NDT.net - February 2003, Vol. 8 No.2

## FINITE DIFFERENCES METHOD APPLIED TO THE ANALYSIS OF THE DETECTION AND LOCALISATION OF PLAN DEFECT

Smail SAHNOUN, Sougrati BELATTAR, Amal TMIRI
Laboratoire d’Energיtique et de Traitement du Signal
Dיpartement de Physique, Facultי des Sciences
e-mail:ssahnoun@hotmail.com
Paper presented at the 8th ECNDT, Barcelona, June 2002

### Summary

The thermal characterisation of material constitutes an important area of the thermal metrology. In this article we use a new approach of non destructive thermal control to analyse and characterise inclusion defects in flat plates. This approach is based on control volumes and the finite differences method . We have been interested particulary in unidirectional transfers l. The obtained responses allow to establish curves between maxima of heat flow differences (flow with defect – flow without defect) with the position and the thickness defect.

### 1 Introduction

The main concern of manufacturers of materials, is to know if these materials that contain defects which can deteriorate their rigidity. It is so, the recourse to the non destructive thermal control methods be essential. Indeed non destructive thermal control is a technique which allows the analysis of various structures by the study of their responses to thermal excitations, and the establishment of a relation between this response and the imperfections which they contain [2]. The defects and the structure have different thermophysical properties; the presence of a defect changes the thermal flow which crosses the same homogeneous structure. If this changing is large, the defect would be easily detectable.

By the help of the numerical finite differences and control volume method [1], we show that the variations of the heat flows and surface temperatures permit the establishment of abacus of characterization of materiel with plane defect.

### 2 Formulation of problem

We consider a plane plate of thickness l = 10mm, and width L = 20mm. The structure is supposed to be at the initial temperature T0, in which is inserted a layer of different nature representing a defect, figure 1.

The defect is placed at a distance l1 from the input face of the plate (x = 0). If we except the defect, the material of the wall is supposed homogeneous and has thermal parameters different from those of the defect. One applies a thermal flow density F0 to the input of the wall. The temperature of output of the wall is supposed to be constant. The faces y = 0 and y = L are insulated and the contact resistances are neglected. To be in conditions of a monodimensional conduction we suppose that the defect has the same section as the plate which contains it (figure 1). We have considered a plane homogeneous concrete plate containing a resistive defect materialized by a plate of polystyrene. Several configurations were considered related to the position and the dimension of the defect.

 Fig 1: Used model.

The choice of these configurations is completely arbitrary but describing encountered practical cases. We adopted the following notations : The temperature and the thermal flow density at the entry of the healthy material are represented by Ts(x, t) and Fs(t) respectively. Equally The temperature and the thermal flow density at the entry of the material with defect are represented by Ti(x, t) and Fd(t) respectively. Under these conditions, the presence of the defect brings a difference in temperature

 (1)

and a difference in flow

 (2)

2.1 Used model
The used model figure 1 is represented by an homogeneous plane of thickness l, conductivity k1= 1.7(w/m.k), density r1 = 2250(kg/m3), and a specific heat c = 827(J/kg.k). The output temperature (x=l) is supposed to be constant T0 . It contains a defect of thickness e=l2-l1, placed perpendicularly to the sens of propagation of heat, of conductivity k2=0.041(w/m.k), density r2=14.7(kg/m3), and specific heat c = 1310(J/kg.k). The whole wall-defect is supposed initially in thermal balance, at T0=25°C, and heated by a source of flows F0 = 1000 W/m2.

The structure can be decomposed into three areas. Areas I and II represent the homogeneous parts, and III the area of the defect. Resistances of contact between layers are supposed negligible.

2.1.1 Regions without defect
The heat transfer in the regions without defect can be described by the equation:

 (3)

the principle of the energy conservation imposes the following boundary conditions:

 (4)

 (5)

where TI(x, t) , (respectively TII(x, t) ) represents the transitory temperature of the concrete in region I located at
0 < x < l1 ,( respectively l2 < x < l).

2.1.2 Region of defect
The heat transfer in this region can be described by:

 (6)

Where Td(x, t) represents the transitory temperature of defect.

The contact being perfect between the defect and the concrete, one can thus write on the interfaces of contact the following temperatures:

 (7)

and the heat flows :

 (8)

the initial conditions are given by :

 (9)

 (10)

The side faces of the plate are insulated

 (11)

2.2 Control volumes Method
The numerical method used is that of volumes control, method for an irregular grid [6]. It consists in cutting out the model of figure 1 in a whole of small volumes of control, on which are integrated the preceding equations of heat transfer by taking into account of boundary and initial conditions.

### 3 Results and discussion

The comparison of the surface temperature T of the inspected material with that of the same material presumably healthy Ts of area I makes it possible to detect a possible characteristic anomaly of the presence of defect. A positive difference will translate the presence of a resistive defect. A negative difference will indicate a capacitive disturbance compared to the structure to be tested.

3.1 Effect of the position and thickness of the defect
3.1.1 On the temperature profile
The curves, figure 2, represent the evolution of the difference in temperature, for three thicknesses and three positions of the defect in the structure. The thickness of defect is taken equal to (e=1mm, 2mm and 4mm), and for each thickness the defect is placed at three different positions (x = 2mm, 3mm and 6mm) in wall of 10 mm as total thickness. The temperature difference is strongly related to the thickness and the position of defect. They take permanent values according the thickness of the defect. It appears clearly that for a given thickness, the differences in temperatures evolve differently in transitory regim according to the position of the defect. A great value would result from a great thickness of the defect. The permanent regim is established according to the thickness of defect. The value of temperature is proportionaly to the thickness as shown in figure 2.

 Fig 2: Variation of the temperature difference according to time for various thicknesses and various position of defect.

3.1.2 On the profile of the heat flow
We represented in figure 3 the differences of the heat flows for three thicknesses of the defect (e=1mm, 3mm and 4mm), in the structure. For each one, the defect is placed at two different positions (x = 2mm and 3mm).

 Fig 3: Variation of the difference in flow according to time for e = 1mm, 3mm and 4mm and xp = 2mm and 3mm.

We can observe that DF increases when the thickness of the defect increases. For a given thickness it appears that DF increase when the defect itself move towards the output of structure.

Finally we can not that DF pass by a maximum which depend on thickness e and on position xp, occurring at the moment tp.

In order to clarify the nondestructive analysis of the considered structure, we considered two essential parameters: The defect position xp and the defect thickness e. A given thickness e ,we note on figure 4 that the maximas fm increases according to a law of the form : y = ax2+ bx + c when the defect moves towards the interior of the structure

 Fig 4: Variation of the maxima of the differences in flow according to time of a resistive defect of whose the thickness varies from 0,2mm to 1 mm.

These results show that, in transient state, the heat flow is further delayed when the thickness of the defect becomes large. From that, for a given thickness e a time envolve according to a law of the form: y = a' x2 + b'x + c' versus the position of the defect xp figure 5.

 Fig 5: Evolution of times tp according to the position of the defect xp .

These curves are very useful in thermal non destructive control. They can be used as curves of reference in order to understand the behavior of a structure in the presence of internal anomalies.

### 4 Conclusion

use of the finite differences method in thermal non destructive control shows that any anomaly in the homogeneous material can be identified by analysis of the temperature distribution profile or the thermal flows compared to those of the same presumably healthy structure. Simulations highlight the effect of various parameters related to the defect of inclusion. A defect would be easily detectable if it is close to the face of inspection, or it has rather a significant thickness.

The influence of the position and thickness of the defect on the evolution of the density flux differences, at the output of the plates with and without defect, was analyzed. We found that the maxima of the differences in flux density and the corresponding time increase when the thickness becomes rather significant on the one hand, and on the other hand when the defect moves towards the interior of the structure.

### 5 References

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