Genetic algorithm reconstruction of orthotropic composite plate
elastic constants from a single non-symmetric plane
ultrasonic velocity data
, C.V. Krishnamurthy
, Krishnan Balasubramaniam
Center for Nondestructive Evaluation, Indian Institute of Technology, Chennai 600 036, India
Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India
Received 12 March 2006; accepted 9 June 2006
Available online 28 September 2006
This paper reports a Genetic Algorithm (GA) based reconstruction procedure to determine the elastic constants of an orthotropic
plate from ultrasonic velocity data. Phase velocity measurements are carried out using ultrasonic back-reﬂection technique on laminated
unidirectional graphite–epoxy (0)
and quasi-isotropic graphite–epoxy (+45,À45,0,90)
ﬁber reinforced composite plates. A forward
model to generate the slowness curves from elastic constants has been used to verify the quality of the reconstruction. The sensitivity of
the chosen GA parameters is studied. As expected, out of 9 orthotropic elastic constants to be determined, the C
found to be
insensitive. The GA based reconstruction using data obtained from multiple planes were evaluated and it is shown that the single plane
reconstruction at a non-symmetric plane was suﬃcient for the computation of the seven elastic constants.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: A. Layered structures; B. Mechanical properties; D. Non-destructive testing; Genetic algorithm
Evaluation of the mechanical behavior of composite
materials  under severe loading conditions requires the
knowledge of elastic constants of the material under con-
sideration. Conventional techniques like tensile and com-
pressive tests are destructive in nature and provide only a
few elastic constants and are diﬃcult to perform on thin
plate like structures. Ultrasonic techniques are advanta-
geous in these aspects over the conventional techniques
and are uniquely qualiﬁed for non-destructive measure-
ment of several of the elastic constants of such materials.
Elastic constants are related to ultrasonic velocities
along diﬀerent propagation directions through the Chris-
toﬀel equation [2–4]. Thus it is possible to evaluate elastic
wave velocities in any direction given the elastic constants.
Similarly, when elastic wave velocities are measured from
experiments, in principle, it is possible to deduce the elastic
In practice, with normal incidence velocity data, only a
few elastic constants can be determined through a simple
inverse procedure. For instance C
can be calculated
directly from the ultrasonic velocity at normal incidence.
But to ﬁnd the other elastic constants, it becomes necessary
to use the oblique incidence velocity data, coupled with a
suitable inversion technique. The inversion problem (dis-
cussed in Appendix A) contains trigonometric terms and
hence it is non-linear and non-unique in nature. The use
of over determined data sets (data obtained from multiple
angles of incidence) cancel out the random error in the
The determination of elastic constants from ultrasonic
bulk wave velocity data along planes of symmetry of the
plate has been studied by Chu and Rokhlin . They have
shown that quasi-longitudinal and quasi shear velocity
1359-8368/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
Corresponding author. Fax: +91 44 2257 0545.
E-mail address: email@example.com (K. Balasubramaniam).
Composites: Part B 38 (2007) 216–227