FUNDAMENTALS OF THE ANISOTROPIC
MATERIAL CONTACT CAUSTICS
Damir Semenski, Ante Bakić
University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, pp 102, HR-10002 Zagreb, Croatia
e-mail: damir.semenski@fsb.hr, ante.bakic@fsb.hr
1. Introduction
Experimental optical method of caustics is established
in fracture mechanics through the stress intensity factor
determination. Recently, the task of experimentation is
to analyze the deformation of material surface and to
assess the structural damage. Mechanism of surface
damaging includes various types of abrasion, adhesion
and surface fatigue. The various surface coatings and
the conditions of lubrication can significantly change
the type of contact. An objective is the force
determination and the analysis of the isotropic body
contact. The extension to anisotropy introduces the
procedure, which enables the treatment of composites
that are mainly of orthotropic structure.
2. Optical principle and presumptions
The principle of the optical method of caustics is based
on illumination of the region of high deformation
gradients of the loaded specimen. The light beams are
reflected from the reflective surface or propagated
through the transparent specimen in such a way
forming a dark spot, which depends on the amount and
inclination of a contact force (Fig. 1).
Fig. 2
The experimental measurements for the normal force
application in laboratory conditions matched quite well
to the numerical simulations in full measurement
range, as shown in Fig. 3.
F
screen
x
dark spot
caustics
y
lightening
area
Fig. 3
incident
light
h/2
h
Fig. 1
 i ,o is
In isotropic body contact caustic coordinate y max
experimentally measured value from the recorded
picture of experimental caustics, as shown in Fig. 2.
The presumptions of theoretical approach to
anisotropic body contact are:
 The equations for the stress-strain state in the region
near the point of force application are taken into
account.
 The elastic constants in composites for the usual
orthotropic mechanical structure have significant
mismatch in the directions of the principal axes of
orthotropy L and T.
 Field of analysis of strains and stresses is near of the
force application but outside the plasticity region.
 Material is homogeneous from macromechanical
point of view. 3.
3. Theoretical background
Light beams conditions are characterized through the
virtual plane at the distance z 0 , m as a scale factor and
h as a thickness of the specimen. The position of the
beam components on the screen can be described as
follows:
x' , y'  f (r,  , h, F ,  , S ij )
where r and  are polar coordinates, h is thickness of
material, F is force intensity and  is force inclination.
Material is characterized by the coefficients Sij  S ji
of a compliance material matrix.
The dark spot will appear, surrounded by the
concentrated light on the edge, at the points where the
Jacobian determinant condition is satisfied:
 ( x , y )
J
0.
 ( r,  )
This defines the initial curve position on the specimen
surface where light beams are reflected and projected
into the singular caustic curve. The solution for the
optical shadow of caustics is given in the suitable for
shadow dark spot simulations.
4. Simulations of the optical effects
Functions are calculated by using MATLAB software
package. Guide user interface has been programmed to
be used interactively and user friendly when changing
material properties and illumination parameters.
The characteristic carbon fiber reinforced material is
taken into account. The shadow formation is simulated
for the material of the following mechanical
characteristics:
EL  149 GPa , ET  30GPa , GLT  15GPa, TL  0,06644 .
Normal force is directed to the axis L of the material,
causing the symmetrical shape of the optical effect
shown in Fig. 4.
Fig. 4.
By changing the direction of the force application there
will be a significant change of the shape of the singular
caustics curve. The shadow optical effect is illustrated
by for the force inclined by 30 degrees to the axis L, as
shown in Fig. 5.
Fig. 5
As the first step in the evaluation of the experimental
caustics, simulations confirm the expectations for the
shadows of caustics as extremely dependent upon
mechanical characteristics of material and the
inclination of the force. Such conclusion corresponds
to the previously done experimentation in fracture
mechanics is possible to be confirmed in contact
problem too.
References
[1] Kalthoff, J.F., “Shadow Optical Method of
Caustics”, chapter in “Handbook on Experimental
Mechanics” (ed. A.S. Kobayashi,), Prentice-Hall,
Engelwood Cliffs-New York, 1987, pp. 430-500.
[2] Lekhnitskii, S.G. “Theory of Elasticity of an
Anisotropic Body,” Mir Publishers (transl.. from
Russian), Moscow, 1981.
[3] Rossmanith, H.P.,Zhang, J., Knassmillner R.E.,
“Kontaktenkaustiken in orthotropen Werkstoffen”,
Österreichische
Ingineur-und
ArchitektenZeitschrift (ÖIAZ), 136. Jg., H. 6, 1991, pp. 214218.
[4] Semenski, D., Jecić, S., Goja, S., Mažar, L., “On
the Method of Caustics in Contact Problem”, Proc.
of 3rd Congress of Croatian Soc. of Mechanics,
Cavtat-Dubrovnik, 2000, pp. 389-396.
[5] Semenski, D. Bakić, A., “Approach to the
Anisotropic Body Contact by the Optical Method
of Caustics”, Proc. of 4th Congress of Croatian
Soc. of Mechanics, Bizovac, 2003, pp. 441-447.