PSI - Issue 10

Ch.F. Markides / Procedia Structural Integrity 10 (2018) 163–170 Ch.F. Markides / Structural Integrity Procedia 00 (2018) 000 – 000

165

3





2 R R R RK  j

j R R R RK 

R RKP

2

1

j 



1





j

frame

2

2 , ( ) p     i

2 2

(1)

Φ

K

( ) 

,

,













 

) R R t

4

4  

 

j

j

j

j

or, taking into account that R j =1.5 R ,

2

2

2 2

i  

RKP

6

 





frame

(2)

Φ

p

( ) 

,

( ) 

,







RK

RK

t

6

3



In Eqs. (1), μ = E /[2(1+ ν )], where E is the Young’s modulus and ν is the Poisson’s ratio, while assuming plane stress conditions, κ =(3 – ν )/(1+ ν ) ( j indicates the jaw). The above expressions can be, also, applied in the case of a ring with relatively small internal radius (Kourkoulis and Markides (2014); Kourkoulis et al. (2015)). 3. Simplified equations of reflected caustics in the contact problem Let the above disc de made of a transparent material and let a red light emitted from a He-Ne laser be impinging on the loaded disc in the contact region of interest. For a divergent light beam, the experimental optical setup is shown in Fig. 2a. Light rays are reflected on the front and rear faces of the disc and received on a reference screen placed a distance Z o from the front face of the disc. Images of light reflections on the are then magnified according to the magnification factor λ m =( Z o ± Z i )/ Z i with Z i being the distance between the focus of the second lens and the disc Using only the first lens, the case adopted next for clarity, Z i →∞ whence λ m =1.

P frame

Camera

Screen

(a)

(b)

P frame

Caustic (front)

Initial curve (front)

y

D

Initial curve (rear)

x

C f

Caustic (rear)

ℓ

O

B

– ℓ

D

Disc (contact region)

C r

t

θ

y ΄

z

E f

Caustic (front) Caustic (rear)

H

r o

Z o

O ΄

A

E r

x ΄

P r ΄

n r

2 ϕ

z ΄

P r

W r

Z i

P

n f

P f

t

w r

Lens 1

2 ϕ

W f

g ( x , y )

P ΄ w f

Lens 2

Screen

Z o

Laser

P f ΄

Parallel, normally impinging light

Fig. 2. (a) The optical experimental set-up; (b) The optical mapping for parallel incident light.

In this context, consider the contact region of the disc, referred to the Oxyz system, used in the previous section (Fig.2b). Let a ray of a parallel light beam incident normally (along the z -direction) on a point of the deformed disc in the contact region. Neglecting refraction within the disc for simplicity, the light ray will travel straight along its thickness t , i.e., along the segment P f P r joining the respective points on its front and rear distorted faces (only the out-of-plane deformation g ( x , y ) is here considered). Under that simplif ying assumption and according to Snell’s law , the reflected light rays form the points P r , P f will for m angles ±2 ϕ with the axis z , where ± ϕ are the angles the relevant unit normal vectors n r , n f , at P r , P f , subtend with the axis z . The images of these reflections are received on the screen where a coordinate system O ΄ x ΄ y ΄ z ΄ , obtained by a parallel translation of Oxyz at a distance Z o , is considered. If P ΄ is the projection of P , P f , P r on the screen and w f , w r are the vectors specifying the deviations of the reflected light rays from P f , P r respectively, the optical mapping between any pair of points P f , P r of the disc and their image points P f ΄ , P r ΄ on the screen will be defined by the vectors (Fig.2b):