·Home ·Table of Contents ·Reliability and Validation 1  Prediction of Weld Quality by the Wave Propagation Modelling on Ultrasonic Testing Simulation
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Abstract
Computer simulation of ultrasonics is considered to be a very helpful tool to get information and physical understanding of wave phenomena inside the material, which can help to a suitable prediction of the service life of the component. This simulation permits to establish a complete relation between different pulses in the Ascan signal.
The ultrasonic testing of steel welds is a commonly used method in NDT. However, the analysis of the received signals, in a satisfactory way, is not sure. There is some uncertainty connected with testing techniques, like errors of characteristic measurement and influence of factors that cannot be taken into account for building up a model.
As these factors cannot be evaluated a priori, and their combination can bring unpredictable influence on the testing results, it is possible to represent them as additional noise response. And, since an incident wave is modulated during travelling through the material, in a similar manner as an input signal being modulated during passing through a system. So the incident and travelled waves can be considered as an input i (t) and output functions o (t) of the system. Such an approach allows describing the material and testing as a united model, like in the case of dynamic mathematical modelling or system theory [1].
In the present paper we try to simulate the propagation of the ultrasonic wave in a homogeneous material with the modelling of the system response, by the use of the system theory.
Key words: Ultrasonic waves, defect interaction, system theory, modelling, FEM, simulations
1. Introduction
The nondestructive evaluation by ultrasonics deals with the materials features characterisation, eventually to pilot their evolution or their degradation, and to detect and evaluate the defects. Modelling becomes a powerful tool for prediction and results interpretation. Generally, its basic aim is the simulation of the response of a measuring system.
By definition a system is a functional set packed into a black box. The input signals operate as system excitation and the resulting signals produce the system behaviour description [2] fig1.
Fig 1: "System environment" 
In the case of ultrasonic system, the incident wave can be adjusted during travelling through the material in a similar manner to an input signal being modulated during passing through a system and the output signals can be regulated by the reflected waves via the Ascan representation.
For simulating the system response, the fig.2 describes the system components and the different parts that must be modelled. The part1 relates to the incident wave and piezoelectric effect, the part2 to the ultrasonic field propagation within material and the part3 to the wave interaction with defect and geometrical limits of the component.
Fig 2: "Ultrasonic system components" 
In ideal case the description of the acousticelastic response of the measuring chain concerns:
 the transducer with the equations of piezoelectricity and wave propagation,
 the electrical and electronic connections with the classical electricity equations,
 the wave propagation between transducer and defect by the basic equations of elastic wave propagation in the considered medium,
 the defect interaction, which produces generally a diffusion or diffraction of the ultrasonic field, can be described by numerical modelling,
 And the back wave propagation to the transducer is usually modelled under the reciprocity principles [3].
In the aim to simplify the calculations, the following hypothesis are drawn:
 The material is homogeneous, and the acoustic interaction is calculated in the case of the plane waves,
 the measured signal is considered proportional to the loads,
 the controlled piece is regular,
 A binary matrix describes the defect.
2. Theory of elasticity.
The tested material under ultrasonic vibrations, is a structure constituted by sinusoidal oscillations of the medium particles. The structure is then simulated by a composition of linear nodes in which each point represents the atom state (the particle velocity v, and the particle displacement x
are material characteristics) [4].
Fig 3: "Particles displacement simulation in the structure" 
Let us look at the fig.4, which describes the structure in (x, y, z) axis, and the formulation of the different constraints. And let us formulate D = {u, v, w} as the displacement vector, q
= {¶
u/¶
x + ¶
v/¶
y + ¶
w/¶
z} as the deformation vector and F={X_{0},Y_{0}, Z_{0}} the resultant volumetric force. According to the Hook law in the elastic domain, the relation between deformations and constrains is as follows [4]:
 (1)

N: constrains; E: Young's modulus; q
: deformations
In the case of the mechanics theory of continuous milieus, the Lamé formulation draws the nominal and tangential constraints equations in (2) & (3). N_{1},N_{2},N_{3} are the nominal constrains, T_{1},T_{2},T_{3} are the tangential ones [5].
 (2)

 (3)

Where: G = 1/2 E / (1+m
) : Modulus of shear. l
= m
E / (12m
) (1+m
) : Lamé coefficient
m
: Poisson coefficient (m
= 0,28 in steel; m
= 0,34 in Aluminium) .
Fig 4: "Nominal and tangential constraints" 
3. Equilibrium conditions formulation
After projections on the axis, the equilibrium conditions between elastic forces and applied forces by volume unit, give the following system of equation (4):
 (4)

In conditions of the vibrations propagation in an infinite milieu, the elemental volume is subject to vibrations, and the inertia forces by volume unit are F_{i} = { j
¶
^{2}u/¶
t^{2} ,  j
¶
^{2}v/¶
^{2}t,  j
¶
^{2}w/¶
t^{2}}, j
is the material density. According to the Alembert principles, the inertia force F_{i} =  j
¶
^{2}D/¶
t^{2} is added to the resultant force F [6]. But if we only consider that the vibration is around the equilibrium position, the system of equations (4) becomes:
 (5)

By substitutions in the equations (5) of the expressions developed in (2) and (3) we obtain, for example, for the first equation of (5) :
 (6)

The same development will be performed for the other equations of (5) and we obtain the following solution:
 (7)

After substitution and calculations, the relation between the constraints and the deformations within the formulation (1) gives the solution (8.a)(8.b).
 (8.a)

 (8.b)

Where [
M]
is the material properties matrix and q
(x, y, z) is the deformation vector.
Knowing that the deformations are derived from the displacements conforming to the finite element method (FEM), the relation (8.b) becomes [6]:
 (8.c) 
Where [A] is the nodal coordinate matrix and [B(x,y,z)] is the deformation matrix in relation with the displacement function.
4. Testing conditions formulation
Let us consider the propagation of ultrasonic oscillations in the tested material, of transverse and longitudinal velocities V_{t } and V_{l} respectively. The displacement vector D is then function of x,y,z & the time t. For an easiest formulation of the longitudinal and transverse waves equations, let us project D into 2 vectors: shear displacements vector D_{t} and longitudinal displacement vector D_{l }, so as D =D_{t} + D_{l}.
4.1. Equilibrium equations
The equilibrium equation (7), can be divided into 2 equations:
 For longitudinal waves:
 For transverse waves :
Knowing the velocity formulations [7]:
 (9) 
The equations for the transverse (10.a) and the longitudinal (10.b) waves become [7].
 (10.a) 
 (10.b) 
4.2. Description into (x,y) plan
For simplifying calculations, let us consider the plan (x,y). The derivation about z become equal to zero, and the shear displacements in the case of transverse waves turn to:
 (10.c) 
The problem is to find a function F
satisfying : j
¶
^{2}F
/ ¶
t^{2} = G D
F
The same reasoning is drawn for longitudinal waves with a function Y.
The displacement vector D in the plan (x,y) is then D (u,v) = D_{t} + D_{l} with:
 (10.d) 
4.3. Conditions at interface (Reflection and transmission waves at the interface)
In the case of transverse wave:
 The normal displacements are equal in the interface between the 2 milieus : u_{1 }= u_{2
}
 The constraints are equal between the milieus : N_{1} = T_{2} = T_{3}
The normal constrain is N_{1} = l
q
+ 2 G ¶
u/¶
x, with q
= {¶
u/¶
x + ¶
v/¶
y + ¶
w/¶
z}.
As we are in (x,y) plan, N_{1} = l
(¶
u/¶
x + ¶
v/¶
y) + 2G ¶
u/¶
x, the tangential constraints T_{2}=0 and T_{3 }= G(¶
u/¶
y + ¶
v/¶
x). So the conditions are N_{1}= T_{3} , and the equations of movement are simplified.
As resolution of the system, the theory of reflection and refraction between 2 materials is well outlined within the Snell law (fig.5). In this junction, Knott has proposed the amplitude equations of the reflected and refracted waves as solution of the equations of movement.
In the case of transverse incident wave, the longitudinal and transverse reflected waves (in the material 1) are characterised by the ensuing equations [5]:
 (11)

While the longitudinal and transverse refracted waves (in the material 2) are:
 (12)

A_{1} ,A_{2} , B_{0}, B_{1}, B_{2} are the amplitudes.
(sin a
_{1})_{ }/ V_{t1} = (sin a
_{2})_{ }/ V_{l1} = (sin a
_{3})_{ }/ V_{t2}= (sin a
_{4})_{ }/ V_{l2} (Snell law).
Fig 5: " Reflected & refracted waves ". 
By conversion of F
_{l} ,F
_{t}Y_{l} Y_{t} with their respective formulation after calculations, we can obtain the following system:
 (13)

The conditions at the interface are translated in the ensuing solution after derivations on F
and Y and the relation N_{1} = T_{3} :
 (14)

5. Example of porosity simulation
In our approach, the defect is considered as a reflective interface. So the porosity is simulated by a gas/solid interface, since the ultrasonic waves of high frequency don't propagate in the air, the reflection is then entirely in the solid.
The amplitude equations in the case of transverse incident wave of amplitude B_{0}, become a system of 2 equations with 2 variables:
Knowing B_{0}, the resolution of the system gives the A_{1} and B_{1} amplitudes.
5. Conclusion
In this paper we have exposed a solution for predicting the Ascan response and a calculation of the reflected amplitudes of the system, with a priori knowledge of the piece and the defect within, and the incident wave state. The system behaviour is modelled under elasticity physical state drawn below, by the finite element simulation. Future work inside this approach is to formulate the calculated data in the same formalism as experimental ones and to calculate the error ratio prediction from measured data. The results will be illustrated in a future contribution.
Acknowledgements
Pr. A. Benchaala is thankful for his comments and suggestions about this paper.
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