Techniques for Evaluating the Depth of a Crack by Means of Laser Spot Thermography ^†

Abstract

Laser Spot Thermography is a useful tool in nondestructive crack detection. Our goal is to estimate the depth of a fracture from external thermal measurements. First we transform a set of real 3D data in a 2D effective one. Then we use the 2D data set as input in different methods for solving an inverse problems for the heat equation. Our guiding idea is that an effort in the direction of the mathematical analysis of the problem, rewards us in term of computational costs.

1. Introduction

The present communication deals with the evaluation of planar fractures in bricks or concrete slabs by means of Laser Spot Thermography (LST) (see for example [1]. It is well known that LST is a kind of active thermography, specific for recovering thin vertical defects, in which a large amount of heat is released in a small area close to the defect.

We introduce a simplified 3D geometry of the system Slab-Crack-Heat source. Let the half space $\Omega =(x,y,z)=(-\infty ,\infty )\times (-\infty ,\infty )\times (0,\infty )$ represent our artifact whose accessible face $z=0$ is heated by a Laser Spot centred in $P_L=(0,0,0)$. Let $\sigma$ be the planar linear crack ${\left\{(x_0,y,z)\right\}}_{y\in \mathbb{R}\phantom{a}and\phantom{a}0<z<b\left(y\right)\le \infty }$ with $x_0>0$ and b is a positive function. Here we assume that the function b is approximatively constant. In this case b is a number that represents the average depth of the fracture. Although the geometry of the slab is clearly the parallelepiped ${\Omega }_{a,R}=(-R,R)\times (-R,R)\times (0,a)$ with $0<a<R$, we assume in what follows that our specimen is the half-space $\Omega$. These assumptions are not restrictive as long as we have in mind materials characterized by low diffusivity [2].

2. The Direct Model and the Inverse Problem

The temperature $u^b$ of ${\Omega }_{\sigma }=\Omega \setminus \sigma$ in presence of a laser source $\varphi (x,y,t)$ solves the following Initial Boundary Value Problem for the heat equation in ${\Omega }_{\sigma }\times (0,t_{max}]$:

$$\rho c{u}_t(x,y,z,t)=\kappa \Delta u(x,y,z,t)$$

(1)

$$\pm \kappa u_x(x_0^{\mp },y,z,t)+\frac{H}{2}(u(x_0^{\mp },y,z,t)-u^0(x_0,y,z,t))=0$$

(2)

(H is the thermal conductance of the crack, see [3]; $\rho$, c and $\kappa$ are density, specific heat and thermal conductivity of the material at hand)

$$-\kappa u_z(x,y,0,t)=\varphi (x,y,t)$$

(3)

$$u(x,y,z,0)=0$$

(4)

while u vanishes together with his gradient at infinity. The laser source is $\varphi (x,y,t)={\varphi }_L\delta (x,y){\chi }_{(0,t_{laser})}\left(t\right)$ where ${\chi }_E\left(x\right)=1$ if $x\in E$ and ${\chi }_E\left(x\right)=0$ elsewhere i.e. the laser is ON from $t=0$ to $t=t_{laser}<t_{max}$. The constant ${\varphi }_L$ (divided by $\rho c$) is the rate at which heat is liberated per unit time from $t=0$ to $t=t_{max}$ at the point $(0,0,0)$. The function $\varphi$ is called Continuous Point Source (CPS). If the real fracture has width $w>0$ and it is filled by a material (air, usually) with conductivity ${\kappa }_c$, we have $H=\frac{{\kappa }_c}{w}$.

Let $u^0(x,y,z,t)$ be the (background) solution of (1)–(4) when $\sigma =\varnothing$ (i.e. $b=0$). Its level sets are circumferences. The CPS defined by $\varphi$ concentrates a large amount of energy in a small area around a point $(0,0,0)$ chosen as close as possible to one side of the fracture (conventionally it is $x_0>0$). The geometry of the level sets of $u^b(x,y,0,t)$ is sensitive to the presence of the fracture. The deviation (whatever is measured) of such level sets from the circumferences corresponding to $u^0$, increases with the depth of the fracture. A measure of this deviation is the normalized gap

$$\beta =\underset{ϵ\to 0}{lim}\frac{u^b(x_0-ϵ,0,0,t)-u^b((x_0+ϵ,0,0,t)}{u^b(x_0-ϵ,0,0,t)+u^b(x_0+ϵ,0,0,t)}$$

(5)

The relation $\beta \left(b\right)$ defined in (5) is invertible in theory, but we observe in simulations that its derivative is almost zero for $b>b_{tr}$. In our applicative context $b_{tr}\approx 1.5$ cm. This was expected from general properties of thermography [4].

3. Analysis of the 3D Problem by Means of 2D Tools

The real problem modeled in previous section is a three dimensional one. More precisely, we have a solid slab (a 3D object), a Continuous laser Point Source to heat it, an infrared camera which collect thermal maps of the surface $z=0$ and a hidden fracture described by a portion of surface perpendicular to the top side of the slab. Our goal is to evaluate the average depth b of the fracture. The usual way to achieve this task is to minimize the distance between measured surface temperature and the temperature computed varying b. To do this it will be necessary to solve many time the three dimensional IBVP introduced in previous section. Thanks to the particular geometry of $\sigma$ we can reduce the computational costs by transforming the 3D model in an equivalent 2D one.

The CPS defined above produces,on the surface of the undamaged slab, a temperature (see [3] section 10.4)

$$u_P^0(x,y,0,t)=\frac{{\varphi }_L}{4{\left(\pi \alpha \right)}^{3/2}}{\int }_0^{t_L}\frac{e^{-\frac{x^2+y^2}{4\alpha (t-t^{\prime })}}}{{(t-t^{\prime })}^{3/2}}d{t}^{\prime }$$

A Continuous Linear Source (supported by the y-axis), defined by

$$\varphi (x,y,t)={\varphi }_L\delta \left(x\right){\chi }_{(0,t_L)}\left(t\right),$$

should produce an “effective” temperature $u_L^0(x,y,0,t)=\frac{{\varphi }_L}{2\pi \alpha }{\int }_0^{t_L}\frac{e^{-\frac{x^2+y^2}{4\alpha (t-t^{\prime })}}}{t-t^{\prime }}d{t}^{\prime }$.

If $u_P^0(x,0,0,t)$ is known from measurements we can write $u_L^0(x,0,0,t)$ as

$$u_L^0(x,0,0,t)=2\sqrt{\pi \alpha }B{u}_P^0(x,0,0,t)$$

(6)

where

$$B(x,t)=\frac{{\int }_0^{t_L}\frac{e^{-\frac{x^2}{4\alpha (t-t^{\prime })}}}{t-t^{\prime }}d{t}^{\prime }}{{\int }_0^{t_L}\frac{e^{-\frac{x^2}{4\alpha (t-t^{\prime })}}}{{(t-t^{\prime })}^{3/2}}d{t}^{\prime }}$$

Actually, we will use the factor $2\sqrt{\pi \alpha }B$ to convert the "real" 3D temperature $u_P^b$ (due to a CPS in presence of the crack $\sigma$) in an "effective" 3D temperature $u_L^b$ (due to a CLS in presence of the crack $\sigma$). Hence, $u_L^b(x,0,0,t)$ can be regarded as a 2D temperature $u_{2P}^b(x,0,t)$ (due to a CPS $\varphi (x,t)={\varphi }_L\delta \left(x\right){\chi }_{(0,t_L)}\left(t\right)$ in presence of the crack $\sigma ={\left\{(x_0,z)\right\}}_{0\le z\le b}$). The factor $2\sqrt{\pi \alpha }B$ can be used to convert $u^b$ also, as long as the simplified linear formula ([5])

$$u_{2P}^b(x,0,t)=\left\{\begin{array}{cc}{u}_{2P}^0(x,t)+\beta u_{2P}^0(2x_0-x,t)\hfill & x<x_0\hfill \\ (1-\beta )u_{2P}^0(x,0,t)\hfill & x>x_0\hfill \end{array}\right.$$

(7)

is approximatively correct. Since $u_P^b(x,0,0,t)$ is known from measurements, $\beta$ can be computed easily [5].

4. Evaluation of the Depth of the Fracture

Transformation (6) gives us the effective surface temperature of a rectangle (2D slab) due to the CPS $\varphi (x,t)={\varphi }_L\delta \left(x\right){\chi }_{(0,t_{laser})}\left(t\right)$. A reliable estimate of b can be obtained in different ways. We report here about three recent methods:

1. $\mathit{\beta }$-tool.

In [5] we construct a “rooler” by means of a sequence of simulations of the 2D system. For each value $0<b_1<b_2<...<b_N<a$ (a is the real thickness of the slab) we solve IBVP (1)–(4) and obtain

$${\beta }_k\left(t\right)=\underset{ϵ\to 0}{lim}\frac{u^{b_k}(x_0-ϵ,0,t)-u^{b_k}(x_0+ϵ,0,t)}{u^{b_k}(x_0-ϵ,0,t)+u^{b_k}(x_0+ϵ,0,t)}.$$

(8)

The functional relation between b and $\beta$ behaves like a Gaussian. More precisely $\beta \left(b\right)={\beta }_1-A{e}^{-\frac{{(b-D)}^2}{s^2}}$. If the measured surface temperature (in 3D) at time t is $U_{meas}(x,y,t)$, first we extract the value $u_P^b(x,t)=2\sqrt{\pi \alpha }B{U}_{meas}(x,0,t)$ and then compute the corresponding ${\beta }_{meas}$ by means of (8). Finally, we have $b=D+s\sqrt{ln\left(\frac{A}{{\beta }_1-{\beta }_{meas}}\right)}$ In [6], $\beta$-tool is used successfully starting from experimental data.

2. Expansion of the external temperature.

In [2] we have proved that, in $Q^{+}=(x_0,\infty )\times (0,\infty )\times (0,t_{max}]$, we have $u^b(x,z,t)=C_H(x-x_0,z,t)+E(x-x_0,z,t)$ where $C_H$ is the known temperature of $Q^{+}$ in presence of an infinite crack ${\{x_0,z\}}_{0<z<\infty }$ with thermal conductance H[3]. E is the solution of a mixed IBVP for the heat equation on $Q^{+}$. The evaluation of the depth comes from the minimization of the discrepancy ${\int }_{x_0}^{\infty }{(u^b(x,0,t)-2\sqrt{\pi \alpha }B{U}_{meas}(x,0,t))}^{2}dx$. Examples based on simulations are in [2].

3. Reciprocity Gap.

A third method for the evaluation of b from the knowledge of $U_{meas}$ is based on the determination of the spatial support of the function $\left[u\right](z,t)={lim}_{ϵ\to 0}{u}^b(x_0-ϵ,z,t)-u^b((x_0+ϵ,z,t)$ (temperature gap between the sides of the crack varying z and t ) [7]. It is a work in progress in collaboration with E. Francini (University of Florence).

5. Conclusions

The average depth b of an emerging planar fracture $\sigma$ in a concrete slab ${\Omega }_{\sigma }$ can be evaluated in different ways. We observe that this inverse problem can be translated in a two dimensional framework reducing dramatically computational costs. At this point, different techniques for determining b in different contexts, are available.