Application of ICAMM to Impact-Echo Testing - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 106

theoretical response of the volumetric wave propagation of impact-echo testing
can be modelled as [ 17 ],

2 u i
ot 2

o T ij
ox j

¼ q 0 o ð 5 : 1 Þ

T ij ¼ c ijkl S kl ð 5 : 2 Þ

where q 0 is the material density; u i is the length elongation with respect to the
starting point in force direction; o T ij
ox j

is the force variation in the i direction due to

deformations in j directions; c ijkl is the elastic constant tensor (Hooke's law); and
S kl is the strain or relative volume change under deformation in side l in direction
k in a unitary cube that represents a material element.
The Eqs. ( 5.1 ) and ( 5.2 ) state that the force variation in the direction i due to the
side stresses in directions j of the material elementary cube is equal to the mass per
volume (density) times the strain acceleration (Newton's third law in tensorial
form). Deriving an analytical solution to problems that involve stress wave
propagation in delimited solids is very difficult, and this is the reason why the
existing bibliography in this field is not very extensive. Thus, these equations are
normally solved numerically by FEM [ 18 ].
The simulation models of this work consisted of parallelepiped-shaped mate-
rials of 0.07 9 0.05 9 0.22 m. (width, height and length), which were supported
at one third and two thirds of the block length (direction z). Simulated finite
models corresponded to one class of homogeneous models and eleven classes of
inhomogeneous models. The dynamic response of the material structure (time-
varying displacements in the structure) under the action of a transient load was
estimated from the transient analysis. The transient load, i.e., the hammer impact
excitation, was simulated by applying a force-time history of a half-sine wave with
a period of 64 ls as a uniform pressure load on two elements at the centre of the
front side of the model.
The elastic material constants for the simulated material (aluminium alloy
series 2,000) were: density 2,700 kg/m3; elasticity modulus 69,500 Mpa.; and
Poisson's ratio 0.22. This material simulation model (including the specific values
for the elasticity constants) was selected to replicate the specimen (aluminium
alloy parallelepipeds) where real experiments were performed afterwards. Cer-
tainly, a myriad of different materials and forms could be selected which could
influence the obtained results. Thus, for example, it has been reported [ 19 ] that
some accepted conclusions about the application of the impact-echo method might
not be true depending on the Poisson ratio value (which is the ratio of transverse
contraction strain to longitudinal extension strain in the direction of stretching
force). However, we consider that the conclusions derived in this work should be
applicable to other kinds and/or forms of material given that different defect
patterns could result in different statistical models. Comparing specimens of the