristics of masonry samples, whose texture is not visible due to plaster, are estimated by means of homogenization technique applied through thermographic images. In particular, three masonry samples with different textures have been purposely built. The chosen textures were periodic, quasi-periodic and random. The images, taken with a thermocamera, are analyzed in order to identify the texture. An homogenization technique, based on the application of appropriate boundary conditions, is used. The mechanical characteristics, obtained using the textures identified using photographic images and thermographic images, are compared. The influence of some parameters (such as the dimensions of the structural element used in morphological operator) are analyzed. The obtained results permit to point out the reliability of the masonry elastic characteristics assessment by the proposed procedure.


Introduction
The assessment of the mechanical characteristics of the masonry assumes an important role with particular attention to structural analysis and restoration of historical constructions and cultural heritages.
Considering the masonry as a bi-phase composite, built by mortar and bricks (or stones), the mechanical behaviour is depending on the actual texture, both in elastic field [1,2] and in plastic one [3,4].
An adequate recognition of the texture is necessary to evaluate the Periodic Unit Cell (PUC) in periodic case [5]. This is also true to asses the Representative Volume Element (RVE) for quasi periodic or random arrangements [6,7]. Moreover, the knowledge of the actual texture takes a fundamental role using the approach that involves the periodization of random media by statically equivalent periodic unit cell [8][9][10][11][12]. Eventually the texture knowledge is necessary for the homogenization residuals analysis [13].
Given the previous observations, it is obvious the importance of developing procedures for identifying the texture of the masonry in order to estimate the mechanical characteristics by non destructive testing.
In recent years, the possibility of estimating the masonry texture by means of digital (photographic) images has been explored by several authors [11,14]; moreover digital images can be acquired by laser scanner [15].
These procedures are not applicable when the masonry walls are covered with plaster or, very frequently for historical building, with frescoes.
To overcome these situations, the thermographic tool have recently been proposed in [16,17] where some methods to enhance the images and subsequently to estimate the masonry characteristics are proposed.
Nevertheless a fundamental aspect to be investigated is the reliability assessment of the thermographic procedure for masonry texture recognition. To analysis this topic an ongoing research has been launched with the building of masonry wall samples with different textures and the assessment of mechanical characteristics by homogenization theory based both on photographic and thermographic images.
This paper reports the first obtained results and it is organized in the following sections. After some aspects of the thermographic theory and the peculiarities of the apparatus used in the experimental tests are described in Sec. 2, the characteristics of masonry samples and how they were built are described in Sec. 3 The homogenization approach used to obtain the elastic masonry characteristics is briefly remembered in Sec. 4, and the procedure to identify the texture is recalled in Sec. 5. Finally, the comparison between photographic results and thermographic results is shown in Sec. 6, with comments on the reliability of the proposed procedure.
The influence of some parameters involved in the proposed approach (such as the dimensions of the structural element used in morphological operators) are analyzed.

Thermographic images
The principle by which the thermography works is that every body emits electromagnetic radiation, some part of it being in infrared range depending on its temperature. If the sample is not in thermal equilibrium, there is an heat flux which pass trough it: since the thermal conductivity of mortar and bricks are different, the two phases have different temperatures which can be detected by an infrared sensor. Therefore, by means of thermography it is possible to measure the infrared radiation emitted from every body without direct contact, so that it can be considered a non-destructive test technique.
A thermographic camera differs from a photographic camera on the wave lengths of the radiation which it can detect: the wave lengths of the radiation which can be measured by a thermographic camera are in the infrared range, between about 700 nm-1 mm, which correspond to 430 THz-300 GHz in frequency, while the wave lengths of the radiation which can be measured by a photographic camera are in the visible range, between about 400 nm-700 nm, which correspond to 190 GHz-430 GHz, see  The image resolution is typically much smaller than that allowed by a photographic camera, and therefore reconstruction technique could be used to enhance the quality of the image [16,17]. In the present paper, two thermocamera models have been used: the thermocamera model B360 from FLIR has been used for the quasi-periodic masonry sample and model 885-2 from Testo has been used for periodic and random masonry samples. Each of the model have a sensor of dimension 320×240 pixels. All the samples have been exposed to direct sunlight in order to improve the heat flux trough the body.

Building of samples
Three samples of masonry walls have been built; for all the samples, UNI bricks of dimensions 250×120×55 mm have been employed, either used as a whole or split in two or in four, as shown in Fig. 2. Mortar joint were made 1 mm in thickness. In what follows, we denote width of the brick its horizontal length, height of the brick its vertical length. The bricks were assembled using three different textures: (i) a periodic texture, made with brick having the same widths and heights. In particular, a running bond texture has been used, where head joints of adjacent rows are half brick width apart; (ii) a quasi-periodic texture, made with bricks having different widths but equal heights, arranged in horizontal rows in such a way to avoid the correspondence between vertical joints ; (iii) a random texture, made using bricks having different widths and heights (achieved by rotating some bricks).
The texture's typologies of the masonry wall are shown in Fig. 3. For periodic and random textures, samples whit 780 x 780 mm dimensions were built. The sample of quasi-periodic masonry has been built larger than the others, with 835 x 780 mm dimensions. In Fig. 3 the actual portion of samples is shown inside the dashed line.
The masonry samples, as built before the application of the plaster, are shown in Fig. 4. As a final step, the samples have been covered with plaster, as can be seen in Fig. 5(a) for the quasi-periodic texture sample. An example of the image of the sample with quasi-periodic masonry obtained bt the thermocamera is shown in Fig. 5

Homogenization of masonry
The mechanical characteristics of the masonry have been estimated as in [6]. The procedure is here recalled briefly. We recall that we are in a case of plain stress, and therefore With the preceding, the relation between stress and strain can be expressed, using Voigt notation, with the following where C is a symmetric square matrix denoted stiffness matrix and Instead of stiffness matrix, the flexibility matrix S can be used Obviously, S = C −1 . In the case of orthotropic material, C 13 = C 23 = S 13 = S 23 = 0.
Since the material is heterogeneous, the relations (2 and (5) are in terms of mean values averaged over the volume Ω as follows where In (7), C H is the effective stiffness matrix and S H is the effective flexibility matrix. In general, if the domain is not the size of a representative volume element, the estimates of C H and S H depends on the boundary conditions applied [18]. Two type of boundary conditions are mainly used: (i) essential boundary conditions, expressed in terms of displacements as where (x, y) are the coordinates of the point on the boundary. By imposing 19] and therefore the first column of the estimate of C H is given by σ . Proceeding in analogous way the other column of the estimate of C H can be found. This estimate of C H is denoted by C E . (ii) natural boundary conditions, expressed in terms of forces as where (n x , n y ) denotes the outward normal vector at the point on the boundary. By imposing σ 0 xx , σ 0 yy , σ 0 and therefore the first column of the estimate of S H is given by . Proceeding in analogous way the other column of the estimate of S H , and therefore the estimate of C H as C H = S H −1 can be found. This estimate of C H is denoted by C N It is possible to prove that [20] C In case that C E and C N are sufficiently close, C H can be estimated through the following arithmetic mean:

Identification of texture
The estimation of the texture from the photographic and the thermographic images follows the same steps.
In particular, the images are converted from color images to black and white images [17], in which black pixels identify mortar elements and white pixels identify brick elements. It is worth noting that, if necessary, the image was corrected in order to compensate the perspective.
An image can be seen as one or more discrete functions, f i (x, y), with x = 1, 2, . . . , N and y = 1, 2, . . . , N , where N is the image width in pixels (it is assumed that the image has equal width and height). In case of color image, there are three functions f i , one for each channel (red, green and blue). In case of gray-scale image, there is only one function, corresponding to the gray level (usually expressed in the range 0-1).
At first the image is converted from color to grayscale by eliminating hue and saturation information while retaining the luminance [21]. In particular the following function is used g(x, y) = 0.299 f R (x, y) + 0.587 f G (x, y)+ + 0.114 f B (x, y) (13) where g(x, y) is the gray level of the pixel at position (x, y) and {f R , f G , f B } are the red, green and blue levels of the same pixel in the color image.
Thereafter a median filter to enhance the quality of image is applied. The median filter is defined by the followinḡ g(x, y) = median g(s, t) for (s, t) in N l (x,y) (14) where N l (x,y) is a structural element, a square of side l pixels center in pixel at (x, y). In the present case, l =3.
The gray-scale image is converted to a black and white image by the following where k is the gray level used as a threshold. In the current application, the value of k is chosen using an adaptive approach [22] in order to compensate the effect of gradient of illumination present in the images. The value 1 (black) is associated with mortar pixels, the value 0 (white) with brick pixels. As final steps, morphological operators are used. At first, mortar (black) region of pixels which are surrounded by brick (white) pixels are removed. Finally erosion and dilation operators are applied (in this succession) in order to smooth the contours of the inclusions.
In particular, erosion is defined by The resulting black and white image has a consistent separation of phases, i.e. each stone is surrounded by mortar joints and unrealistic conjunction of inclusions is reduce as much as possible.
A portion with eight rows of stones and with a width equivalent to two entire bricks has been chosen, together with the corresponding head and bed joints, therefore the dimensions of the portion are 520 mm in width and 520 mm in height. The corresponding image size in pixel is 200×200.
In Figs. 6-8, the obtained results for both the thermographic image and the photographic image of the same portion of sample is shown. In particular, in the case of thermographic images two different black and white images image have been obtained, differing in the value of l used for dilation operator: l = 3 is the same used for erosion, l = 5 is slightly larger and used in order to obtain a concentration ratio, defined as the percentage of brick phase, closer to the one of photographic image.  Table 2 Concentration ratio (non-dimensional) and components of estimated stiffness matrix for periodic texture (in MPa).  Table 3 Concentration ratio (non-dimensional) and components of estimated stiffness matrix for quasi-periodic texture (in MPa).

Numerical results
In order to obtain the estimates of C E and C N , the boundary value problems are solved through finite element method using 4-node elements with plane stress formulation, assuming that each pixel of the binary image is a finite element. The mechanical characteristics used for the constituent phases are reported in Tab. 1.
The results in terms on components of stiffness matrix are shown in Tab. 2-4. In all of the cases the difference between the estimates in natural and essential boundary conditions are Fig. 6 Identified texture of periodic masonry sample: photographic image image (a) and its identified texture, thermographic image (c) and its identified texture with l = 3 (d) and l = 5 (e). Fig. 7 Identified texture of quasi-periodic masonry sample: photographic image image (a) and its identified texture, thermographic image (c) and its identified texture with l = 3 (d) and l = 5 (e). below the 3%, and therefore the portion of the sample is close to the representative volume element. As expected, the results show a dependency not only on the texture but also on the concentration ratio, c 1 . The difference is about 25% for l = 3 and 12% for l = 5 in the case of periodic texture, 11% for l = 3 and 5% for l = 5 in the case of quasi-periodic texture, while is about 17% for l = 3 and 4% for l = 5 in the case of random texture.
The information contained in the stiffness matrix, and in its inverse the flexibility matrix, can be used to estimate the value of the engineering constants involved in orthotropic materials, E x , E y , ν xy and G xy and their mutual relation. In particular, in plane stress the flexibility matrix can be written as and the stiffness matrix can be written as The values of the engineering constants E x , E y and ν xy can be found by the following  The results are show in Tab. 5 Moreover, the following relation between the engineering constants can be established and the results are show in Tab. 6. As can be seen, the results in terms of ratio of engineering constants are in good agreement, since the effect of phase concentration is greatly reduced. Therefore, in terms of characterization of the orthotropic behavior the estimates that can be made using thermographic images can be considered quite reliable.

Conclusions
In the present paper we present the first results of an ongoing research relative to estimation of the mechanical characteristics of masonry walls covered with plaster  Table 6 Relation between engineering constants.
using nondestructive techniques. The method is based on a thermographic image of the masonry, to which digital image techniques are applied in order to obtain a black and white image with a consistent separation of the phases (brick and mortar in the present case study). Then, the mechanical characteristics are estimated by means of an homogenization procedure based on natural and essential boundary conditions. These mechanical characteristics have been compared with those obtained using a photographic image of the same sample with a good agreement; then the obtained results highlight the reliability of thermographic procedure. As a further development, it is planned that the research will continue with laboratory tests on the masonry walls samples described in the present paper to experimentally estimate their elastic characteristics, in order to provide a validation of the proposed procedures.